University of Southern California Dissertations and Theses

On stochastic integro-differential equations

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On stochastic integro-differential equations

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Content ON STOCHASTIC INTEGRO-DIFFERENTIAL EQUATIONS by Chukiat Phonsom A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY APPLIED MATHEMATICS May 2020 Copyright 2020 Chukiat Phonsom For my family ii Acknowledgments IwouldliketothankmyacademicadvisorProfessorRemigijusMikulevičiusforhisguidanceinmathemat- ics and academic life. My Ph.D. was an amazing journey because of his persistent caring in my mathematical development and professional growth. I would like to acknowledge my family for moral support throughout this long Ph.D. journey. My successes would not be possible without their encouragement. Last but not least, I thank my fiancée Nicha, for always supporting me through ups and downs during my time at USC. iii Disclosure This manuscript is an accumulation and an extension of the author’s work in [MPh17, MPh19, MPh20]. The author actively conducted original research in these papers. More precisely, Chapters 1-5, 7, and 11 of this manuscript are written based on results and ideas in [MPh17, MPh19, MPh20] with necessary modifications. Chapters 6, 8, 9, and 10 contain some further considerations of the main problem. In order to present this manuscript in a self-contained manner, we fully include all necessary content from [MPh17, MPh19, MPh20]. Any other references to results and ideas in literature outside the author’s work [MPh17, MPh19, MPh20] will be stated explicitly with cited sources. iv Table of Contents Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Disclosure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Manuscript Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Relevant Work and Our Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Notation, Function Spaces, Main Results, and Examples . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Notation and Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Some Estimates of O-RV Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Function Spaces - Norm Equivalence and Characterizations . . . . . . . . . . . . . . . . . . . . . . 23 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Proof of Proposition 15 (i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.4 Proof of Proposition 15 (ii) and (iii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.5 Other Characterizations of Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.6 Dense Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5 Probability Density Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.1 Probability Density Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.2 Fractional Operators on Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3 Elliptic Equation for Smooth Input Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.4 Embedding into the Space of Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . 64 6 L p -norm Estimates of Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.1 Difference Estimate by B ;1=p pp R d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.2 Difference Estimate by H ;1=2 p R d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.2.1 Notation, Assumptions and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.2.2 More Probability Density Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.2.3 Estimate ofG f;1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.2.4 Estimate ofG f;2 : Square Function Operators . . . . . . . . . . . . . . . . . . . . . . . 86 7 Proof of the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.1 Elliptic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.2 Parabolic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.2.1 Existence, Uniqueness, and Apriori Estimates for Smooth Input Functions . . . . . . . 96 7.2.2 Estimate of T g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 v TABLE OF CONTENTS TABLE OF CONTENTS 7.2.3 Estimates of R f; ~ R : verification of Hörmander Conditions . . . . . . . . . . . . . . 105 7.2.4 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 8 Time Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 9 Stochastic Variable Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 10 Quasilinear Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 10.1 Proof of the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 10.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Appendix A: Gaussian Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Appendix B: Moment Estimate of Lévy Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Appendix C: A Non-Degeneracy Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Appendix D: Stochastic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 vi TABLE OF CONTENTS TABLE OF CONTENTS Abstract We consider the following stochastic parabolic integro-differential equation in the scale of L p - spaces of functions whose regularity is defined by a Lévy measure with O-regulary varying radial profile du (t;x) = [L u (t;x)u (t;x) +f (t;x)]dt + Z U (t;x;z)q (dt;dz); u (0;x) = g (x); (t;x)2E = [0;T ]R d ; with 0 and an integro-differential operator L ' (x) = Z R d 0 [' (x +y)' (x) (y)yr' (x)] (dy);'2C 1 0 R d ; where is a Lévy measure with Blumenthal-Getoor index of 2 (0; 2), (y) = 0 if 2 (0; 1); (y) = 1 fjyj1g (y) if = 1 and (y) = 1 if 2 (1; 2): Existence and uniqueness of the solution is proved by deriving apriori estimates. Some probability density estimates of the associated Lévy process are used as well. We also study some quasilinear and stochastic variable coefficient models. vii Chapter 1 Introduction and Background 1.1 Introduction Stochastic integro-differential equations are natural counterparts of stochastic partial differential equa- tions where integral operators substitute derivative operators. Integro-differential equations appear naturally in physics, engineering (e.g. [CV10, LYS14, GIK10, MLZ19]), in finances (e.g. [CT04, SS08, FLM12]) and in population dynamics (e.g. [C10].) In [CT04], the authors give a comprehensive introduction to option pricing in models with jumps (Chapters 8-13.) In non-linear filtering problems, the source of non-local integral operators lies in the use of Lévy noise instead of Brownian noise, which in turn leads to non-local operators in Zakai equations for filtering density. We refer to [MP12, FH18] for examples of Zakai equations. Due to a variety of applications, as mentioned above, theoretical aspects of these equations need to be investigated. In this manuscript, we investigate the well-posedness of the Cauchy problem (1.1) with integro-differential operators in L p -spaces. Some extensions of (1.1) will also be considered. Let2 (0; 2) andA be the class of all nonnegative measures onR d 0 =R d nf0g such that R jyj 2 ^1d < 1 and = inf ( < 2 : Z jyj1 jyj d <1 ) : In addition, we assume that for 2A ; Z jyj>1 jyjd < 1 if 2 (1; 2); Z R<jyjR 0 yd = 0 if = 1 for all 0<R<R 0 <1: 1 1.1. INTRODUCTION CHAPTER 1. INTRODUCTION AND BACKGROUND Let ( ;F;P) be a complete probability space with a filtration of algebras on F = (F t ;t 0) sat- isfying the usual conditions. LetR (F) be the progressive algebra on [0;1) : Let (U;U; ) be a measurable space with finite measure ;R d 0 = R d nf0g: Let p (dt;dz) be Fadapted point measures on ([0;1)U;B ([0;1)) U) with compensator (dz)dt: We denote the martingale measure q (dt;dz) = p (dt;dz) (dz)dt: In this manuscript, we consider the stochastic parabolic Cauchy problem du (t;x) = [L u (t;x)u (t;x) +f (t;x)]dt (1.1) + Z U (t;x;z)q (dt;dz); u (0;x) = g (x); (t;x)2E = [0;T ]R d ; with 0 and integro-differential operator L ' (x) = Z R d 0 [' (x +y)' (x) (y)yr' (x)] (dy);'2C 1 0 R d ; where2A ; (y) = 0 if2 (0; 1); (y) = 1 fjyj1g (y) if = 1 and (y) = 1 if2 (1; 2): The symbol of L is () = Z R d 0 e i2y 1i2 (y)y (dy);2R d : Note that (dy) = dy=jyj d+ 2 A and, in this case, L = L = c (;d) () =2 , where () =2 is a fractional Laplacian. The equation (1.1) is forward Kolmogorov equation for the Lévy process associated to . We assume that g;f and areF 0 B R d -,R (F) B R d - ,R (F) B R d U-measurable respectively. We define for 2A its radial distribution function (r) = (r) = x2R d :jxj>r ;r> 0; and w (r) =w (r) = (r) 1 ;r> 0: 2 1.2. MANUSCRIPT STRUCTURE CHAPTER 1. INTRODUCTION AND BACKGROUND One of our main assumptions is thatw (r) is an O-RV (O-Regularly Varying) function both at zero and at infinity. That is r 1 (x) = lim sup !0 w (x) w () <1; r 2 (x) = lim sup !1 w (x) w () <1;x> 0: The regular variation functions were introduced in [Kar30] and used in tauberian theorems which were extended to O-RV functions as well (see [AA77], [BGT87], and references therein.) They are very convenient for the derivation of our main estimates. Given 2A ;p2 [1;1);s2R, we denote by H s p (E) =H ;s p (E) the closure in L p (E) of C 1 0 (E) with respect to the norm jfj H ;s p (E) = F 1 (1 Re ) s Ff Lp(E) ; whereF is the Fourier transform in space variable. We denote byH s p (E) =H ;s p (E) the corresponding space of random functions u (t;x); (t;x)2E. In this manuscript, under O-RV preperties of w and nondegeneracy assumptions (see assumptions A;B below), we prove the existence and uniqueness of solutions to (1.1) in the scale of spacesH ;s p . Moreover, we have the solution estimate juj H ;s p (E) C h jfj H ;s1 p (E) +jgj B ;s1=p pp (R d ) +jj H ;s1=2 2;p (E) +jj B ;s1=p p;pp (E) i if p 2; whereB ;s pp is the Besov counterpart ofH ;s p . 1.2 Manuscript Structure This manuscript will be organized as follows. In Chapter 2, we state the main result and assumptions for (1.1). Examples of Lévy measure that satisfies the main assumptions are given. In Chapter 3, we derive some essential estimates of O-RV functions. In Chapter 4, we discuss function spaces. In particular, we provide some characterizations of function spaces that are needed for the main result. In Chapter 5, we provide some probability density estimates, which are important for deriving apriori estimates for (1.1). In Chapter 6, we derive estimates for hypersingular integrals defined with respect to Lévy measure. As an application, the estimates are used for the stochastic variable model in Chapter 9. In Chapter 7, we give the main proof for (1.1). 3 1.3. RELEVANT WORK AND OUR CONTRIBUTION CHAPTER 1. INTRODUCTION AND BACKGROUND In Chapter 8, we investigate the time regularity of the solution. In Chapter 9, we discuss a stochastic variable model. In Chapter 10, we extend the linear model to a quasilinear model. In Chapter 11, we collect and prove auxiliary results which are needed throughout this manuscript. 1.3 Relevant Work and Our Contribution In this manuscript, we study equation (1.1) in Bessel potential spaces- proving existence, uniqueness, and regularity of the solutions in both space and time. The main challenge lies in the fact that the symbol () is not smooth in , the standard Fourier multiplier results do not apply in this case. As an example, we consider 2A defined in radial and angular coordinates r =jyj;z =y=r; as () = Z 1 0 Z jzj=1 (rz)a (r;z)j (r)r d1 S (dz)dr; 2B R d 0 ; (1.2) where S (dz) is a finite measure on the unit sphere onR d . Here we explore some existing literature that is closely related this work in the context of equation (1.1) in L p -spaces. In [Zha13], (1.1) was considered in the standard fractional Sobolev spaces using L 1 -BMO type estimate, with g = 0, = 0 and a nondegenerate in the form (1.2) with a = 1;j (r) =r d In [DK12], an elliptic problem in the whole space with L was studied for in the form (1.2) with S (dz) = dz being a Lebesgue measure on the unit sphere in R d , with 0 < c 1 a c 2 , and with j (r) =r d . The authors use purely analytical method by using the local Hölder norm estimate of the solution to derive the mean oscillation estimate In [KK16], an elliptic problem in the whole space with L was studied for in the form (1.2) with S (dz) = dz being a Lebesgue measure on the unit sphere inR d , with 0 < c 1 a c 2 , and a set of technical assumptions onj (r). A sharp function estimate based on the solution Hölder norm estimate (following the idea in [DK12] ) was used The case of fractional Laplacian (dy) = dy=jyj d+ has been extensively studied by the authors of [MP12, MP13, MP14]. Their results in [MP14] include some variable coefficient models. We also refer to [CL12, KK12] for some other variations. Techniques presented in these papers require restrictive or technical assumptions for the permissible class of Lévy measures. More precisely, in form (1.2), papers listed above either require the radial profile of the form j (r) = r d or the Lebesgue measure on unit sphere. To the best of our knowledge, our assumptions in 4 1.3. RELEVANT WORK AND OUR CONTRIBUTION CHAPTER 1. INTRODUCTION AND BACKGROUND this manuscript are more natural and allow the more general examples (see in particular, example 4.) The O-RV properties imposed on the radial profile of show up naturally when we derive L p -estimates of the solution. In this manuscript, we follow a classical route. More specifically, we provide a probabilistic representation of the solution, which can be seen as singular integral operators. The key steps involve using the Fourier transform to derive L 2 estimates. To achieve L p estimates, we verify the appropriate version of Hörmander conditions and apply Calderón-Zygmund theorems by associating L to a family of balls. Frequency of generators of Lévy processes varies in general rendering standard fractional Bessel potential spaces corresponding to fractional Laplacian (4) insufficient as solution spaces. It is necessary to find alternative spaces where the solution maximal optimality aligns with the classical case of the heat equation. We define and characterize Besov spaces and Bessel potential spaces whose smoothness is captured by scaling of Lévy measures. These concepts are of interest in the field of function spaces of generalized smoothness (see [FJS01, FL06].) Additionally, we extend results to a quasilinear model (Chapter 10), and a stochastic variable coefficient model (Chapter 9.) We also acknowledge that a similar equation was studied in [MX19] in generalized Hölder spaces for a similar class of Lévy measures. We will follow some of their estimates regarding O-RV functions. Moreover, L p -theory for the classical case of Laplacian operator i.e. in place of L has been inves- tigated by Krylov; for example, in [Kry94, Kry96, Kry99]. This manuscript can be considered a non-local generalization of Krylov’s results. 5 Chapter 2 Notation, Function Spaces, Main Results, and Examples 2.1 Notation and Definition The following notation, definition and convention will be used in the manuscript. E = [0;T ]R d , T 0 R d 0 =R d nf0g We denote by C 1 0 R d the set of all infinitely differentiable functions onR d with compact support We denote byS R d the Schwartz space onR d . Let V be a Banach space, we denote byS 0 R d ;V the space of Vvalued tempered distribution. We simply writeS 0 R d when V =R. LetN =f1; 2;:::g;N 0 =f0; 1;:::g;R d 0 =R d nf0g: If x;y2R d ; we write xy = d X i=1 x i y i ;jxj = p xx For z2R d =f0g, ^ z =z=jzj The standard Fourier transform and its inverse are denoted by Fh (t;) = ^ h (t;) = Z R d e i2x h (t;x)dx; F 1 h (t;x) = Z R d e i2x h (t;)d for h2S R d We denote the partial derivatives in x of a function u (t;x) on R d+1 by @ i u = @u=@x i , @ 2 ij u = @ 2 u=@x i @x j , etc.; Du = ru = (@ 1 u;:::;@ d u) denotes the gradient of u with respect to x; for a 6 2.1. NOTATION AND DEFINITION CHAPTER 2. NOTATION, FUNCTION SPACES, MAIN RESULTS, AND EXAMPLES multiindex 2N d 0 we denote D x u (t;x) = @ j j u (t;x) @x 1 1 :::@x d d For 2 (0; 2] and a function u (t;x) onR d+1 , we write @ u (t;x) =F 1 [jj Fu (t;)] (x) We denote A =[ 2(0;2) A , where A is the class of all non-negative measures onR d 0 =R d nf0g such that R jyj 2 ^ 1d <1 and = inf ( < 2 : Z jyj1 jyj d <1 ) For2A , we denoteZ t ;t 0; the Lévy process associated toL , i.e.,Z t is càdlàg with independent increments and its characteristic function is given by Ee i2Z t = expf ()tg;2R d ;t 0 where the symbol () = R R d 0 e i2y 1i2 (y)y (dy);2R d : Equivalently, Z t = Z t = R t 0 R R d 0 (y)yq(ds;dy) + R t 0 R R d 0 (1 (y))yp(ds;dy);t 0 where p is the Poisson point measure on [0;1)R d 0 such thatEp (dt;dy) = (dy)dt and q (dt;dy) =p (dt;dy) (dy)dt. For 2 A , we will generally denote by (r) = (r) = x2R d :jxj>r ;r > 0 and w (r) = w (r) = (r) 1 ;r > 0, a (r) = infft> 0 :w (t)rg;r > 0 the generalized inverse of w and a 1 (r);r> 0 the generalized inverse of a For 2A , we denote (dy) = (dy) and sym (dy) = 1 2 [ (dy) + (dy)] The lettersc =c (;:::;),C =C (;:::;),C 1 =C 1 (;:::;),C =C denote constants depending only on quantities appearing in parentheses or subscripts. In a given context the same letter will generally be used to denote different constants depending on the same set of arguments. These constants can change from line to line. An integral with respect to any Lévy measures and random measures onR d 0 and its subspaces excludes 0. This is always the case even when we do not include the omission of 0 explicitly in integral notation We will use notation fug , for functions f andg when there arec;C > 0 independent of inputs such that cgfCg 7 2.2. FUNCTION SPACES CHAPTER 2. NOTATION, FUNCTION SPACES, MAIN RESULTS, AND EXAMPLES We use standard abbreviation LHS and RHS for left-hand side and right-hand side of equations or inequalities respectively 2.2 Function Spaces ForE = [0;T ]R d ;p 1, we denote by ~ C 1 (E) the space of all measurable functions f onE such that for any multiindex 2N d 0 and all p 1; sup (t;x)2E jD f (t;x)j + sup t2[0;T] jD f (t;)j Lp(R d ) <1: Similar space of functions onR d is denoted ~ C 1 R d . Let V be a Banach space with normjj V . LetS R d ;V be the Schwartz space of V-valued rapidly decreasing functions. We use standard notationS R d when V =R. For a Vvalued measurable function h onR d and p 1 we denote jhj p Lp(R d ;V ) =jhj p V;p = Z R d jh (x)j p V dx: We fix 2A . Obviously, Re = sym , where sym (dy) = 1 2 [ (dy) + (dy)]: Let Jf =J f = (IL sym )f =fL sym f;f2S R d ;V : For s2R set J s f =J ;s f = (IL sym ) s f =F 1 [(1 sym ) s ^ f];f2S R d ;V : L ;s f =F 1 ( sym ) s ^ f ;f2S R d ;V : Note that L ;1 f =L sym f;f2S R d : 8 2.2. FUNCTION SPACES CHAPTER 2. NOTATION, FUNCTION SPACES, MAIN RESULTS, AND EXAMPLES Forp2 [1;1);s2R; we define, following [FJS01], the Bessel potential spaceH s p R d ;V =H ;s p R d ;V as the closure ofS R d ;V in the norm jfj H ;s p (R d ;V ) = jJ ;s fj Lp(R d ;V ) = F 1 h (1 sym ) s ^ f i Lp(R d ;V ) = j(IL sym ) s fj Lp(R d ;V ) : We drop V when V =R. According to Theorem 2.3.1 in [FJS01], H t p R d H s p R d is continuously embedded ifp2 (1;1);s< t, H 0 p R d = L p R d . For s 0;p2 [1;1); the normjfj H ;s p (R d ) is equivalent to (see Theorem 2.2.7 in [FJS01]) jfj H ;s p (R d ) =jfj Lp(R d ) + F 1 [( sym ) s Ff] Lp(R d ) : (2.1) Further, for characterizations of our function spaces we will use the following construction (see [BL76].) Remark 1. For an integer N > 1 there exists a function = N 2 C 1 0 (R d ) (see Lemma 6.1.7 in [BL76]), such that supp = : 1 N jjN , ()> 0 if N 1 <jj<N and 1 X j=1 (N j ) = 1 if 6= 0: Let ~ () = (N) + () + N 1 ;2R d : Note that supp ~ N 2 jjN 2 and ~ =. Let ' k =' N k =F 1 N k ;k 1; and ' 0 =' N 0 2 S R d is defined as ' 0 =F 1 " 1 1 X k=1 N k # : Let 0 () =F' 0 (); ~ 0 () =F' 0 () +F' 1 ();2R d ; ~ ' =F 1 ~ ;' =F 1 ; and ~ ' k = 1 X l=1 ' k+l ;k 1; ~ ' 0 =' 0 +' 1 that is F ~ ' k () = N k+1 + N k + N k1 = ~ N k ;2R d ;k 1: 9 2.2. FUNCTION SPACES CHAPTER 2. NOTATION, FUNCTION SPACES, MAIN RESULTS, AND EXAMPLES Note that ' k = ~ ' k ' k ;k 0. Obviously, f = P 1 k=0 f' k inS 0 R d for f2S R d : Let s2R and p;q 1. For 2A , we introduce the Besov space B s pq R d ;V = B ;s pq R d ;V as the closure ofS R d ;V in the norm jfj B s pq (R d ;V ) =jfj B ;s pq (R d ;V ) = 0 @ 1 X j=0 jJ ;s ' j fj q Lp(R d ;V ) 1 A 1=q : We introduce the corresponding spaces of functions on E = [0;T ]R d : The spaces B s pq (E;V ) = B ;s pq (E;V ) (resp. H s p (E;V ) = H ;s p (E;V )) consist of all measurable B ;s pq R d ;V (resp. H ;s p R d ;V ) -valued functions f on [0;T ] with finite corresponding norms: jfj B s pq (E;V ) = jfj B ;s pq (E;V ) = Z T 0 jf(t;)j q B ;s pq (R d ;V ) dt ! 1=q ; jfj H s p (E;V ) = jfj H ;s p (E;V ) = Z T 0 jf(t;)j p H ;s p (R d ;V ) dt ! 1=p : Similarly we introduce the corresponding spaces of random functions. Let ( ;F;P) be a complete probability spaces with a filtration of algebrasF = (F t ) satisfying the usual conditions. LetR (F) be the progressive algebra on [0;1) . The spaces B s pp R d ;V = B ;s pp R d ;V and H s p R d ;V = H ;s p R d ;V consists of allFmeasurable random functions f with values in B ;s pp R d ;V and H ;s p R d ;V with finite norms jfj B ;s pp (R d ;V ) = n Ejfj p B ;s pp (R d ;V ) o 1=p and jfj H ;s p (R d ;V ) = n Ejfj p H ;s p (R d ;V ) o 1=p : The spaces B s pp (E;V ) = B ;s pp (E;V ) and H s p (E;V ) = H ;s p (E;V ) consist of all R (F)measurable random functions with values in B ;s pp (E;V ) and H ;s p (E;V ) with finite norms jfj B ;s pp (E;V ) = n Ejfj p B ;s pp (E;V ) o 1=p and jfj H ;s p (E;V ) = n Ejfj p H ;s p (E;V ) o 1=p : 10 2.3. MAIN RESULTS CHAPTER 2. NOTATION, FUNCTION SPACES, MAIN RESULTS, AND EXAMPLES If V r =L r (U;U; );r 1, the space of rintegrable measurable functions on U, and V 0 =R, we write B ;s r;pp (A) = B ;s pp (A;V r ); B ;s r;pp (A) =B ;s pp (A;V r ); H ;s r;p (A) = H ;s p (A;V r ); H ;s r;p (A) =H ;s p (A;V r ); and L r;p (A) =H 0 r;p (A); L r;p (A) =H 0 r;p (A); where A =R d or E. For scalar functions we drop V in the notation of function spaces. LetU n 2U;U n U n+1 ;n 1;[ n U n =U and (U n )<1;n 1:Wedenoteby ~ C 1 r:p (E); 1p<1;the space of allR (F) B R d -measurableV r -valued random functions onE such that for every multiindex 2N d 0 , E Z T 0 sup x2R d jD (t;x)j p Vr dt +E h jD j p Lp(E;Vr ) i <1; (2.2) and = Un for some n if r = 2;p: Similarly we define the space ~ C 1 r:p R d by replacingR (F) and E byF andR d respectively in the definition of ~ C 1 r;p (E). Corresponding deterministic spaces are denoted by ~ C 1 r;p R d and ~ C 1 r;p (E) (droppingE in (2.2).) When the context is clear we sometimes drop andR d from notation of function spaces. 2.3 Main Results We set for 2A , (r) = (r) = x2R d :jxj>r ;r> 0 w (r) =w (r) = (r) 1 ;r> 0: Our main assumption is that w =w (r) is an O-RV function both at zero and at infinity. That is w is a positive, finite, measurable function and r 1 (x) = lim sup !0 w (x) w () <1; r 2 (x) = lim sup !1 w (x) w () <1;x> 0: By Theorem 2 in [AA77], the following limits exist: p 1 =p w 1 = lim !0 logr 1 () log q 1 =q w 1 = lim !1 logr 1 () log (2.3) 11 2.3. MAIN RESULTS CHAPTER 2. NOTATION, FUNCTION SPACES, MAIN RESULTS, AND EXAMPLES and p 2 =p w 2 = lim !0 logr 2 () log q 2 =q w 2 = lim !1 logr 2 () log : (2.4) Note that p 1 q 1 (see Remark 6 of [MX19].) We will assume throughout this paper that p 1 ;p 2 ;q 1 ;q 2 > 0. The numbers p 1 ;p 2 are called lower indices and q 1 ;q 2 are called upper indices of O-RV function. When the context is clear, for a function f which is both O-RV at zero and infinity we always denote its lower index at zero by p 1 , upper index at zero by q 1 , lower index at infinity by p 2 and upper index at infinity by q 2 . If we wish to be precise, we will write p f 1 ;p f 2 ;q f 1 ;q f 2 . For brevity, we say that f is an O-RV function if it is both O-RV at zero and infinity. If = 1, we assume throughout this manuscript that R R<jyjR 0 y (dy) = 0 for any 0<R<R 0 <1. 12 2.3. MAIN RESULTS CHAPTER 2. NOTATION, FUNCTION SPACES, MAIN RESULTS, AND EXAMPLES The main results for (1.1) is the following statement. Theorem 2. Let p2 (1;1);s2R: Let 2 A ; and w = w be continuous O-RV function at zero and infinity with p i ;q i ;i = 1; 2; defined in (2.3), (2.4). Assume A. For i = 1; 2 0 0; where ~ R (dy) =w (R) (Rdy);R2 (0;1): Then for each f 2 H ;s p (E);g2 B ;s+11=p pp R d , 2 B ;s+11=p p;pp (E)\H ;s+1=2 2;p (E) if p2 [2;1) and 2B ;s+11=p p;pp (E) if p2 (1; 2), there is a unique u2H ;s+1 p (E) solving (1.1). Moreover, there is C =C (d;p;) such that for p2 [2;1), jL uj H ;s p (E) C h jfj H ;s p (E) +jgj B ;s+11=p pp (R d ) +jj B ;s+11=p p;pp (E) +jj H ;s+1=2 2;p (E) i ; juj H ;s p (E) C h jfj H ;s p (E) + 1=p jgj H ;s p (R d ) + 1=p jj H ;s p;p (E) + 1=2 jj H ;s 2;p (E) i ; and for p2 (1; 2), jL uj H ;s p (E) C h jfj H ;s p (E) +jgj B ;s+11=p pp (R d ) +jj B ;s+11=p p;pp (E) i ; juj H ;s p (E) C h jfj H ;s p (E) + 1=p jgj H ;s p (R d ) + 1=p jj H ;s p;p (E) i ; where = 1 ^T: Note that the assumptionA depends on only through w =w . 13 2.4. EXAMPLES CHAPTER 2. NOTATION, FUNCTION SPACES, MAIN RESULTS, AND EXAMPLES 2.4 Examples Example 3. We list examples of functions that are O-RV both at infinity and at zero. Examples (i)-(iv) are taken from p.237 of [KSV14]. Example (v) is taken from p.41 of [KSV18]. (i) (r) =r +r ; 0<< < 1 (ii) (r) = (r +r ) ; ;2 (0; 1) (iii) (r) =r (log (1 +r)) ; 2 (0; 1);2 (0;) (iv) (r) = (log (cosh ( p r)) log ( p r)) ; 2 (0; 1) (v) (r) = r +m 1= ; m;2 (0; 1);m> 0. Next we provide some concrete examples of permissible Lévy measures. The following examples are provided in Corollary 6 of [MX19] and the Remark before it (see also [DM78] p.78-III, 70-74.) Example 4. (Corollary 6 of [MX19]) Any Lévy measure 2A can be disintegrated as () = Z 1 0 Z S d1 (rz) (r;dz)d (r); 2B R d 0 ; where = x2R d :jxj>r ;r > 0, and (r;dz);r > 0 is a measurable family of measures on the unit sphere S d1 with (r;S d1 ) = 1;r > 0: If w = 1 is continuous, O-RV at zero and infinity, and satisfies AssumptionA. Assume thatjfs2 [0; 1] :r i (s)< 1gj> 0;i = 1; 2; and inf j ^ j=1 Z S d1 ^ z 2 (r;dz)c 0 > 0; r> 0; hold, then all assumptions of Theorem 2 holds. (cf. Lemma 92 in the Appendix.) The next example is motivated by Example 2.12 of [KK16]. The main difference is thatS (dz) (see below) can be more general than the Lebesgue measure. Example 5. Consider Lévy measure in radial and angular coordinate in the form (B) = Z 1 0 Z jzj=1 1 B (rz)a (r;z)j (r)r d1 S (dz)dr;B2B R d 0 ; where S (dz) is a finite measure on the unit sphere. Assume (i) There is C > 1;c> 0; 0< 1 2 < 1 such that C 1 r 2 j (r)r d C r 2 14 2.4. EXAMPLES CHAPTER 2. NOTATION, FUNCTION SPACES, MAIN RESULTS, AND EXAMPLES and for all 0<rR, c 1 R r 1 (R) (r) c R r 2 : (ii) There is a function 0 (z) defined on the unit sphere such that 0 (z)a (r;z) 1;r> 0;z2S d1 , and for all ^ = 1, Z S d1 ^ z 2 0 (z)S (dz)c> 0: Under these assumptions it can be shown thatB holds, and w is an O-RV function with 2 1 p i q i 2 2 ;i = 1; 2: 15 Chapter 3 Some Estimates of O-RV Functions We start with a simple but useful corollaries about functions that are O-RV at both zero and infinity. We will use estimates in this chapter extensively in this manuscript. For2A ;R> 0, we denote ~ R (dy) = w (R) (Rdy). Since the authors of [MX19] study a similar class of Lévy measures, some estimates are provided in their Appendix. We will summarize and extend the results from [MX19]. Lemma 6. Let 2A ; and w = w be an O-RV function at zero and infinity with p i ;q i ;i = 1; 2; defined in (2.3), (2.4), and Assumption A holds. Then for any 1 > q 1 _q 2 , 0 < 2 0 such that c 1 y x 2 w (y) w (x) c 2 y x 1 ; 0<xy<1: Proof. Due to similarity, we only show the right hand side of the inequality. By Karamata characterization (see (1.7) of [AA77]) of O-RV functions, there are 0< 1 < 2 such that the RHS inequality holds if either x_y 1 or x^y 2 . If y 2 and x 1 ; w (y) w (x) = w (y) w ( 2 ) w ( 2 ) w ( 1 ) w ( 1 ) w (x) c y x 1 w ( 2 ) w ( 1 ) ; and similarly we consider other cases. Remark 7. It trivially follows from Lemma 6 that lim r!0 w (r) = 0 and lim r!1 w (r) =1: For the rest of manuscript, we will always interpret w (0) = 0. The following Lemma is taken from Lemma 8 of [MX19]. 16 CHAPTER 3. SOME ESTIMATES OF O-RV FUNCTIONS Lemma 8. (provided in Lemma 8 of [MX19]) Assume that w (r);r > 0; is an O-RV function at zero with lower and upper indices p 1 ;q 1 ; that is, r 1 (x) = lim "!0 w (x) w () <1;x> 0; and p 1 = lim !0 logr 1 () log q 1 = lim !1 logr 1 () log () : (i) Let > 0 and >p 1 : There is C > 0 so that Z x 0 t w (t) dt t Cx w (x) ;x2 (0; 1]; and lim x!0 x w (x) = 0: (ii) Let > 0 and <q 1 . There is C > 0 so that Z 1 x t w (t) dt t Cx w (x) ;x2 (0; 1]; and lim x!0 x w (x) =1: (iii) Let < 0 and >q 1 . There is C > 0 so that Z x 0 t w (t) dt t Cx w (x) ;x2 (0; 1]; and lim x!0 x w (x) = 0: (iv) Let < 0 and 0 so that Z 1 x t w (t) dt t Cx w (x) ;x2 (0; 1]; and lim x!0 x w (x) =1: Similar statement holds for O-RV functions at infinity. Lemma 9. Assume that w (r);r > 0; is an O-RV function at infinity with lower and upper indices p 2 ;q 2 ; that is, r 2 (x) = lim "!1 w (x) w () <1;x> 0; and p 2 = lim !0 logr 2 () log q 2 = lim !1 logr 2 () log () : 17 CHAPTER 3. SOME ESTIMATES OF O-RV FUNCTIONS (i) Let > 0 and >q 2 : There is C > 0 so that Z 1 x t w (t) dt t Cx w (x) ;x2 [1;1); and lim x!1 x w (x) = 0: (ii) Let > 0 and 0 so that Z x 1 t w (t) dt t Cx w (x) ;x2 [1;1); and lim x!1 x w (x) =1: (iii) Let < 0 and 0 so that Z 1 x t w (t) dt t Cx w (x) ;x2 [1;1); and lim x!1 x w (x) = 0: (iv) Let < 0 and >q 2 . There is C > 0 so that Z x 1 t w (t) dt t Cx w (x) ;x2 [1;1); and lim x!1 x w (x) =1: Proof. Following the proof of Lemma 8 in [MX19]. The claims follow easily by Theorems 3, 4 in [AA77]. We will prove (iv) only, other cases follow the same line of argument. Let < 0 and >q 2 , then lim t!1 w ("t) w (t) = lim t!1 w " 1 "t w ("t) =r 2 " 1 <1;"> 0: Therefore, by definition, w (t) ;t 1, is an O-RV function at infinity with lower index p and upper index q such that p = lim "!0 logr 2 " 1 log" =q 2 p 2 = lim "!0 logr 2 (") log" =q: Then by Theorems 3 and 4 in [AA77], for >q 2 ; Z x 1 t w (t) dt t Cx w (x) ;x 1; and lim x!1 x w (x) =1. 18 CHAPTER 3. SOME ESTIMATES OF O-RV FUNCTIONS Combining Lemmas 8 and 9 yields Corollary 10. Assume that w (r);r> 0; is an O-RV at zero and infinity with indices p 1 ;q 1 ;p 2 ;q 2 . (i) For any > 0, and >p 1 _p 2 . There is C > 0 so that Z r 0 t w (t) dt t Cr w (r) ;r> 0 For any < 0, and >q 1 _q 2 . There is C > 0 so that Z r 0 t w (t) dt t Cr w (r) ;r> 0 (ii) For any > 0, and <q 1 ^q 2 . There is C > 0 so that Z 1 r t w (t) dt t Cr w (r) ;r> 0 For any < 0, and 0 so that Z 1 r t w (t) dt t Cr w (r) ;r> 0: Lemma 11. Let 2A ,w =w be an O-RV function at zero and infinity with indices p 1 ;q 1 ;p 2 ;q 2 defined in (2.3), (2.4). Then for any 1 >q 1 _q 2 and 0< 2 0 such that Z jyj1 jyj 1 ~ r (dy) + Z jyj>1 jyj 2 ~ r (dy)C;r> 0: Proof. Using similar computation to Lemma 9 of [MX19]. First, Z jyj1 jyj 1 ~ r (dy) = (r) 1 r 1 Z jyjr jyj 1 (dy) = (r) 1 r 1 1 Z r 0 s 1 [ (s) (r)] ds s = (r) 1 r 1 1 Z r 0 (s)s 11 ds 1; and similarly, 19 CHAPTER 3. SOME ESTIMATES OF O-RV FUNCTIONS Z jyj>1 jyj 2 ~ r (dy) = w (r)r 2 2 Z 1 0 (s_r)s 2 ds s = 1 +w (r)r 2 2 Z 1 r (s)s 2 ds s : Thus Z jyj1 jyj 1 ~ r (dy) + Z jyj>1 jyj 2 ~ r (dy) = w (r)r 1 1 Z r 0 w (s) 1 s 1 ds s +w (r)r 2 2 Z 1 r w (s) 1 s 2 ds s = I 1 +I 2 : By Lemma 9, there is C so that I 2 =w (r)r 2 2 Z 1 r w (s) 1 s 2 ds s C;r 1: By Lemma 8, there is C so that w (r)r 2 2 Z 1 r w (s) 1 s 2 ds s C;r2 (0; 1): Hence there is C so that I 2 C for all r> 0: Similarly, using Lemmas 8-9, we estimate I 1 : Remark 12. In particular, if w = w satisfies assumption A, then we may choose in Lemma 11, 1 ; 2 so that (i) 1; 2 2 (0; 1) if 2 (0; 1); (ii) 1 ; 2 2 (1; 2) if 2 (1; 2); (iii) 1 2 (1; 2] and 2 2 (0; 1) if = 1: Lemma 13. Let 2A ,w =w (r);r> 0; be a continuous O-RV function at zero and infinity with indices p 1 ;q 1 ;p 2 ;q 2 defined in (2.3), (2.4), and p 1 ;p 2 > 0: Let a (r) = infft> 0 :w (t)rg;r> 0: Then (i) a is left-continuous, a (t)2 (0;1), w (a (t)) =t;t> 0; and a (w (t)) ta (w (t) +);t> 0: 20 CHAPTER 3. SOME ESTIMATES OF O-RV FUNCTIONS Moreover, if a 1 is a generalized inverse of a, i.e. a 1 (r) = infft> 0 :a (t)rg;r> 0, then a 1 =w. (ii) a is O-RV at zero and infinity with lower indices p; p and upper indices q; q respectively so that 1 q 1 pq 1 p 1 ; 1 q 2 p q 1 p 2 : Proof. (i) follow easily from the definitions (see also Remark 7) (ii) By Theorem 3 in [AA77], for any 2 (0;p 1 ) there is C > 0 so that w (x) x C w (y) y ; 0<xy 1: Hence a (y) y 1 C a (x) x 1 ; 0<xy 1; and, by Karamata characterization (1.7) in [AA77] and Theorem 3 in [AA77], a is O-RV at zero with upper index, q 1 p 1 : Similarly we find that the lower index at zero p 1 q1 , and determine that a is O-RV at infinity with indices p q so that 1 q 2 p q 1 p 2 : The following claim is an obvious consequence of Lemmas 8, 9, 13. Corollary 14. Let 2 A ,w = w (r);r > 0; be a continuous O-RV function at zero and infinity with indices p 1 ;q 1 ;p 2 ;q 2 defined in (2.3), (2.4), and p 1 ;p 2 > 0: Let a (r) = infft> 0 :w (t)rg;r> 0: (i) For any > 0 and < q1 ^ q2 there is C > 0 such that Z r 0 t a (t) dt t Cr a (r) ;r> 0; lim r!0 r a (r) = 0; lim r!1 r a (r) =1; 21 CHAPTER 3. SOME ESTIMATES OF O-RV FUNCTIONS and for any < 0; > p1 _ p2 there is C > 0 such that Z r 0 t a (t) dt t Cr a (r) ;r> 0; lim r!0 r a (r) = 0; lim r!1 r a (r) =1: (ii) For any > 0 and > p1 _ p2 there is C > 0 such that Z 1 r t a (t) dt t Cr a (r) ;r> 0; lim r!0 r a (r) = 1; lim r!1 r a (r) = 0; and for any < 0 and < q1 ^ q2 there is C > 0 such that Z 1 r t a (t) dt t Cr a (r) ;r> 0; lim r!0 r a (r) = 1; lim r!1 r a (r) = 0: 22 Chapter 4 Function Spaces - Norm Equivalence and Characterizations 4.1 Introduction In this chapter, we discuss characterizations of Bessel potential spaces and Besov spaces. Let ~ C 1 R d be the space of all functionsf onR d such that for any multiindex 2N d 0 and for all 1p<1 sup x2R d jD f (x)j +jD fj Lp(R d ) <1: For a separable Hilbert spaceG andr 1, we denotel r (G) the space of all sequencesa = (a j );a j 2G, with finite norm jaj lr (G) = 0 @ 1 X j=0 ja j j r G 1 A 1=r : We denote l r =l r (R). Let L p R d ;G be the space of allG-valued measurable functions f such that jfj Lp(R d ;G) = Z jf (x)j p G dx 1=p <1: For 2A ;N > 1, we introduce Besov spaces ~ B ;s pq R d to be closure ofS R d in the norm jfj ~ B ;s pq (R d ) = 0 @ 1 X j=0 w N j sq jf' j j q Lp(R d ) 1 A 1=q : 23 4.2. AUXILIARY RESULTS CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS where ' j ;j 0 is the system provided in Remark 1. Let ~ H ;s p R d be closure ofS R d in the norm, jfj ~ H ;s p (R d ) = 0 @ 1 X j=0 w N j s f' j 2 1 A 1=2 Lp(R d ) : Let ~ H ;s p R d ;l 2 be space of all sequences f = (f k ) k0 with f k 2 ~ H ;s p R d and finite norm jfj ~ H ;s p (R d ;l2) = 0 @ 1 X k;j=0 w N j s f k ' j 2 1 A 1=2 Lp(R d ) : We will prove the following characterization of Bessel potential spaces and Besov spaces. Proposition 15. Let 2A ;w =w be an O-RV function and A, B hold. Let s2R;p;q2 (1;1). (i) ~ B ;s pq R d =B ;s pq R d and the norms are equivalent (ii) ~ H ;s p R d =H ;s p R d and the norms are equivalent (iii) ~ H ;s p R d ;l 2 =H ;s p R d ;l 2 and the norms are equivalent. Moreover, for all s;s 0 2R, J s : A s 0 ! A s 0 s is an isomorphism where A s = B ;s pq R d ;H ;s p R d or H ;s p R d ;l 2 : The proof will rely on a series of auxiliary results. 4.2 Auxiliary Results In this section, we present some results regarding L p estimates of vector-valued functions.We start with Lemma 16. Let N > 1, and j (x);x2R d ;j 0; be a sequence of measurable functions. Assume (i) There is > 0 so that Z jxj j j (x)jdxA;j 0: (ii) There is a non-negative increasing function w (r);r2 [0; 1]; so that P 1 k=0 w N k <1 and Z j j (x +y) j (x)jdxw (jyj);jyj 1;j 0: Then for K j (x) =N jd j N j x ;x2R d ;j 0, we have 1 X j=0 Z jxj>4jyj jK j (x +y)K j (x)jdxB;y2R d ; (4.1) 24 4.2. AUXILIARY RESULTS CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS for some constant B: Proof. For any y2R d ; 1 X k=0 Z jxj>4jyj jK k (x +y)K k (x)jdx = 1 X k=0 Z jxj>N k 4jyj k x +N k y k (x) dx 1 X k=0 sup j0 Z jxj>N k 4jyj j x +N k y j (x) dx = 1 X k=0 F N k y ; where F (z) = sup j0 Z jxj>4jzj j j (x +z) j (x)jdx;z2R d : Let G (y) = 1 X k=1 F N k y ;y2R d : Since G (Ny) =G (y);y2R d , it is enough to prove that G (y)B; 1=Njyj 1; for some B > 0: We split the sum G (y) = 1 X k=1 F N k y = 1 X k=0 ::: + 1 X k=1 ::: = G 1 (y) +G 2 (y); 1=Njyj 1: With 1=Njyj 1;k 0; by Chebyshev’s inequality, Z jxj>N k 4jyj j x +N k y j (x) dx Z jxj>N k 4jyj j x +N k y dx + Z jxj>N k 4jyj j j (x)jdx C Z jxj>N k 3jyj j j (x)jdxC Z jxj>N k1 3 j j (x)jdx CN k Z jxj j j (x)jdxCAN k ; and G 1 (y)CA 1 X k=0 N k ; 1=Njyj 1: 25 4.2. AUXILIARY RESULTS CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS For 1=Njyj 1;k< 0; Z jxj>N k 4jyj j x +N k y j (x) dx Z j x +N k y j (x) dxw N k ; and G 2 (y) 1 X k=1 w N k ; 1=Njyj 1: The claim is proved. Corollary 17. Let the assumptions of Lemma 16 hold and sup j; ^ j () <1, and let G be a separable Hilbert space. Then (i) For any 1<p;r<1 there is a constant C p;r so that 0 @ X j jK j f j j r 1 A 1=r Lp(R d ) C p;r 0 @ X j jf j j r 1 A 1=r Lp(R d ) : for all f = (f j )2L p R d ;l r : (ii) For any 1<p<1 there is a constant C > 0 such that 0 @ X j jK j f j j 2 G 1 A 1=2 Lp(R d ) C p 0 @ X j jf j j 2 G 1 A 1=2 Lp(R d ) for all f = (f j )2L p R d ;l 2 (G) . Proof. (i) Since (4.1) holds according to Lemma 16, the statement follows by Theorem V.3.11 in [GR85]. (ii) SinceG is isomorphic to l 2 , the statement follows by Theorem V.3.9 in [GR85]. As the first application we have Corollary 18. Let ; 0 2S R d , ~ =F 1 ;j 1; ~ 0 =F 1 0 . Let N > 1; ~ j (x) = N jd ~ N j x ;x2 R d ;j 1: Then for each 1<p;r<1 there is a constant C p;r so that for all f = (f j )2L p R d ;l r 0 @ X j f j ~ j r 1 A 1=r Lp(R d ) C p;r 0 @ X j jf j j r 1 A 1=r Lp(R d ) : (4.2) 26 4.2. AUXILIARY RESULTS CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS Ifr = 2, then (4.2) holds for a separable Hilbert spaceG-valued sequencesf = (f j )2L p R d ;l r (G) (simply absolute value in (4.2) is replaced byG-norm). Proof. WeapplypreviousCorollary17with 0 = ~ 0 ; j (x) = (x) = ~ (x);j 1;K j (x) =N jd j N j x ;x2 R d ;j 0: Obviously, sup [j ()j +j 0 ()j] < 1; Z jxj [j (x)j +j 0 (x)j]dx < 1; and Z j (x +y) (x)jdx Z Z 1 0 jr (x +sy)jdsjyjdx jyj Z jr (x)jdx;y2R d : Similarly, Z j 0 (x +y) 0 (x)jdxjyj Z jr 0 (x)jdx;y2R d : The statement follows by Corollary 17. We will need the following auxiliary statements about estimates of convolution of smooth functions and probability density of Z R t =Z ~ R t . As a reminder, for R > 0, Z R t = Z ~ R t ;t > 0 is the stochastic process with independent increments associated with ~ R =w (R) R , i.e., Ee i2Z R t = exp ~ R ()t with ~ R () = Z e i2y 1i2 (y)y d~ R ;2R d : Lemma 19. Let 2A ;w = w be an O-RV function and A, B hold. Let ; 0 2 C 1 0 R d be such that 0 = 2 supp () . Let ~ =F 1 ; ~ 0 =F 1 0 , and for R> 0, H R (t;x) = E ~ x +Z R t ;t 0;x2R d ; H R 0 (t;x) = E ~ 0 x +Z R t ;t 0;x2R d : 27 4.2. AUXILIARY RESULTS CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS (i) For any 0< 2 0 independent of R such that Z (1 +jxj 2 ) H R (t;x) dx C 1 e C2t ;t 0; Z jxj 2 H R 0 (t;x) dx C 0 (1 +t);t 0; Z H R 0 (t;x) dx C 0 ;t 0: (ii) There are constants C 1 ;C 2 > 0 independent of R so that for y2R d ; Z H R (t;x +y)H R (t;x) dx C 1 jyje C2t ; Z H R 0 (t;x +y)H R 0 (t;x) dx jyj Z r ~ 0 (x) dx: Proof. (i) Note that FH R (t;) = exp ~ R ()t ();2R d : (4.3) Step 1. This step is taken from the proof of Lemma 6 in [MX19] with only minor changes. We include the details here for completeness. We fix > 0 such that supp () :jj 1 . Let ~ R; (dy) = fjyjg ~ R (dy);R> 0, then for 2 supp (), andjyj, 1 cos (y) 1 jyj 2 = jj 2 ^ y 2 ; ^ ==jj; and for some c 0 =c 0 ("); changing the variable of integration, applying Lemma 6, Re ~ R; () = Z jyj (1 cos (y)) ~ R; (dy) jj 2 Z jyj ^ y 2 ~ R (dy) (4.4) = jj 2 Z jyj1 2 ^ y 2 w (R) w (R) ~ R (dy)c 0 jj 2 for all 2 supp (): Step 2. The rest of the proof follows our argument in Lemma 2 of [MPh19]. Taking inverse Fourier transform in (4.3), H R (t;) =F (t;)P t ; where F (t;x) =F 1 exp ~ R; t (x) =E ~ x +Z ~ R; t ;t 0;x2R d ; 28 4.2. AUXILIARY RESULTS CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS and P t (dy) is the distribution of Z ~ R ~ R; t . By Plancherel’s theorem, (4.4), and 0 = 2 supp (), for any multiindex 2N d 0 , Z jx F (t;x)j 2 dx C Z D () exp ~ R; ()t 2 d C 1 e C2t ;t 0: By Cauchy-Schwarz inequality, with d 0 = d 2 + 1; Z 1 +jxj 2 jF (t;x)jdx = Z 1 +jxj 2 (1 +jxj) d0 jF (t;x)j (1 +jxj) d0 dx Z (1 +jxj) 2d0 dx 1=2 Z (1 +jxj) 4 jF (t;x)j 2 (1 +jxj) 2d0 dx 1=2 C Z F (t;x) 2 1 +jxj 2 d0+2 dx 1=2 C 1 e C2t ;t 0: Let 0< 2 0 so that E h Z ~ R ~ R; t 2 i = Z jyj 2 P t (dy)C (1 +t);t 0: Hence there are constants C 1 ;C 2 so that Z (1 +jxj 2 ) H R (t;x) dx = Z (1 +jxj 2 ) Z F (t;xy)P t (dy) dx Z Z (1 +jxyj 2 )jF (t;xy)jP t (dy)dx + Z Z jyj 2 jF (t;xy)jP t (dy)dx C 1 e C2t ;t 0; and Z jxj 2 H R 0 (t;x) dx = Z jxj 2 E ~ 0 x +Z R t dx E Z x +Z R t 2 ~ 0 x +Z R t dx +E h Z R t 2 i Z ~ 0 (x) dx C (1 +t): The last inequality is trivial. 29 4.2. AUXILIARY RESULTS CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS (ii) Similarly as in part (i), for y2R d ; Z H R (t;x +y)H R (t;x) dx = Z Z Z 1 0 rF (t;x +syz)ydsP t (dz) dx jyj Z jDF (t;x)jdxC 1 jyje C2t ;t> 0; and directly Z H R 0 (t;x +y)H R 0 (t;x) dxjyj Z r ~ 0 (x) dx: Lemma 20. Let w be a non-decreasing O-RV function at zero and infinity, with p w 1 ;p w 2 > 0. Let 2A and define ~ R (dy) =w (R) (Rdy), R> 0: (i) Assume there is N 2 > 0 so that Z (jyj^ 1) ~ R (dy) N 2 if 2 (0; 1); Z jyj 2 ^ 1 ~ R (dy) N 2 if = 1; Z jyj1 jyj 2 ~ R (dy) + Z jyj>1 jyj ~ R (dy) N 2 if 2 (1; 2) for any R> 0: Then there is a constant C 1 so that for all 2R d ; Z [1 cos (2y)] (dy) C 1 N 2 w jj 1 1 ; Z jsin (2y) 2 (y)yj (dy) C 1 N 2 w jj 1 1 ; assuming w jj 1 1 = 0 if = 0: (ii) Let inf R2(0;1);j ^ j=1 Z jyj1 ^ y 2 ~ R (dy) =c 0 > 0: Then there is a constant c 2 =c 2 (w;c 0 )> 0 such that Z [1 cos (2y)] (dy)c 2 w jj 1 1 for all 2R d ; assuming w jj 1 1 = 0 if = 0: 30 4.2. AUXILIARY RESULTS CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS Proof. The following simple trigonometric estimates hold: jsinxxj jxj 3 6 ; 1 cosx 1 2 x 2 ;x2R; (4.5) 1 cosx x 2 if jxj=2: (i) Let 6= 0. Denoting ^ ==jj; and using (4.5), Z 1 cos 2 ^ jjy (dy) = w jj 1 1 Z 1 cos 2 ^ y ~ jj 1 (dy) w jj 1 1 2 2 Z jyj 2 ^ 1 ~ jj 1 (dy); and there is C 1 so that Z jsin (2y) 2 (y)yj (dy) = w jj 1 1 Z sin 2 ^ y 2 (y) ^ y ~ jj 1 (dy) C 1 N 2 w jj 1 1 for all 2R d . (ii) By (4.5) for all 2R d ; Z [1 cos (2y)] (dy) = Z [1 cos 2 ^ y ] jj 1 (dy) Z jyj 1 4 4 ^ y 2 jj 1 (dy) = 4 1 Z j4yj1 ^ 4y 2 jj 1 (dy) = 4 1 w j4j 1 1 Z jyj1 ^ y 2 ~ j4j 1 (dy) cw j4j 1 1 cw jj 1 1 : Lemma 21. Let 2A ;w =w be an O-RV function and A, B hold. Let Z j t =Z ~ N j t be the Lévy process associated to L ~ N j ;j 1, and Z t =Z t . Let ; 0 2C 1 0 R d be such that 0 = 2supp(): Let j (x) = Z 1 0 e w(N j )t E ~ x +Z j t dt;j 1;x2R d ; 0 (x) = Z 1 0 e t E ~ 0 (x +Z t )dt;x2R d ; 31 4.2. AUXILIARY RESULTS CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS where ~ =F 1 ; ~ 0 =F 1 0 . Let K j (x) =N jd j N j x ;j 0;x2R d : Then for 1<p;r<1 there is a constant C such that for all f = (f j )2L p R d ;l r 0 @ 1 X j=0 jK j f j j r 1 A 1=r Lp(R d ) C 0 @ 1 X j=0 f j ~ j r 1 A 1=r Lp(R d ) (4.6) C 0 @ 1 X j=0 jf j j r 1 A 1=r Lp(R d ) ; where ~ j =F 1 N j ;j 1: If r = 2, then (4.6) holds for a separable Hilbert space G-valued sequences f = (f j )2 L p R d ;l r (G) (simply absolute value in (4.6) is replaced byG-norm). In particular, there is a constant C so that jK j fj Lp(R d ;G) Cjfj Lp(R d ;G) ;j 0;f2L p R d ;G : (4.7) Proof. Let ; 0 2C 1 0 R d be such that =; 0 0 = 0 ; and 0 = 2supp(). Let ~ =F 1 ; ~ 0 =F 1 0 , and ~ j (x) = Z 1 0 e w(N j )t E~ x +Z j t dt;j 1;x2R d ; ~ 0 (x) = Z 1 0 e t E~ 0 (x +Z t )dt;x2R d ; Let ~ K j (x) =N jd ~ j N j x ;j 0;x2R d : Obviously, K j f = ~ K j f ~ j ;j 0: We will check the assumptions of Lemma 16 for ~ j ;j 0. (i) We will prove that Z jxj 2 ~ j (x) dxA;j 0; 32 4.2. AUXILIARY RESULTS CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS where 0< 2 <p ! 1 ^p ! 2 . By Lemma 19, Z jxj 2 ~ 0 (x) dx Z Z 1 0 e t jxj 2 jE~ 0 (x +Z t )jdtdx C Z 1 0 e t (1 +t)dt; and Z jxj 2 ~ j (x) dx Z 1 0 Z jxj 2 E~ x +Z j t dxdt C Z 1 0 C 1 e C2t dt;j 1: (ii) We prove Z ~ j (x +y) ~ j (x) dxAjyj;jyj 1;j 0: By Lemma 19, for any y2R d ; Z ~ 0 (x +y) ~ 0 (x) Cjyj Z 1 0 e t dt; and Z ~ j (x +y) ~ j (x) dx Z 1 0 Z E~ x +y +Z j t E~ x +Z j t dxdtC 1 jyj Z 1 0 e C2t dt;j 1: (iii) We prove that F ~ j () A;j 1;2R d : Indeed, by Lemma 20, there is c> 0 so that for 2supp() exp ~ N j ()t e ct ;t> 0: Hence F ~ j () Z 1 0 e ct ()dtA;2R d ;j 1; and, obviously, F ~ 0 () Z 1 0 e t jexpf ()tg 0 ()jdtA: 33 4.2. AUXILIARY RESULTS CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS Therefore (4.6) follows from Corollaries 17 and 18. (iv) The estimate (4.7) is an obvious consequence of (4.6) (take f = (f k ) with f k = 0 if k6=j: The statement is proved. Lemma 22. Let 2 A ;w = w be an O-RV function and A, B hold. Let N > 1;; 0 2 C 1 0 R d , ~ =F 1 ; ~ 0 =F 1 0 , and j (x) = w N j ~ (x)L ~ N j ~ (x);x2R d ;j 1; 0 (x) = ~ 0 (x)L ~ 0 (x);x2R d : Let K j (x) =N jd j N j x ;j 0;x2R d : Then for 1<p;r<1 there is a constant C such that for all f = (f j )2L p R d ;l r 0 @ 1 X j=0 jK j f j j r 1 A 1=r Lp(R d ) C 0 @ 1 X j=0 f j ~ j r 1 A 1=r Lp(R d ) (4.8) C 0 @ 1 X j=0 jf j j r 1 A 1=r Lp(R d ) ; (4.9) where ~ j =F 1 N j ;j 1: If r = 2, then ((4.8), (4.9) hold for a separable Hilbert space G-valued sequences f = (f j )2L p R d ;l r (G) (simply absolute value is replaced byG-norm). In particular, there is a constant C so that jK j fj Lp(R d ;G) Cjfj Lp(R d ;G) ;j 0;f2L p R d ;G : (4.10) Proof. Let ; 0 2C 1 0 R d and =; 0 0 = 0 ; ~ =F 1 ; ~ 0 =F 1 0 , and ~ j (x) = w N j ~ (x)L ~ N j ~ (x);x2R d ;j 1; ~ 0 (x) = ~ 0 (x)L ~ 0 (x);x2R d : 34 4.2. AUXILIARY RESULTS CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS Let ~ K j (x) =N jd ~ j N j x ;j 0;x2R d : Again, K j f = ~ K j f ~ j ;j 0: We will check the assumptions of Corollary 17 for ~ j ;j 0. We choose 0< 2 <p ! 1 ^p ! 2 . (i) First we prove that Z jxj 2 ~ j (x) dxA;j 0: Obviously, Z w N j jxj 2 j~ (x)jdx w (1) Z jxj 2 j~ (x)jdx<1; Z jxj 2 j~ 0 (x)jdx < 1: We split L ~ N j ~ (x) = Z jzj1 [~ (x +z) ~ (x) (z)zr~ (x)]~ N j (dz) + Z jzj>1 [~ (x +z) ~ (x) (z)zr~ (x)]~ N j (dz) = A j (x) +B j (x);x2R d ;j 1: For 2 [1; 2); A j (x) = Z jzj1 Z 1 0 (1s)~ xixj (x +sz)z i z j ds~ N j (dz); and Z jxj 2 jA j (x)jdx C Z 1 0 Z Z jzj1 jx +szj 2 D 2 ~ (x +sz) jzj 2 ~ N j (dz)dxds +C Z 1 0 Z Z jzj1 D 2 ~ (x +sz) jzj 2+2 ~ N j (dz)dxds C Z (jxj 2 + 1) D 2 ~ (x) dx;j 1; For 2 (0; 1); A j (x) = Z jzj1 Z 1 0 r~ (x +sz)z~ N j (dz);x2R d ; 35 4.2. AUXILIARY RESULTS CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS and Z jxj 2 jA j (x)jdx Z Z jzj1 Z 1 0 jx +szj 2 jr~ (x +sz)jdsjzj ~ N j (dz)dx + Z 1 0 Z Z jzj1 jzj 2 jr~ (x +sz)jjzj ~ N j (dz)dxds C Z (jxj 2 + 1)jr~ (x)jdx Z jzj1 jzj ~ N j (dz); C Z (jxj 2 + 1)jr~ (x)jdx;j 1: Now; Z jxj 2 jB j (x)jdx Z Z jzj>1 jx +zj 2 j~ (x +z)j ~ N j (dz)dx + Z Z jzj>1 jzj 2 j~ (x +z)j ~ N j (dz)dx + Z Z jzj>1 jxj 2 j~ (x)j ~ N j (dz)dx + Z jxj 2 jr~ (x)jdx Z jzj>1 (z) ~ N j (dz) C;j 1: Similarly, by splitting we show that Z jxj 2 jL ~ 0 (x)jdx<1: (ii) Now we prove that Z ~ j (x +y) ~ j (x) dxAjyj;jyj 1;j 0: First obviously, w N j Z j~ (x +y) ~ (x)jdx w N j Z Z 1 0 jr~ (x +sy)jjyjdsdx w (1)jyj Z jr~ (x)jdx and, similarly, Z j~ 0 (x +y) ~ 0 (x)jdxjyj Z jr~ 0 (x)jdx: 36 4.3. PROOF OF PROPOSITION ?? (I) CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS Now, Z L ~ N j ~ (x +y)L ~ N j ~ (x) dx Z Z 1 0 L ~ N j r~ (x +sy) jyjdsdxjyj Z Z 1 0 L ~ N j r~ (x) dsdx Cjyj;j 1;y2R d ; and, similarly, Z jL ~ 0 (x +y)L ~ 0 (x)jdx Cjyj;y2R d : (iii) We prove that F ~ j () A;j 1;2R d : Indeed, by Lemma 20, there is a constant C independent of j so that F[L ~ N j ~ ] () = ~ N j () () C;j 1;2R d : Similarly, jF[L ~ 0 ] ()j =j () 0 ()jC;2R d : (iv) We have (4.8) by Corollary 17, and (4.9) follows by Corollary 18. The estimate (4.10) follows, obviously, from (4.9). 4.3 Proof of Proposition 15 (i) Proof. Let p2 (1;1);f2S 0 R d and f' j 2L p R d ;j 0: It is enough to prove that for each s2R there are constants C;c (independent of f and j) so that jJ s f' j j Lp(R d ) Cw N j s jf' j j Lp(R d ) ; (4.11) and w N j s jf' j j Lp(R d ) cjJ s f' j j Lp(R d ) : (4.12) 37 4.3. PROOF OF PROPOSITION ?? (I) CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS First, denoting = sym ; Jf' j = F 1 h (1 ) N j ^ f i ;j 1; Jf' 0 = F 1 h (1 ) 0 ^ f i ; and for 2R d ; (1 ()) N j = (1 N j N j ) N j = w N j 1 (w N j ~ N j N j ) N j : Hence (4.11) withs = 1 follows by Lemma 22. Applying repeatedly (4.11) with s = 1, we see that (4.11) holds for any integer s 0: On the other hand, for j 1 (recall ' =F 1 ), J 1 ' j = Z 1 0 e t E' j ( +Z t )dt =F 1 Z 1 0 e t e ()t N j dt = w N j F 1 Z 1 0 e w(N j )t e ~ N j (N j )t N j dt = w N j N jd Z 1 0 e w(N j )t E' N j +Z j t dt; where Z j t =Z ~ N j t is the Lévy process associated to L ~ N j . For j = 0; J 1 ' 0 = Z 1 0 e t E' 0 ( +Z t )dt; where Z t = Z t ;t > 0. Hence (4.11) with s =1 follows by Lemma 21. Applying repeatedly (4.11) with s =1, we see that (4.11) holds for any negative integer s: Applying interpolation inequality we get (4.11) for all s2R. Letk2Z =f0;1;:::g ands = (1)k + (k + 1)2 (k;k + 1) with 2 (0; 1). According to Theorem 2.4.6 in [FJS01], H s p R d = H k p ;H k+1 p , H s p is the complex interpolation space between H k p and H k+1 p : By Theorem 1.9.3 in [Tri78], there is a constant C; independent of f;j, so that jJ s f' j j Lp(R d ) = jf' j j H s p (R d ) Cw N j (1)k(k+1) jf' j j Lp(R d ) = Cw N j s jf' j j Lp(R d ) : 38 4.4. PROOF OF PROPOSITION ?? (II) AND (III) CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS Now, we prove (4.12). Let f' j 2L p R d ;j 0;s2R. By (4.11), J s f' j 2L p R d ;s2R, and jf' j j Lp(R d ) = J s J s f' j Lp(R d ) Cw N j s jJ s f' j j Lp(R d ) and (4.12) follows. Thus for s2 R;p;q 2 (1;1); we have ~ B ;s pq R d = B ;s pq R d and the norms are equivalent. In addition, for anyt;s2R; the mapping J t :B ;s pq R d !B ;st pq R d is an isomorphism. 4.4 Proof of Proposition 15 (ii) and (iii) We start with Lemma 23. Let p;q2 (1;1). Then for each integer m;s there is a constant C so that for all f = (f j )2 L p R d ;l q , 0 @ 1 X j=0 jw N j m J s f j ' j j q 1 A 1=q Lp(R d ) (4.13) C 0 @ 1 X j=0 w N j ms f j ' j q 1 A 1=q Lp(R d ) ; 0 @ 1 X j=0 jw N j m f j ' j j q 1 A 1=q Lp(R d ) (4.14) C 0 @ 1 X j=0 w N j m+s J s f j ' j q 1 A 1=q Lp(R d ) : Ifq = 2, then (4.13), (4.14) hold for a separable Hilbert spaceG-valued sequencesf = (f j )2L p R d ;l 2 (G) (simply absolute values in (4.13), (4.14) are replaced byG-norms). Proof. Denote = sym , let K j (x) =N jd j N j x ;j 0, with j (x) = w N j ' (x)L ~ N j ' (x);x2R d ;j 1; 0 (x) = ' 0 (x)L ' 0 (x);x2R d : 39 4.4. PROOF OF PROPOSITION ?? (II) AND (III) CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS For f2 ~ C 1 R d , Jf' j = F 1 h (1 ) N j ^ f i = F 1 h (w N j ~ N j N j )w N j 1 N j ^ f i = w N j 1 K j f;j 1; Jf' 0 = K 0 f: By Lemma 22, for f = (f j ) with f j 2 ~ C 1 R d ; 0 @ 1 X j=0 jw N j m Jf j ' j j q 1 A 1=q Lp(R d ) = 0 @ 1 X j=0 jw N j m1 K j f j j q 1 A 1=q Lp(R d ) C 0 @ 1 X j=0 w N j m1 f j ' j q 1 A 1=q Lp(R d ) : Applying this inequality repeatedly we find that (4.13) holds for any s2N;m2Z: LetZ j t be the Lévy process associated toL ~ N j ;j 1, andZ t be the Lévy process associated toL . Let K j (x) =N jd j N j x ;j 0;x2R d ; with j (x) = Z 1 0 e w(N j )t E' x +Z j t dt;j 1;x2R d ; and 0 (x) = Z 1 0 e t E' 0 (x +Z t )dt;x2R d : Then for f2 ~ C 1 R d , J 1 f' j = F 1 Z 1 0 e t expf tg N j ^ fdt = F 1 w N j Z 1 0 e w(N j )t exp ~ N j N j t N j ^ fdt = w N j K j f; j 1; J 1 f' 0 = K 0 f: 40 4.4. PROOF OF PROPOSITION ?? (II) AND (III) CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS By Lemma 21, for f = (f j ) with f j 2 ~ C 1 R d ; 0 @ 1 X j=0 jw N j m J 1 f j ' j j q 1 A 1=q Lp(R d ) C 0 @ 1 X j=0 w N j m+1 f j ' j q 1 A 1=q Lp(R d ) : Applying this inequality repeatedly we find that (4.14) holds for any negative integer s and m2Z: Proof. Proof of (ii) and (iii). First we prove that ~ H ;s p R d = H ;s p R d and the norms are equivalent in the scalar case, i.e. the sequence (f k ) k0 has one nonzero componentf 1 =f. Ifs2Z (s is an integer), then by well known characterization of L p and Lemma 23, jfj H s p (R d ) = jJ s fj Lp(R d ) C 0 @ 1 X j=0 jJ s f' j j 2 1 A 1=2 Lp(R d ) (4.15) C 0 @ 1 X j=0 w N j s f' j 2 1 A 1=2 Lp(R d ) ;f2 ~ C 1 R d : (4.16) On the other hand, by Lemma 23 and characterization of L p again, 0 @ 1 X j=0 w N j s f' j 2 1 A 1=2 Lp(R d ) C 0 @ 1 X j=0 jJ s f' j j 2 1 A 1=2 Lp(R d ) (4.17) CjJ s fj Lp(R d ) : (4.18) for all f2 ~ C 1 R d : We use interpolation to prove equivalence for all s2R. Assume s2 (m;m + 1) and s = (1)m + (m + 1) with m2Z: Let a 0 j =w N j m ;j 0;a 1 j =w N j (m+1) ;j 0;a j =w N j s ;j 0: 41 4.4. PROOF OF PROPOSITION ?? (II) AND (III) CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS Set l k p = 8 > < > : x = (x j ) :jxj a k ;p = 0 @ X j a k j jx j j p 1 A 1=p <1 9 > = > ; ;k = 0; 1;: By Theorem 2.4.6 in [FJS01] ( is continuous negative definite function), H l p = H m p ;H m+1 p ; (4.19) the complex interpolation space betweenH m p andH m+1 p . By Theorem 5.5.3 in [BL76],l 2 = l 0 2 ;l 1 2 , complex interpolation space between l 0 p and l 1 p . Hence by Theorem 1.18.4 in [Tri78], L p R d ;l 0 2 ;L p R d ;l 1 2 =L p R d ;l 2 ; (4.20) the complex interpolation space between L p R d ;l 0 2 and L p R d ;l 1 2 : Consider the mapping S :H m p R d f7! (f' j ) j0 2L p R d ;l 0 2 : According to Lemma 23 (see (4.17), (4.18)),S maps continuouslyH m p R d intoL p R d ;l 0 2 andH m+1 p R d into L p R d ;l 1 2 (note H m+1 p R d H m p R d ). Consider the continuous mapping R :L p R d ;l 0 2 f = (f j ) j0 7! 1 X j=0 f j ~ ' j 2H m p R d : Indeed, if f = (f j ) j0 2L p R d ;l 0 2 , then g = P 1 j=0 f j ~ ' j 2H m p R d , and g' j = 2 X k=2 f j+k ~ ' j+k ' j ;j 2; g' 1 = 2 X k=1 f 1+k ~ ' 1+k ' 1 ;g' 0 = 2 X k=0 f k ~ ' k ' 0 : Let ~ f j = 2 X k=2 f j+k ~ ' j+k ;j 2; ~ f 1 = 2 X k=1 f 1+k ~ ' 1+k ; ~ f 0 = 2 X k=0 f k ~ ' k : 42 4.4. PROOF OF PROPOSITION ?? (II) AND (III) CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS Hence by Corollary 18, 0 @ 1 X j=0 w N j m g ~ ' j 2 1 A 1=2 Lp(R d ) = 0 @ 1 X j=0 w N j m ~ f j ~ ' j 2 1 A 1=2 Lp(R d ) C 0 @ 1 X j=0 w N j m ~ f j 2 1 A 1=2 Lp(R d ) C 0 @ 1 X j=0 w N j m f j ~ ' j 2 1 A 1=2 Lp(R d ) C 0 @ 1 X j=0 w N j m f j 2 1 A 1=2 Lp(R d ) ; i.e. the mapping R : L p R d ;l 0 2 ! H m p R d is continuous. Similarly we prove that R : L p R d ;l 1 2 ! H m+1 p R d is continuous. Obviously, RS = I (identity map on H m p R d ). Now by (4.19), (4.20) and Theorem 1.2.4 in [Tri78], S :H s p R d !L p R d ;l 2 is isomorphic mapping onto a subspace of L p R d ;l 2 , i.e., there are constants 0<c 1 <c 2 so that c 1 jSvj Lp(R d ;l 2 ) jvj H s p (R d ) c 2 jSvj Lp(R d ;l 2 ) : Hence (4.15)-(4.18) hold for any s2R. Now we prove that ~ H ;s p R d ;l 2 =H ;s p R d ;l 2 and the norms are equivalent by reducing it to a scalar case. Let f = (f k ) k0 with f k 2 ~ C 1 R d and only finite number of f k be nonzero: Let k ;k 0; be a sequence of independent standard normal r.v. and (x) = 1 X k=0 k f k (x);x2R d : According to (4.15)-(4.18), there are constants 0<c 1 <c 2 so thatP-a.s. c 1 jj p ~ H ;s p (R d ) jj p H ;s p (R d ) c 2 jj p ~ H ;s p (R d ) ; and c 1 Ejj p ~ H ;s p (R d ) Ejj p H ;s p (R d ) c 2 Ejj p ~ H ;s p (R d ) : All the equivalences follow easily from Lemma 90. 43 4.5. OTHER CHARACTERIZATIONS OF FUNCTION SPACES CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS Remark 24. Let N > 1, then by Lemma 6, 1 X k=0 w N k " <1;"> 0: Hence for any "> 0 we have the following continuous embedding: ~ H ;s+" p R d ~ B ;s pp R d ~ H ;s" p R d ;p> 1: For the second inclusion, for any p2 (1; 2];2 ~ H ;s p R d , jj ~ H ;s p (R d ) = 0 @ 1 X j=0 w N 1 s ' j 2 1 A 1=2 Lp(R d ) 0 @ 1 X j=0 w N 1 s ' j p 1 A 1=p Lp(R d ) =jj ~ B ;s pp (R d ) ; and for p> 2, with any > 0 using Lemma 6, 0 @ 1 X j=0 w N 1 s ' j 2 w N j 2 1 A 1=2 0 @ 1 X j=0 w N 1 s ' j p 1 A 1=p 0 @ 1 X j=0 w N j 2p p2 1 A 1 2 1 p and jj ~ H ;s" p (R d ) Cjj ~ B ;s pp (R d ) : In fact, for p 2, H ;s p R d is continuously embedded into B ;s pp R d . 4.5 Other Characterizations of Function Spaces Let 2 A ;w = w be an O-RV function and A, B hold. Let N > 1, for p;q2 (1;1);s > 0; let L s p;2 R d , (resp. B s p;q R d ) be the set of all functions f2 L p R d that can be represented by a series of entire functions f k of exponential type N k =N k+1 ;k 0; converging in L p f = 1 X k=0 f k in L p (4.21) 44 4.5. OTHER CHARACTERIZATIONS OF FUNCTION SPACES CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS such that jfj L s p;2 (R d ) = 1 X k=0 w N k s f k 2 ! 1=2 Lp(R d ) <1 (4.22) (or resp. jfj B s p;q (R d ) = 1 X k=0 w N k s f k q Lp(R d ) ! 1=q <1): (4.23) Recall that by Paley-Wiener-Schwartz theorem a function g2 L p R d is entire analytic of type t iff supp(Ff)fjj :jjtg (see [Tri78], 2.5.4, p.197). The normjfj L s p;2 (resp.jfj B s p;q ) is defined as a sum of jfj Lp(R d ) and infimum of (4.22) (resp. (4.23) over all series (4.21). The function spacesL s p;2 R d ,B s p;q R d belong to the class of spaces of generalized smoothness (see e.g. [KL87].) The following statement holds. Proposition 25. Let 2A ;w =w be an O-RV function andA, B hold. Let s> 0;p;q2 (1;1);N > 1. Then H ;s p R d =L s p;2 R d , B ;s p;q R d =B s p;q R d ; and the norms are equivalent. Proof. Letf2 ~ H ;s p R d , sincef = P 1 j=0 f j (withf j =f' j ; see description of the sequence' j in Remark 1) converges in L p , and supp(Ff j ) jjN j+1 , it follows that jfj L s p;2 (R d ) jfj ~ H ;s p (R d ) : Let f2L s p;q R d , and K j = N j+1 ;N j+1 d n N j ;N j d ;j 1;K 0 = [N;N] d : Let h j =F 1 Kj ;j 0. By Theorem 1 in [Kal80], f = 1 X k=0 fh k in L p ; and the normjfj L s p;2 (R d ) is equivalent to the norm jfj ~ L s p;2 (R d ) =jfj Lp(R d ) + 0 @ 1 X j=0 w N j s fh j 2 1 A 1=2 Lp(R d ) : Now, f' j = j X k=(j2p0)_0 fh k ' j = ~ f j ' j ;j 0; with ~ f j = j X k=(j2p0)_0 fh k ;j 0: 45 4.5. OTHER CHARACTERIZATIONS OF FUNCTION SPACES CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS where p 0 is the smallest positive integer so that p d=N p0 1: Since for (j 2p 0 )_ 0kj we have by Lemma 6, w N k cw N j , where c is independent of j;k. It follows by Corollary 18, jfj ~ H ;s p (R d ) C p;q 0 @ 1 X j=0 w N j s ~ f j 2 1 A 1=2 Lp(R d ) Cjfj ~ L s p;2 (R d ) : Similarly we prove that B s p;q R d =B s p;q R d and the norms are equivalent. We apply the results in [KL87] to describe the norms by averaged local oscillations. Given f :R d !R and y2R d , let y f (x) = 1 y f (x) =f (x +y)f (x);x2R d : Then m y f (x) = m X j=0 (1) j m j f (x +jy);x;y2R d : Let Q m t f (x) = Z jyj1 m ty f (x) dy;x2R d ;t> 0: A simple consequence of Theorem 4.2 in [KL87] is the following statement. Proposition 26. Let 2A ;w = w be an O-RV function and A, B hold. Let s > 0;p;q2 (1;1). Let 1 >q 1 _q 2 and m 0 be the least integer m such that m>s 1 . Then, (i) For any mm 0 the norm of H ;s p R d is equivalent to the norm kfk H ;s p (R d ) =jfj Lp(R d ) + Z 1 0 jQ m t f (x)j 2 dt tw (t) 2s ! 1=2 Lp(R d ) : (ii) For any mm 0 the norm of B ;s p;q R d is equivalent to the norm kfk B ;s p;q (R d ) =jfj Lp(R d ) + Z 1 0 jQ m t fj q Lp(R d ) dt tw (t) qs 1=q : Proof. In order to apply Theorem 4.2 in [KL87], we will show that for any integer mm 0 the sequence k = N km w(N k ) s is strongly decreasing, 46 4.6. DENSE SUBSPACES CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS that is k c j for allkj and somec> 0, and there isk 0 so that k 2 1 j for all kj +k 0 . Indeed, both conditions follows easily from Lemma 6 k j = N m(kj) w N j s w (N k ) s N (jk)(m1s) : The claim follows by Theorem 4.2 in [KL87]. We would like to mention another application of the results in [KL87]. Remark 27. Let j 0 1;let f 2 H ;s p R d ;s > j0 p w 1 ^p w 2 ;p2 (1;1). It can be shown by checking the assumptions of Theorem 3.5 in [KL87] that D j f2L p R d ;jj 0 : 4.6 Dense Subspaces In this section, we explicitly construct sequences of smooth functions that converge to functions in Besov and Bessel potential spaces. Let U n 2U;U n U n+1 ;n 1;[ n U n = U and (U n ) <1;n 1: We denote by ~ C 1 r:p (E); 1 p<1; the space of allR (F) B R d measurableV r -valued random functions onE such that for every 2N d 0 , E Z T 0 sup x2R d jD (t;x)j p Vr dt +E h jD j p Lp(E;Vr ) i <1; and = Un for some n if r = 2;p: Similarly we define the space ~ C 1 r:p R d by replacingR (F) and E by F andR d respectively in the definition of ~ C 1 r;p (E). Let 2 A ;N > 1;w = w is an O-RV function and A, B hold. We now discuss an approximating sequence of the input , based on Proposition 15. Lemma 28. Let 2 A ;w = w be an O-RV function and A, B hold. Let U n 2U;U n U n+1 ;n 1;[ n U n = U and (U n ) <1;n 1: Let s2R;p2 (1;1), 2D r;p , where D r;p =D r;p (A) =B ;s r;pp (A) with r = 0;p; orD r;p =H ;s r;p (A) with r = 0; 2, A =R d or E. For 2D r;p we set n = n X j=0 ' j Un ; if r = 2;p; n = n X j=0 ' j ; if r = 0: Then there is C > 0 so that j n j Dr;p Cjj Dr;p ; 2D r;p ;n 1, 47 4.6. DENSE SUBSPACES CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS andj n j Dr;p ! 0 as n!1: Moreover, for r = 0; 2;p; every n and multiindex 2N d 0 , E Z T 0 sup x jD n j p Vr dt +jD n j p Lr;p(E) < 1 if A =E; E[ sup x jD n j p Vr ] +jD n j p Lr;p(R d ) < 1 if A =R d ; Proof. Let ~ n = Un ;n 1: Since ' k = 1 X l=1 ' k+l ' k ;k 1;' 0 = (' 0 +' 1 )' 0 ; we have for n> 1; ~ n n ' k = 0;k<n; ~ n n ' k = ~ n ' k1 + ~ n ' k + ~ n ' k+1 ' k ;k>n + 1; ~ n n ' n = ~ n ' n+1 ' n ; ~ n n ' n+1 = ~ n ' n+1 + ~ n ' n+2 ' n+1 : Let V r = L r (U;U; );r = 2;p and V 0 = R. By Corollary 18, there is a constant C independent of 2H ;s 2;p (E) so that 0 @ 1 X j=0 w N j s ~ n n ' j 2 Vr 1 A 1=2 Lp(E) C 0 @ 1 X j=n w N j s ' j 2 Vr 1 A 1=2 Lp(E) ! 0;r = 0; 2; as n!1: Obviously ~ n n ' j Lr;p(E) C j+1 X k=j1 ~ n ' k Lr;p(E) ;jn; ~ n n ' j Lr;p(E) = 0;j <n;r = 0;p; and j n j Dr;p(E) Cjj Dr;p(E) ; 2D r;p ;n 1;r = 0; 2;p. Thusj n j Dr;p(E) ! 0 as n!1;r = 0; 2;p: 48 4.6. DENSE SUBSPACES CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS Let r = 0; 2;p, 2D r;p (E). Obviously, for any k 0; E Z E j' k j p Vr dxdt<1: Since for any multiindex 2N d 0 ; ' k = ' k ~ ' k ;D ' k = ' k D ~ ' k ; andP-a.s. for all s;x; with 1 q + 1 p = 1; jD ' k (s;x)j Vr Z j' k (s;xy)j Vr jD ~ ' k (y)jdy; sup x jD ' k (s;x)j Vr Z j' k (s;)j p Vr dx 1=p jD ~ ' k j Lq (R d ) ; we have for any multiindex ; jD ' k j Lr;p(E) <1; and E Z T 0 sup x jD ' k j p Vr dt<1;r = 0; 2;p: The proof for the case of A =R d is a repeat with obvious changes: The statement follows. In the next Corollary, although the statement is obvious from the definition of function spaces, we construct convergent sequences of C 1 0 R d ;V r functions. Corollary 29. Let 2A ;w =w be an O-RV function and A, B hold. Let s2R;p2 (1;1). The space C 1 0 R d ;V r ofV r -valued infinitely differentiable functions with compact support is dense in D r;p R d where D r;p R d =B ;s r;pp R d with r = 0;p or D r;p R d =H ;s r;p R d with r = 0; 2. Proof. In the view of Lemma 28, it suffices to show that for any V =V r -valued function f such that for all multiindex 2N d 0 , sup x jD f (x)j Vr +jD fj Lp(R d ;Vr ) <1; there exists f n 2 C 1 0 R d ;V r so that f n ! f in D r R d . Let g2 C 1 0 R d with 0 g (x) 1;x2R d , g (x) = 1 forjxj 1, and g (x) = 0 forjxj 2. Let f n (x) :=f (x)g (x=n);x2R d : 49 4.6. DENSE SUBSPACES CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS Obviously f n 2C 1 0 R d ;V r , and for any multiindex ; D f n (x) = D f (x)g (x=n) + X 1+2=;j2j1 n j2j D 1 f (x) D 2 g (x=n);x2R d ; D f n Lp(R d ;Vr ) C (jj) sup 0 jj D 0 f Lp(R d ;Vr ) ; and D f n D f Lp(R d ;Vr ) ! 0. Since for any multiindex we have R y ' j (y)dy = 0; it follows for m> 0;j 1, by Taylor remainder theorem, for x2R d ; f n ' j (x) = Z ' j (y) 8 < : f n (xy) X :jjm D f n (x) ! (y) 9 = ; dy = Z ' j (y) X :jj=m+1 Z 1 0 (1t) m+1 (m + 1)! D f n (xty) (y) dtdy = N j(m+1) X :jj=m+1 Z ' (y) Z 1 0 (1t) m+1 (m + 1)! D f n xtN j y dt (y) dy: By Lemma 6, there exists 0 > 0 such that w N j s CN js 0 . Let m > 1 be such that t = N 0 s N m < 1. Hence there is a constant C =C (m;w ) (independent of n) so that w N j s jf n ' j j Lp(R d ;Vr ) C (m;w )t j ;j 0: Now, for r = 0; 2 and any k 0; 0 @ 1 X j=k w N j s f n ' j (x) 2 Vr 1 A 1=2 Lp(R d ) 1 X j=k w N j s f n ' j (x) Vr Lp(R d ) X jk w N j s f n ' j (x) Lp(R d ;Vr ) C (m;w ) X jk t j : 50 4.6. DENSE SUBSPACES CHAPTER 4. FUNCTION SPACES - NORM EQUIVALENCE AND CHARACTERIZATIONS Since the same estimate holds for f; 0 @ X jk w N j s f' j (x) 2 Vr 1 A 1=2 Lp(R d ) C (m;w ) X jk t j ; and 0 @ X j<k w N j s (ff n )' j (x) 2 Vr 1 A 1=2 Lp(R d ) ! 0; it follows that 0 @ X j w N j s (ff n )' j (x) 2 Vr 1 A 1=2 Lp(R d ) ! 0 as n!1. Likewise, for r = 0;p; lim n!1 jf n fj B ;s r;pp (R d ) = lim n!1 0 @ 1 X j=0 w N j s (f n f)' j p Lp(R d ;Vr ) 1 A 1=p = 0: An obvious consequence of Lemma 28 (the form of the approximating sequence is identical for different V r ) is the following Lemma 30. Let 2A ;w =w be an O-RV function and A, B hold. Let p 1 and s;s 0 2R. Then the set ~ C 1 p;p (E) is a dense subset inB ;s 0 p;pp (E); ~ C 1 0;p R d is a dense subset ofB ;s 0 pp R d ; and ~ C 1 r;p (E) is dense inH ;s r;p (E);r = 0; 2: Moreover, the set ~ C 1 2;p (E)\ ~ C 1 p;p (E) is a dense subset ofB ;s 0 p;pp (E)\H ;s 2;p (E). We will need this Lemma when we prove the existence of the solution of (1.1) by passing to the limit. 51 Chapter 5 Probability Density Estimates In this chapter, we derive some estimates of probability density of Z ~ R t (see below for detailed descrip- tion), these preliminary estimates will be used in verifying Hörmander condition and stochastic Hörmander condition (see [KK17]) to derive apriori estimates for (1.1). Let 2A and p (dt;dy) be a Poisson point measure on [0;1)R d 0 such thatEp (dt;dy) = (dy)dt: Letq (dt;dy) =p (dt;dy) (dy)dt. We associate toL the stochastic process with independent increments Z t =Z t = Z t 0 Z R d 0 (y)yq(ds;dy) + Z t 0 Z R d 0 (1 (y))yp(ds;dy);t 0: By Itô’s formula, Ee i2Z t = expf ()tg;t 0;2R d ; where () = Z e i2y 1i2 (y)y (dy): For R > 0, let Z R t = Z ~ R t ;t > 0 be the stochastic process with independent increments associated with ~ R =w (R) R , i.e., Ee i2Z R t = exp ~ R ()t with ~ R () = Z e i2y 1i2 (y)y d~ R ;2R d : Note Z R t =Z ~ R t and R 1 Z ! (R)t ;t> 0; have the same distribution. When well-defined, we denote byp (t;) the probability density of the processZ t ,t> 0; andp ~ R (t;) = p R (t;) the probability density of Z ~ R t ;t> 0. 52 5.1. PROBABILITY DENSITY ESTIMATES CHAPTER 5. PROBABILITY DENSITY ESTIMATES For R> 0, consider Lévy measures R;0 (dy) = fjyj1g ~ R (dy), i.e., Z (y) R;0 (dy) = Z jyj1 (y) ~ R (dy) =w (R) Z jyjR (y=R) (dy); 2B 0 R d : Let = R 0 be a random variable with characteristic function exp R;0 () given below, denote ^ = =jj;2R d 0 , R;0 () = Z jyj1 e i2y 1i2 (y)y ~ R (dy) = Z jyj1 h e i2jj ^ y 1i2 (y) ^ yjj i ~ R (dy) = w (R) w Rjj 1 Z jyjjj h e i2 ^ y 1i2 (y) ^ y i ~ Rjj 1 (dy): (5.1) 5.1 Probability Density Estimates In this section, we present some results on probability density estimates. Lemma 31. Let 2A ;w =w be an O-RV function and A, B hold. Then (i) Re R;0 ()cjj ;jj 1; with some c;> 0 independent of R. (ii) = R 0 has a pdf p R;0 (x);x2R d ; such that for any multiindex 2N d 0 , and a positive integer n 0, there exists C =C (;)> 0 independent of R such that sup x @ p R;0 (x) + Z (1 +jxj 2 ) n @ p R;0 (x) dxC;R> 0: Proof. (i) Let 2 2 (0;p ! 1 ^p ! 2 ). By Lemma 6, there is C > 0 so that w (R) w Rjj 1 Cjj 2 ;R> 0;jj 1: Hence, according to (5.1), forjj 1; ^ ==jj, Re R;0 ()cjj 2 Z jyj1=8 h cos 2 ^ y 1 i ~ Rjj 1 (dy) cjj 2 inf R2(0;1);j ^ j=1 Z jyj1 ^ y 2 ~ R (dy) =cjj 2 ; 53 5.1. PROBABILITY DENSITY ESTIMATES CHAPTER 5. PROBABILITY DENSITY ESTIMATES and Z exp Re R;0 () d<1: (5.2) (ii) Since (5.2) holds, by Proposition I.2.5 in [Sat99], = R 0 has a continuous bounded density p R;0 (x) = Z e i2x exp R;0 () d;x2R d : Moreover, by (i), for any multiindex 2N d 0 ; @ p R;0 (x) = Z e i2x (i2) exp R;0 () d;x2R d ; is a bounded continuous function. The function 1 +jxj 2 n @ p R;0 is integrable if (i2x j ) l (i2x k ) 2n @ p R;0 (x) = Z @ l j @ 2n k [e i2x ] (i2) exp R;0 () d = (1) l+2n Z e i2x @ l j @ 2n k [(i2) exp R;0 () ]d is bounded for all j;k; and for ld + 1. Since @ R;0 () is bounded forjj 2 and jr R;0 ()jC (1 + ());2R d ; with () = R jyj1 (y)jyj [(jjjyj)^ 1] ~ R (dy). The boundedness follows from (i) and Lemma 11 (see also Remark 12.) Lemma 32. Let 2A ;w =w be an O-RV function and A, B hold. Then for R> 0, Z R 1 has a bounded continuous probability density of the form p R (1;x) = Z p R;0 (xy)P R (dy);x2R d ; where P R is a probability distribution. Moreover, for any 0< 2 0 independent of R such that sup x @ p R (1;x) + Z (1 +jxj 2 ) D p R (1;x) dxC: 54 5.1. PROBABILITY DENSITY ESTIMATES CHAPTER 5. PROBABILITY DENSITY ESTIMATES Proof. We have ~ R = ~ R;0 + ~ R;2 ; where ~ R;0 (dy) = jyj1 ~ R (dy) and hence, ~ R () = ~ R;0 () + ~ R;2 ();2R d : Denote R 0 and R 2 independentrandomvariableswithcharacteristicexponent exp ~ R;0 () and exp ~ R;2 () respectively. Obviously the distribution ofZ R 1 coincides with the distribution of the sum R 0 + R 2 . Therefore, p R (1;x) = Z p R;0 (xy)P R (dy) (5.3) where P R (dy) is the probability distribution of R 2 . It is clear from (5.3) that for any multiindex 2N d 0 , @ p R (1;x) = Z @ p R;0 (xy)P R (dy) and from Lemma 31, Z @ p R (1;x) dxC; and sup x @ p R (1;x) C Moreover, using Lemma 31 and Lemma 91 in the Appendix, Z jxj 2 D p R (1;x) dxC Z Z [jxyj 2 +jyj 2 ] @ p R;0 (xy) P R (dy)dx C +C Z jyj 2 P R (dy)C: We write 2A sign =A A if = with ;2A , and L = L L . Given 2A sign , we denotejj its variation measure. Obviously,jj2A : Based on Lemma 32, we now extend the probability density estimates to t > 0. Motivated by the idea of scaling the probability at t> 0 to the probability density at t = 1 (see density estimates in [KKK13]), we start with the following elementary computation. 55 5.1. PROBABILITY DENSITY ESTIMATES CHAPTER 5. PROBABILITY DENSITY ESTIMATES Lemma 33. Let ;2A sign . Let 2A ;w =w be a continuous O-RV function and A, B hold. Then p (t;x) = a (t) d p ~ a(t) 1;xa (t) 1 ;x2R d ;t> 0; L p (t;x) = 1 t a (t) d (L ~ a(t) p ~ a(t) ) 1;xa (t) 1 ;x2R d ;t> 0; L L p (t;x) = 1 t 2 a (t) d (L ~ a(t) L ~ a(t) p ~ a(t) ) 1;xa (t) 1 ;x2R d ;t> 0; where a (t) = inffr 0 :! (r)tg;t> 0: Proof. Indeed, by Lemma 32, for each t > 0 and r > 0, the density p ~ a(t) (r;x);x 2 R d ; is infinitely differentiable in x, all derivatives are bounded and integrable. Obviously, expf ()tg = exp ~ a(t) (a (t)) ;t> 0;2R d ; and () expf ()tg = 1 t ~ a(t) (a (t)) exp ~ a(t) (a (t)) ;t> 0;2R d : We derive the first two equalities by taking Fourier inverse. Similarly, the third equality can be derived. Lemma 34. Let 2A ;w =w be an O-RV function and A, B hold. Let 1 >q w 1 _q w 2 ; 0< 2 1 if 2 (1; 2); 1 1 if 2 (0; 1); 1 2 if 2 [1; 2): Let 2A sign and assume that Z jyj1 jyj 1 d f jj R + Z jyj>1 jyj 2 d f jj R M;R> 0: Then for any multiindex k2N d 0 there is C =C (k;)> 0 such that 56 5.1. PROBABILITY DENSITY ESTIMATES CHAPTER 5. PROBABILITY DENSITY ESTIMATES Z (1 +jxj 2 ) D k L ~ R p R (1;x) dx CM; and there is C =C ()> 0 such that Z (1 +jxj 2 ) L ~ R L ~ R p R (1;x) dxCM: Proof. Let 2 (0; 1). Then by Lemma 32, Z (1 +jxj 2 ) D k L ~ R p R (1;x) dx Z Z jyj1 Z 1 0 [1 +jx +syj 2 +jyj 2 ]jyj D k+1 p R (1;x +sy) dsd f jj R dx + Z Z jyj>1 [1 +jx +yj 2 +jyj 2 ] D k p R (1;x +y) d f jj R dx +M Z (1 +jxj 2 ) D k p R (1;x) dx CM: We derive the second estimate similarly, using Lemma 32, Z (1 +jxj 2 ) L ~ R L ~ R p R (1;x) dx Z Z jyj1 Z 1 0 (1 +jx +syj 2 +jyj 2 )jyj L ~ R rp R (1;x +sy) ds f jj R (dy)dx + Z Z jyj>1 (1 +jx +yj 2 +jyj 2 ) L ~ R p R (1;x +y) f jj R (dy)dx +M Z (1 +jxj 2 ) L ~ R p R (1;x) dx CM: Similarly, we handle the cases 2 (1; 2) and = 1: Lemma 35. Let 2A ;w =w be a continuous O-RV function and A, B hold. Let 1 >q w 1 _q w 2 ; 0< 2 1 if 2 (1; 2); 1 1 if 2 (0; 1); 1 2 if 2 [1; 2): 57 5.1. PROBABILITY DENSITY ESTIMATES CHAPTER 5. PROBABILITY DENSITY ESTIMATES Let 2A sign and assume that Z jyj1 jyj 1 d f jj R + Z jyj>1 jyj 2 d f jj R M;R> 0: (5.4) Denote a (t) = inffr :w (r)tg;t> 0: (i) For any multiindex k2N d 0 and 2 [0; 2 ], there is C =C (k;;)> 0 such that Z jzj>c L D k p (t;x) dx CMt 1 a (t) jkj c ; Z L D k p (t;x) dx CMt 1 a (t) jkj : (ii) There is C =C ()> 0 such that Z R d jL p (t;xy)L p (t;x)jdxCM jyj ta (t) ;t> 0;y2R d : (iii) There is C =C ()> 0 such that Z 1 2b Z jL p (ts;x)L p (t;x)jdxdt CM;jsjb<1: Proof. (i) Indeed, by Lemma 33, Chebyshev’s inequality, and Lemma 34, for k2N d 0 ;2 [0; 2 ]; Z jxj>c L D k p (t;x) dx = 1 t a (t) dk Z jxj>c L ~ a(t) D k p ~ a(t) 1; x a (t) dx a (t) k c t Z jxj L ~ a(t) D k p ~ a(t) (1;x) dxCM a (t) k c t : (ii) By Lemma 33, and Lemma 34, Z R d jL p (t;xy)L p (t;x)jdx = 1 t Z L ~ a(t) p ~ a(t) 1;x y a (t) L ~ a(t) p ~ a(t) (1;x) dx 1 t Z 1 0 Z rL ~ a(t) p ~ a(t) 1;xs y a (t) jyj a (t) dxds C jyj ta (t) Z L ~ a(t) rp ~ a(t) (1;x) dxCM jyj ta (t) : 58 5.2. FRACTIONAL OPERATORS ON PROBABILITY DENSITY CHAPTER 5. PROBABILITY DENSITY ESTIMATES (iii) By Lemma 33, and Lemma 34, Z 1 2b Z jL p (ts;x)L p (t;x)jdxdt jsj Z 1 2b Z Z 1 0 jL @ t p (ts;x)jddxdt = jsj Z 1 2b Z Z 1 0 L L p (ts;x) ddxdt = jsj Z 1 2b Z Z 1 0 (ts) 2 (L ~ a(ts) L ~ a(ts) p ~ a(ts) ) (1;x) ddxdt CMjsj Z 1 0 1 (2bs) dCM: 5.2 Fractional Operators on Probability Density In order to deal with the stochastic term we need some finer estimates of probability density. More specifically, we need some estimates of fractional operators applied to probability density. Thefractionaloperatorisdefinedinthefollowingway. Let2A sym =f2A : is symmetric, = sym g: Then for 2 (0; 1) and f2S R d , we have ( ()) ^ f () = c Z 1 0 t [exp ( ()t) 1] dt t ^ f ();2R d ; and therefore, L ; f (x) = F 1 h ( ) ^ f i (x) (5.5) = c E Z 1 0 t [f (x +Z t )f (x)] dt t ;x2R d : For 2A ;2 (0; 1) we let sym = (dy)+(dy) 2 and L ; =L sym; . Lemma 36. Let 2A sym ;2 (0; 1): Let 2A ;w =w be a continuous O-RV function and A, B hold. Assume 1 >q w 1 _q w 2 ; 0< 2 1 if 2 (1; 2); 1 1 if 2 (0; 1); 1 2 if 2 [1; 2); and Z jyj1 jyj 1 de R + Z jyj>1 jyj 2 de R M;R> 0: Let p (t;x);x2R d , be the pdf of Z t ;t> 0 and a (t) = inffr :w (r)tg;t> 0: (i) For any p 1; and "> 0 there is C > 0 such that L ; f Lp(R d ) "jL fj Lp(R d ) +C jfj Lp(R d ) ;f2S R d : (ii) There is C =C ()> 0 such that Z R d L ; p (t;xy)L ; p (t;x) dxCM jyj t a (t) ;t> 0;y2R d : (iii) There is C =C ()> 0 such that Z 1 2b Z L ; 1 2 p (ts;x)L ; 1 2 p (t;x) dx 2 dt CM;jsjb<1: (iv) For any multiindex k2N d 0 and 2 [0; 2 ) there is C =C (k;;)> 0 such that Z jxj>c L ; D k p (t;x) dx CMt a (t) jkj c ; Z L ; D k p (t;x) dx CMt a (t) jkj : For the proof of Lemma 36, we will need the following Lemmas 37, 38. Lemma 37. Let 2 (0; 1); 2A sym and 2A sign : Let 2A ;w = w be a continuous O-RV function and A, B hold. Then p (t;x) = a (t) d p ~ a(t) 1;xa (t) 1 ;x2R d ;t> 0; L ; p (t;x) = 1 t a (t) d (L ~ a(t) ; p ~ a(t) ) 1;xa (t) 1 ;x2R d ;t> 0; L L ; p (t;x) = 1 t 1+ a (t) d (L ~ a(t) L ~ a(t) ; p ~ a(t) ) 1;xa (t) 1 ;x2R d ;t> 0; 60 5.2. FRACTIONAL OPERATORS ON PROBABILITY DENSITY CHAPTER 5. PROBABILITY DENSITY ESTIMATES where a (t) = inffr :! (r)tg;t> 0: Proof. Indeed, by Lemma 32, for each t > 0 and r > 0, the density p ~ a(t) (r;x);x 2 R d ; is infinitely differentiable in x, all derivatives are bounded and integrable. Obviously, expf ()tg = exp ~ a(t) (a (t)) ;t> 0;2R d ; and ( ()) expf ()tg = 1 t ~ a(t) (a (t)) exp ~ a(t) (a (t)) ;t> 0;2R d : We derive the first two equalities by taking Fourier inverse. Similarly, the third equality can be derived. Lemma 38. Let , 2A satisfy assumptions of Lemma 36, 2 (0; 1): (i) For any p 1; and "> 0 there is C > 0 such that L ; f Lp(R d ) "jL fj Lp(R d ) +C jfj Lp(R d ) ;f2S R d : (ii) Let p R (t;x) =p ~ R (t;x);x2R d , be the pdf of Z ~ R t ;t> 0;R> 0: Then for each multiindex k2N d 0 , and 2 [0; 2 ), there is C =C (k;;)> 0 such that, Z 1 +jxj D k L ~ R ; p R (1;x) dx CM; Z (1 +jxj ) L ~ R ; L ~ R p R (1;x) dx CM: Proof. Indeed for any a> 0;f2S R d ;x2R d ; by Itô formula and (5.5), L ; f (x) = cE Z a 0 t Z t 0 L f (x +Z r )dr dt t (5.6) +cE Z 1 a t [f (x +Z t )f (x)] dt t : The statement (i) follows by Minkowski’s inequality. 61 5.2. FRACTIONAL OPERATORS ON PROBABILITY DENSITY CHAPTER 5. PROBABILITY DENSITY ESTIMATES By (5.6), for any multiindex k, Z 1 +jxj L ~ R ; D k p R (1;x) dx CE Z 1 0 t Z t 0 Z 1 +jxj L ~ R D k p R (1;x +Z r ) dxdr dt t +CE Z 1 1 t Z 1 +jxj [ D k p R (1;x +Z t ) + D k p R (1;x) ]dx dt t = A 1 +A 2 : Now, by Lemma 34, and Lemma 91, A 1 C Z 1 0 t Z 1 +jxj L ~ R D k p R (1;x) dxdt +C Z 1 0 t Z t 0 E jZ r j dr Z L ~ R D k p R (1;x) dx dt t C; and A 2 C Z 1 1 t Z 1 +jxj D k p R (1;x) ]dx dt t +C Z 1 1 t E jZ t j Z D k p R (1;x) dx dt t C 1 + Z 1 1 t t 2 dt t C: Similarly, the second inequality of part (ii) is proved. Proof. Proof of Lemma 36 (i) is proved in Lemma 38 (i). (ii) By Lemma 37 and Lemma 38, Z R d L ; p (t;xy)L ; p (t;x) dx = 1 t Z L ~ a(t) ; p ~ a(t) 1;x y a (t) L ~ a(t) ; p ~ a(t) (1;x) dx 1 t Z 1 0 Z rL ~ a(t) ; p ~ a(t) 1;xs y a (t) jyj a (t) dxds C jyj t a (t) Z L ~ a(t) ; rp ~ a(t) (1;x) dxCM jyj t a (t) : 62 5.3. ELLIPTIC EQUATION FOR SMOOTH INPUT FUNCTIONS CHAPTER 5. PROBABILITY DENSITY ESTIMATES (iii) By Lemma 37 and Lemma 38, Z 1 2a Z L ; 1 2 p (ts;x)L ; 1 2 p (t;x) dx 2 dt jsj 2 Z 1 2b Z 1 0 Z L ; 1 2 L p (trs;x) dxdr 2 dt Cjsj 2 Z 1 2b Z 1 0 dr (trs) 1+ 1 2 ! 2 dtC Z 1 2b 1 (ts) 1 2 1 t 1 2 2 dt Cjsj Z 1 2b dt (ts)t C: (iv) Indeed, by Lemma 37, Chebyshev’s inequality, and Lemma 38, Z jxj>c L ; D k p (t;x) dx = 1 t a (t) dk Z jxj>c L ~ a(t) ; D k p ~ a(t) 1; x a (t) dx a (t) k c t Z jxj L ~ a(t) ; D k p ~ a(t) (1;x) dxCM a (t) k c t : Similarly, we derive the second estimate of the claim. 5.3 Elliptic Equation for Smooth Input Functions In this section, we digress and study the elliptic problem with 2A , > 0, u (x)L u (x) =f (x);x2R d ; (5.7) We will need a representation of the solution for the next section. Lemma 39. Let f2 ~ C 1 (R d ) then there is unique u2 ~ C 1 R d solving (5.7). Moreover, u(x) = Z 1 0 e t Ef (x +Z t )dt;x2R d ; and for p2 [1;1] and any mutiindex 2N d 0 ; jD uj Lp(R d ) (1=)jD fj Lp(R d ) : (5.8) Proof. Denote Z t =Z t ;t 0: 63 5.4. EMBEDDING INTO THE SPACE OF CONTINUOUS FUNCTIONS CHAPTER 5. PROBABILITY DENSITY ESTIMATES Uniqueness. Let u 1 ;u 2 2 ~ C 1 (E) solve (5.7) and u =u 1 u 2 . Then u solves (5.7) with f = 0. Let x2R d . By Itô’s formula for e t u (x +Z t ); 0st, we have e t Eu (x +Z t )u (x) =E Z t 0 e s [L u (x +Z s )u (x +Z s )]ds = 0: Passing to the limit as t!1 we obtain that u (x) = 0 for all x2R d . Existence. Let f2 ~ C 1 R d . Set u (x) = Z 1 0 e t Ef (x +Z t )dt;x2R d : By direct estimate using Minkowski’s inequality (5.8) readily follows, i.e. u2 ~ C 1 R d . We fix x2R d , and applying Itô’s formula with e t f(x +Z t ); we have for all t> 0;x2R d ; e t Ef(x +Z t ) =f(x) + Z t 0 e s E (L )f (s;x +Z s )ds Passing to the limit as t!1 we have L u(x)u (x) +f (x) = 0;x2R d : The statement follows. An obvious consequence of Lemma 39 is the following identity for f2 ~ C 1 R d . Corollary 40. Let f2 ~ C 1 R d and u(x) = Z 1 0 e t Ef (x +Z t )dt;x2R d : Then u2 ~ C 1 R d , and f (x) = (L )u(x) = (L ) Z 1 0 e t Ef (x +Z t )dt = Z 1 0 e t E (L )f (x +Z t )dt;x2R d : 5.4 Embedding into the Space of Continuous Functions Inthissectionwewilluseestimatesofprobabilitydensitytoderiveanintegralrepresentationofincrement f(x +y)-f(x);x;y2R d and consequently prove some embedding results from such representation. 64 5.4. EMBEDDING INTO THE SPACE OF CONTINUOUS FUNCTIONS CHAPTER 5. PROBABILITY DENSITY ESTIMATES Lemma 41. Let 2A ;w =w be a continuous O-RV function and A, B hold. Let p jzj (t;) be the pdf of the Lévy process Z ~ jzj t ;t> 0; associated to the Lévy measure ~ jzj (dy) =w (jzj) (jzjdy);z2R d 0 . (i) Let q 1 and dd=q p w 1 _ dd=q p w 2 << d + 1d=q q w 1 ^ d + 1d=q q w 2 : Define b ; (y;z) = jzj d Z 1 0 t p jzj t; y jzj + ^ z p jzj t; y jzj dt t ;y2R d ;z6= 0: Then there exists C so that Z b ; (y;z) q dy 1=q Cjzj d+d=q ;z2R d 0 : (ii) Let q 1 and d + 1d=q p w 1 _ d + 1d=q p w 2 << d + 2d=q q w 1 ^ d + 2d=q q w 2 : Define d ; (y;z) = jzj d Z 1 0 t p jzj t; y jzj + ^ z p jzj t; y jzj rp jzj t; y jzj ^ z dt t ;y2R d ;z6= 0; Then there exists C so that Z d ; (y;z) q dy 1=q Cjzj d+d=q ;z2R d 0 : Proof. From Lemma 37, p (t;x) =a (t) d p ~ a(t) 1;xa (t) 1 , hence, p jzj (t;y) =jzj d p (w (jzj)t;jzjy) =jzj d a (w (z)t) d p ~ a(w(jzj)t) 1;a (w (jzj)t) 1 jzjy and rp jzj (t;y) =jzj d+1 a (w (z)t) d1 rp ~ a(w(jzj)t) 1;a (w (jzj)t) 1 jzjy r 2 p jzj (t;y) =jzj d+2 a (w (z)t) d2 r 2 p ~ a(w(jzj)t) 1;a (w (jzj)t) 1 jzjy 65 5.4. EMBEDDING INTO THE SPACE OF CONTINUOUS FUNCTIONS CHAPTER 5. PROBABILITY DENSITY ESTIMATES ByLemma32,p R (1;)isboundedandintegrablewithanupperboundindependentofR,thus p R (1;) Lq (R d ) C with C independent of R> 0. Thus by Minkowski’s inequality, p jzj (t;) Lq (R d ) jzj dd=q a (w (jzj)t) d+d=q : Similarly, rp jzj (t;) Lq (R d ) jzj d+1d=q a (w (jzj)t) d1+d=q r 2 p jzj (t;) Lq (R d ) jzj d+2d=q a (w (jzj)t) d2+d=q : We split b ; =jzj d Z 1 0 :::dt +jzj d Z 1 1 :::dt =b 1 +b 2 d ; =jzj d Z 1 0 :::dt +jzj d Z 1 1 :::dt =d 1 +d 2 : By Minkowski’s inequality, Z jb 1 (y;z)j q dy 1=q Cjzj d+d=q Z 1 0 t Z p jzj (t;y) q dy 1=q dt t ;z6= 0: Therefore, by Corollary 14 with > dd=q p1 _ dd=q p2 , > 0, Z jb 1 (y;z)j q dy 1=q Cjzj d+d=q jzj dd=q Z 1 0 t a (w (jzj)t) d+d=q dt t =Cw (jzj) Z w(jzj) 0 t a (t) d+d=q dt t Cw (jzj) Z w(jzj)+ 0 t a (t) d+d=q dt t C w (jzj) + w (jzj) a (w (jzj) +) d+d=q C w (jzj) + w (jzj) jzj d+d=q Taking ! 0, R jb 1 (y;z)j q dy 1=q Cjzj d+d=q . 66 5.4. EMBEDDING INTO THE SPACE OF CONTINUOUS FUNCTIONS CHAPTER 5. PROBABILITY DENSITY ESTIMATES By similar computation, with > d+1d=q p1 _ d+1d=q p2 ;the term without derivative in d 1 can be estimated by b 1 , Z jd 1 (y;z)j q dy 1=q Cjzj d+d=q +Cjzj d+d=q jzj d+1d=q Z 1 0 t a (w (jzj)t) d1+d=q dt t Cjzj d+d=q +Cjzjjzj d1+d=q =Cjzj d+d=q : Since for y2R d ;z6= 0; jb 2 (y;z)jjzj d Z 1 1 t Z 1 0 rp jzj t; y jzj +s^ z ds dt t ; it follows by Minkowski’s inequality, jb 2 (;z)j Lq (R d ) Cjzj d+d=q Z 1 1 t Z rp jzj (t;y) q dy 1=q dt t Cjzj d+d=q jzj d+1d=q Z 1 1 t a (w (jzj)t) d1+d=q dt t Cjzjw (jzj) Z 1 w(jzj) t a (t) d1+d=q dt t Therefore, by Corollary 14 with < d+1d=q q1 ^ d+1d=q q2 , > 0, jzjw (jzj) Z 1 w(jzj)+ t a (t) d1+d=q dt t Cjzj w (jzj) + w (jzj) a (w (jzj) +) d1+d=q Cjzj d+d=q w (jzj) + w (jzj) Taking ! 0,jb 2 (;z)j Lq (R d ) Cjzj d+d=q . By similar computation, with < d+2d=q q1 ^ d+2d=q q2 ; 67 5.4. EMBEDDING INTO THE SPACE OF CONTINUOUS FUNCTIONS CHAPTER 5. PROBABILITY DENSITY ESTIMATES jd 2 (;z)j Lq (R d ) Cjzj d+d=q Z 1 1 t Z r 2 p jzj (t;y) q dy 1=q dt t Cjzj d+d=q jzj d+2d=q Z 1 1 t a (w (jzj)t) d2+d=q dt t Cjzj d+d=q concluding the proof. According to Lemma 41, Let 2A ;w =w be a continuous O-RV function and A, B hold. (i) If dd=q p w 1 _ dd=q p w 2 << d+1d=q q w 1 ^ d+1d=q q w 2 for some q 1 then the following function is well defined k ; (y;z) (5.9) = w (jzj) jzj d Z 1 0 t p jzj t; y jzj + ^ z p jzj t; y jzj dt t ;y2R d ;z6= 0; (ii) If d+1d=q p w 1 _ d+1d=q p w 2 << d+2d=q q w 1 ^ d+2d=q q w 2 for someq 1 then the following function is well defined k ; (y;z) (5.10) = w (jzj) jzj d Z 1 0 t p jzj t; y jzj + ^ z p jzj t; y jzj rp jzj t; y jzj ^ z dt t ;y2R d ;z6= 0; wherep jzj isthepdfoftheLévyprocessZ ~ jzj t ;t> 0;associatedtotheLévymeasure ~ jzj (dy) =w (jzj) (jzjdy);z6= 0: Now we derive a representation of an increment of f2S R d . Lemma 42. Let 2A ;w =w be a continuous O-RV function and A, B hold. For 2 (0; 2): Define ; = 8 > < > : if = 1; ( Re ) if 2 (0; 2);6= 1; and L ; f =F 1 ; Ff ;f2S R d (in particular, L ;1 =L .) (i) If there exists q 1 such that dd=q p w 1 _ dd=q p w 2 << d + 1d=q q w 1 ^ d + 1d=q q w 2 : 68 5.4. EMBEDDING INTO THE SPACE OF CONTINUOUS FUNCTIONS CHAPTER 5. PROBABILITY DENSITY ESTIMATES Then there exists a constant c> 0 so that for any f2S R d f (x +z)f (x) =c Z L ; f (xy)k ; (y;z)dy;x2R d ;z6= 0; (5.11) (ii) If there exists q 1 such that d + 1d=q p w 1 _ d + 1d=q p w 2 << d + 2d=q q w 1 ^ d + 2d=q q w 2 : Then there exists a constant c> 0 so that for any f2S R d f (x +z)f (x)rf (x)z =c Z L ; f (xy)k ; (y;z)dy;x2R d ;z6= 0; (5.12) wherek ;1 = k ;1 ;k ; = k sym; ;2 (0; 2) and6= 1; are functions defined by (5.9) in (5.11), and by (5.10) in (5.12), sym (dy) = 1 2 [ (dy) + (dy)]: Proof. The proof is essentially the same for both cases, therefore, we only prove (i). Let "> 0;f2S R d and ~ F " = [" ] ^ f if = 1, and ~ F " = [" Re ] ^ f if 6= 1: For 6= 1; F [f ( +z)f] () = e i2z 1 (" Re ()) (" Re ()) ^ f () = c Z 1 0 t e i2z 1 expf(Re ()")tg ~ F " () dt t ;2R d ;z6= 0; and by Corollary 40 for = 1; F [f ( +z)f] () = Z 1 0 e i2z 1 expf( ()")tg ~ F " ()dt;2R d ;z6= 0: Changing the variable of integration and denoting ^ z =z=jzj, we find for 2R d ;z6= 0; F [f ( +z)f] () = cw (jzj) Z 1 0 e "w(jzj)t t e i2jzj^ z 1 exp Re ~ jzj (jzj)t ~ F " () dt t 69 5.4. EMBEDDING INTO THE SPACE OF CONTINUOUS FUNCTIONS CHAPTER 5. PROBABILITY DENSITY ESTIMATES if 6= 1, and similar formula with obvious changes holds for = 1: Hence f (x +z)f (x) (5.13) = c Z H " (xy)k ; " (y;z)dy;x2R d ;z6= 0; where H " =F 1 ~ F " , and for y2R d ;z6= 0; k ; " (y;z) = w (jzj) jzj d Z 1 0 e "w(jzj)t t p jzj t; y jzj + ^ z p jzj t; y jzj dt t ; (5.14) with p jzj defined with respect to sym if 6= 1. Since Re = sym , it follows for 6= 1; (" Re ) =c Z 1 0 t expf sym ()t"tg 1 1 2(1;2) ( sym ()")t dt t ;2R d : Hence for f2S R d and 2 (0; 2);6= 1; H " (x) =c Z 1 0 t E e "t f x +Z sym t f (x) 1 2(1;2) (L sym f (x)")t dt t ;x2R d : (5.15) For f2S R d and = 1; obviously, H " (x) ="f (x)L f (x);x2R d : (5.16) The statement follows by passing to the limit in (5.13), (5.14), (5.15) and (5.16) as "! 0 , using Hölder’s inequality in (5.13) and applying Lemma 41 (with q chosen as in Lemma 41.) The following obvious consequence holds (take = 1 in Lemma 42). Corollary 43. Let 2A ;2 (0; 1);w =w be a continuous O-RV function and A, B hold. Then there is c> 0 so that for f2S R d f (x +z)f (x) =c Z L f (xy)k ;1 (y;z)dy;x2R d ;z6= 0; and there is C > 0 so that jf (x +z)f (x)jCjL fj 1 w (jzj);x2R d ;z6= 0;f2S R d : 70 5.4. EMBEDDING INTO THE SPACE OF CONTINUOUS FUNCTIONS CHAPTER 5. PROBABILITY DENSITY ESTIMATES Now, we prove an embedding statement. Proposition 44. Let 2A ;w = w be a continuous O-RV function and A, B hold. Let 2 (0; 2);p2 (1;1) such that d=p p w 1 _ d=p p w 2 << d=p + 1 q w 1 ^ d=p + 1 q w 2 : Then there is C 1 so that sup x jf (x +z)f (x)jC 1 w (jzj) jzj d=p L ; f Lp(R d ) ;f2S R d : (5.17) Moreover, there is C 1 so that for any f2S R d ; sup x jf (x)jjfj Lp(R d ) +C 1 L ; f Lp(R d ) : (5.18) Proof. According to Lemma 42, the representation (5.11) holds. Applying Hölder’s inequality and Lemma 41 with 1=p + 1=q = 1; jf (x +z)f (x)j C L ; f Lp(R d ) Z k ; (y;z) q dy 1=q C 1 w (jzj) jzj d=p L ; f Lp(R d ) ;x2R d ;z6= 0: Let x2R d . Then for any z2R d ; jf (x)jjf (x +z)f (x)j +jf (x +z)j: Integrating both sides over the unit ball B 1 = z2R d :jzj 1 ; jf (x)j 1 jB 1 j Z B1 jf (x +z)f (x)jdz + 1 jB 1 j Z B1 jf (x +z)jdz; Corollary 10 implies that R jzj1 w (jzj) jzj d=p dzC; and (5.18) follows by Hölder’s inequality and (5.17). 71 Chapter 6 L p -norm Estimates of Differences This chapter is devoted to the following estimates which will be needed for the stochastic variable coefficient model in Chapter 9. Let 2A , Z R d 0 jf ( +y)f ()j p (dy) !1 p Lp(R d ) Cjfj B ;1=p pp (R d ) ; f2B ;1=p pp R d (6.1) Z R d 0 jf ( +y)f ()j 2 (dy) !1 2 Lp(R d ) Cjfj H ;1=2 p (R d ) ; f2H ;1=2 p R d (6.2) The first inequality holds under the Assumptions A and B. However, we can only deal with specific cases for of for the second inequality. The integral in (6.2) is relevant to what appears in literature as hypersingular integrals and Marcinkiewicz integrals (e.g. [Whe72, Dor85, Str67, HL17]) and are of significant interest in harmonic analysis and potential function spaces community. 6.1 Difference Estimate by B ;1=p pp R d We establish the estimate (6.1) by following Triebel [Tri83] (Section 2.5.12) closely. Lemma45. Let2A ,w =w be an O-RV function andA,B hold,p2 (1;1). Assume thatq w 1 _q w 2 < p<1, then there exists C > 0 such that Z R d 0 jf ( +y)f ()j p (dy) ! 1=p Lp(R d ) Cjfj B ;1=p pp (R d ) ; f2B ;1=p pp R d : Proof. For brevity we will use the notation y f (x) =f (x +y)f (x) to denote an increment. 72 6.1. DIFFERENCE ESTIMATE BY B ;1=P PP R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES Clearly, R jyj>1 jf ( +y)f ()j p (dy) 1=p Lp(R d ) Cjfj Lp(R d ) . Therefore, we only need to show that Z 0<jyj1 jf (x +y)f (x)j p (dy) ! 1=p Lp(R d ) Cjfj B ;1=p pp (R d ) (6.3) We follow closely steps of the proof in [Tri83]-Theorem of Section 2.5.12. Using the equivalent norm ~ B ;1=p pp R d , it is easy to check that for f2B ;1=p pp R d , f = P 1 j=0 F 1 j Ff where RHS converges in L p where j is defined as in Remark 1. Z 0<jyj1 j y f (x)j p (dy) = 1 X k=0 Z N (k+1) <jyjN k jf (x +y)f (x)j p (dy) = 1 X k=0 Z N 1 <jyj1 f x +N k y f (x) p N k dy = 1 X k=0 Z N 1 <jyj1 f x +N k y f (x) p w N k 1 ~ N k (dy) Integrating in x, note that R jyj>1=N ~ R (dy)w (R) R jyj>R=N (dy)C;R> 0, Z 0<jyj1 Z j y f (x)j p dx (dy) 1 X k=0 w N k 1 sup jyjN k j y fj p Lp(R d ) ~ N k jyj>N 1 C 1 X k=0 w N k 1 sup jyjN k 0 @ 1 X j=0 y F 1 j Ff Lp(R d ) 1 A p C 1 X k=0 w N k 1 0 @ 1 X j=0 sup jyjN k y F 1 j Ff Lp(R d ) 1 A p := 1 X k=0 0 @ 1 X j=0 G j;k 1 A p with G j;k :=w N k 1=p sup jyjN k 1 y F 1 j Ff Lp(R d ) . Define j f (x) = sup y2R d j(F 1 jFf)(xy)j 1+jN j yj a some a > 0 sufficiently large (see definition 2 on p.53 of [Tri83].) The following estimate is provided on p.111 of [Tri83], sup jyj2 k 1 y F 1 j Ff Lp(R d ) C min 1;N (jk) j f Lp(R d ) (6.4) where C is independent of f. 73 6.1. DIFFERENCE ESTIMATE BY B ;1=P PP R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES By triangle inequality, and (6.4), 2 4 1 X k=0 0 @ 1 X j=0 G j;k 1 A p 3 5 1=p 0 @ 1 X k=0 0 @ k X j=0 G j;k 1 A p 1 A 1=p + 0 @ 1 X k=0 0 @ 1 X j=k+1 G j;k 1 A p 1 A 1=p 0 @ 1 X k=0 0 @ k X j=0 w N k 1=p N (jk) j f Lp(R d ) 1 A p 1 A 1=p + 0 @ 1 X k=0 0 @ 1 X j=k+1 w N k 1=p j f Lp(R d ) 1 A p 1 A 1=p =S 1 +S 2 : Let q 1 _q 2 < 1 < p, by Lemma 6, w(N j ) 1=p w(N k ) 1=p CN (kj) 1 p ;j k. Also, we fix > 0 such that 1 p +p< 0, S 1 = 0 @ 1 X k=0 0 @ k X j=0 w N k 1=p N (jk) j f Lp(R d ) 1 A p 1 A 1=p = 0 @ 1 X k=0 0 @ k X j=0 w N k 1=p w N j 1=p w (N j ) 1=p N (jk) j f Lp(R d ) 1 A p 1 A 1=p C 0 @ 1 X k=0 0 @ k X j=0 w N j 1=p N (kj)( 1 p 1) j f Lp(R d ) 1 A p 1 A 1=p C 0 B @ 1 X k=0 0 @ k X j=0 w N j 1 N (kj)(1p+p) j f p Lp(R d ) 1 A 0 @ k X j=0 N (kj)q 1 A p=q 1 C A 1=p C 0 @ 1 X k=0 0 @ k X j=0 w N j 1 N (jk)(p1p) j f p Lp(R d ) 1 A 1 A 1=p C 0 @ 1 X j=0 w N j 1 j f p Lp(R d ) 1 A 1=p : 74 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES Now for 0< 2 <p 1 ^p 2 , applying Lemma 6. We fix 0<< 2 p then S 2 = 0 @ 1 X k=0 0 @ 1 X j=k+1 w N k 1=p j f Lp(R d ) 1 A p 1 A 1=p = 0 @ 1 X k=0 0 @ 1 X j=k+1 w N k 1=p w N j 1=p w (N j ) 1=p j f Lp(R d ) 1 A p 1 A 1=p C 0 @ 1 X k=0 0 @ 1 X j=k+1 w N j 1=p N (j+k)( 2 p ) j f Lp(R d ) 1 A p 1 A 1=p C 0 B @ 1 X k=0 0 @ 1 X j=k+1 w N j 1 N (j+k)(2p) j f p Lp(R d ) 1 A 0 @ 1 X j=k+1 N (j+k)q 1 A p=q 1 C A 1=p C 0 @ 1 X j=0 w N j 1 j f p Lp(R d ) 1 A 1=p : Clearly, (6.3) holds provided that j f Lp(R d ) Cjf' j j Lp(R d ) : (6.5) However, this is given in [Tri83](p.56), concluding the proof. Remark 46. Estimates in (6.4) and (6.5) are provided with N = 2. However, it is clear that they also hold with any N > 1 with possibly a different constant. 6.2 Difference Estimate by H ;1=2 p R d 6.2.1 Notation, Assumptions and Main Results First, we introduce some notations. (i) B t denotes the open ball inR d centered at the origin with radius t> 0 (ii) Q t;x denote the open ball inR d centered at x2R d with radius t> 0 (iii) Q t denote an open ball inR d with radius t> 0 when a center is not specified (iv) f Q = 1 jQj R Q f (y)dy, the mean of f in the ball Q with respect to Lebesgue measure We use 0 subscription to denote open balls that omits the origin i.e. B 0 t =B t f0g. For the estimate (6.2), we will need additional assumptions on w . Denoting (dy) = dy w (jyj) 1 2jyj d , and (y) = (jyj) = 1 w (jyj) 1 2jyj d the density of of . 75 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES C. (r) is absolutely continuous and 0 (r) r is non-increasing on r> 0 D. (dy) = a(y)dy ! (jyj)jyj d where 0<ca (y)C Note immediately that assumption D implies assumption B in Theorem 2. Remark 47. According to Proposition 15 the equivalent norms in H ;s p R d ;B ;s pq R d ;s2R are given by jfj ~ H ;s p (R d ) u 0 @ 1 X j=0 w N j s f' j 2 1 A 1=2 Lp(R d ) (6.6) jfj ~ B ;s pq (R d ) u 0 @ 1 X j=0 w N j sq jf' j j q Lp(R d ) 1 A 1=q : Therefore, trivial consequences are (i) If there is another Lévy measure ~ 2A as in Proposition 15 such that! u! ~ then the spaceH ;s p R d = H ~ ;s p R d (norms are equivalent). (ii) In particular, H ;1=2 p R d = H ;1 p R d where (dy) = dy w (jyj) 1 2jyj d . It will be convenient from now to use an equivalent norm jfj H ;1=2 p (R d ) ujfj Lp(R d ) + L ;1 f Lp(R d ) =jfj Lp(R d ) +jL fj Lp(R d ) where the last equivalent is due to symmetry of . In this section we establish some results on Bessel norm. In particular, we verify inequality (6.2). We define a measure onR d 0 , (dy) :=! (jyj)jyj d (dy) Define for (t;x)2 (0;1)R d , f2C 1 0 R d ;r> 0, oscillation operators G f;r (x;t) := " 1 t d Z B 0 t jf (x +z)f (x)j r (dz) # 1=r T r f (x) := Z 1 0 w (t) 1 G f;r (x;t) 2 dt t 1=2 The main results are following theorems. 76 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES Theorem 48. Let 2 A , w = w be an O-RV function and A, B, C hold. Let (dy) = dy w (jyj) 1 2jyj d , p2 (1;1): Then there exists C > 0 such that for f2H ;1 p R d , jT 1 fj Lp(R d ) C L ;1 f Lp(R d ) : Theorem 49. Let 2A , w =w be an O-RV function and A, C, D hold. Let (dy) = dy w (jyj) 1 2jyj d and p2 [2;1). Then there exists C > 0 such that for f2H ;1 p R d , jT 2 fj Lp(R d ) C L ;1 f Lp(R d ) Remark 50. (6.2) follows directly from Theorem 49 (under all Assumptions of the theorem), we defer the details to Section 6.2.3. The proof of Theorems 48, 49 will follow a series of results which are organized in subsections. 6.2.2 More Probability Density Estimates Throughout this section we let 2A , w =w be an O-RV function and A, B, C hold. Let Y t ; t> 0 be Lévy rotationally symmetric Lévy process inR d associated with Lévy measure (dy) = dy w (jyj) 1 2jyj d . We apply the Corollary 7 of [BGR14], for estimate of the density of Y t and Corollary 1.7 of [KR16], for the estimate of gradient of density of Y. For Corollary 1.7 of [KR16], one may check assumptions therein (with the density (y) = 1=w (jyj) 1 2 jyj d ) routinely using property of O-RV functions and our Assumption C. Note that, for r> 0, 0 (r) r = 1 r d dr w (r) 1 2 r d = 1 r 1 2 w (r) 3 2 w 0 (r)r d dw (r) 1 2 r d1 = 1 2 w (r) 3 2 w 0 (r)r d1 +dw (r) 1 2 r d2 : The last expression is clearly non-increasing provided thatw (r) 3 2 w 0 (r)r d1 is non-increasing, suggesting that Assumption C is a mild assumption. We summarize the estimates from Corollary 7 of [BGR14] and Corollary 1.7 of [KR16] (see also 20) for the density of Y t denoted by p (t;x);t> 0;x2R d ; jp (t;x)jC (d) t w (jxj) 1 2 jxj d 77 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES jrp (t;x)jC (d) t w (jxj) 1 2 jxj d+1 We also have from Lemmas 32, 33 and that jp (t;x)jC (d)a (t) d jrp (t;x)jC (d)a (t) d1 where a (t) = inf n s> 0 :w (s) 1=2 t o is the generalized inverse of w 1=2 . We have the following probabilistic representation for J ; f =F 1 ( ) ^ f for f 2 C 1 0 R d , > 0, > 0, which one may check easily by taking Fourier transform. J ; f (x) =c Z 1 0 t 1 e t fp (t)dt =fc Z 1 0 t 1 e t p (t)dt =cfK ; with K ; = R 1 0 t 1 e t p (t)dt. Indeed, for > 0, FJ ; f () = ( ()) ^ f =c Z 1 0 t 1 e ( ())t dt ^ f: Lemma 51. The following estimates hold for K a; = R 1 0 t 1 e t p (t)dt. For 0<< 2d q w 1 ^ 2d q w 2 , and any > 0, there exists C =C (;)> 0 independent of such that K ; (x) C w (jxj) 2 jxj d ! and rK ; (x) C w (jxj) 2 jxj d+1 ! ; x2R d : Proof. From p (t;x)C a (t) d ^ t jxj d w (jxj) 1 2 for > 0, we split the integral, K ; (x)C " 1 jxj d w (jxj) 1 2 Z 0 t e t dt ! + Z 1 t 1 e t a (t) d dt # Using the assumption that 0<< 2d q1 ^ 2d q2 and Corollary 14, Z 1 t 1 e t a (t) d dt Z 1 t 1 a (t) d dt a () d : Therefore, 78 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES K ; (x)C " 1 jxj d w (jxj) 1 2 +1 Z 1 0 t e t dt ! + a () d # C " 1 jxj d w (jxj) 1 2 +1 ! + a () d # Plugging in =w (jxj) 1 2 +; > 0 using the fact that a w (t) 1 2 + t; t> 0 and taking ! 0, we derive the desired estimate K ; (x)C w (jxj) 2 jxj d ! : The estimate for the gradient is derived similarly. 6.2.3 Estimate ofG f;1 Again throughout this section we let2A ,w =w be an O-RV function andA,B,C hold. Using our characterization(6.6), seeRemark6.6,H ; 1 2 p andH ;1 p haveequivalentnormswhere (dy) =dy=w (jyj) 1 2 jyj d - this allows us to exploit precise probability density estimates p (t;x);t> 0;x2R d of Section 6.2.2. To ease notation, we will use (dy) =w (jyj)jyj d (dy). For example, in the case of (dy) =dy=jyj d+ , (dy) =dy is simply the Lebesgue measure. We want to mention immediately that that B 0 t ut d ;t> 0, where B t is the open ball with radius t centered at the origin. Indeed, for any a> 0, by Lemma 6, (a<jzj 2a) = (a) (2a) =w (a) 1 w (2a) 1 =w (a) 1 1 w (a) w (2a) w (a) 1 1 2 with some > 0. Hence, (a<jzj 2a)w (a) 1 for some > 0. B 0 t = Z 0<jzjt jzj d w (jzj) (dz) = 1 X k=0 Z 2 k1 t<jzj2 k t jzj d w (jzj) (dz) 1 X k=0 2 k1 t d w 2 k1 t 2 k1 t<jzj 2 k t 1 X k=0 2 k1 t d w 2 k1 t w 2 k1 t 1 =ct d : 79 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES The reverse inequality B 0 t Ct d follows from similar but simpler computation. Recall the following oscillation functions. G f;r (x;t) = " 1 t d Z B 0 t jf (x +z)f (x)j r (dz) # 1=r ;r> 0 First we observe, by Lemma 10, Z R d 0 jf (x +z)f (x)j 2 (dz) = Z R d 0 jf (x +z)f (x)j 2 w (jzj) 1 jzj d w (jzj)jzj d (dz) = Z R d 0 jf (x +z)f (x)j 2 w (jzj) 1 jzj d (dz) C Z R d 0 jf (x +z)f (x)j 2 Z 1 jzj w (t) 1 t d dt t (dz) =C Z 1 0 w (t) 1 t d Z jzjt jf (x +z)f (x)j 2 (dz) dt t =C Z 1 0 w (t) 1 G f;2 (x;t) 2 dt t : To prove (6.2), it is therefore sufficient to show that Z 1 0 w (t) 1 G f;2 (x;t) 2 dt t 1 2 Lp(R d ) Cjfj H ;1 p (R d ) ; f2H ;1 p R d : which in turn will follows from Z 1 0 w (t) 1 2 G J ;1 f;2 (x;t) 2 dt t 1 2 Lp(R d ) Cjfj Lp(R d ) ; f2L p R d : (6.7) In order to show (6.7), we first replaceG J ;1 f;2 (x;t) withG J ;1 f;1 (x;t). Then, we establish the estimate by the following the classical route of Calderón-Zygmund theorem. Finally, we will extend the estimate, with additional Assumption D to the case ofG J ;1 f;2 (x;t) following the argument of [Dor85]. In this section we establish the estimate forG J ;1 f;1 (x;t) and the result forG J ;1 f;2 (x;t) will be established in the next section. We remind the reader for convenience that for f2C 1 0 R d , T 1 J ;1 f (x) = Z 1 0 w (t) 1 G J ;1 f;1 (x;t) 2 dt t 1 2 ;x2R d : 80 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES Lemma 52. Let 2A , w = w be an O-RV function and A, B, C hold. Let (dy) = dy w (jyj) 1 2jyj d , and p2 (1;1). The following L 2 -boundedness holds, for some C > 0; T 1 J ;1 f L2(R d ) Cjfj L2(R d ) ; f2L 2 R d : Remark 53. In this Lemma we showL 2 boundedness,L p boundedness will follow from the next Lemma via the verification of the Hörmander condition. Proof. We argue similar to Theorem 3 [Whe72]- following their steps with necessary modifications. In this Lemma we show L 2 -boundedness, L p -boundedness will follow from the next Lemma (the verification of Hörmander condition.) For p = 2, by Cauchy-Schwart inequality and using B 0 t ut d , denote for convenience F =J ;1 f, G F;1 (x;t) 2 = 1 t d Z B 0 t jF (x +z)F (x)jd (z) ! 2 C t d Z B 0 t jF (x +z)F (x)j 2 d (z): Integrating both sides and applying Plancherel’s theorem, Z Z 1 0 w (t) 1 2 G F;1 (x;t) 2 dt t dxC Z Z 1 0 w (t) 1 t d dt t Z B 0 t jF (x +z)F (x)j 2 (dz)dx =C Z ^ f (x) 2 (1 (x)) 2 I (x)dx where I (x) = Z 1 0 w (t) 1 t d dt t Z B 0 t e i2xz 1 2 (dz): For any t> 0, R B 0 t e i2xz 1 2 d (z) 4 B 0 t 4Ct d . Moreover, if 0<t< 1 jxj then Z B 0 t e i2xz 1 2 d (z)Cjxj 2 Z B 0 t jzj 2 (dz)Cjxj 2 t 2 B 0 t : Hence, by Corollary 10, I (x)Cjxj 2 Z 1 jxj 0 w (t) 1 tdt +C Z 1 1 jxj w (t) 1 t 1 dt Cjxj 2 jxj 2 w jxj 1 1 +Cw jxj 1 1 Cw jxj 1 1 81 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES and therefore, by Lemma 20, (x)uw jxj 1 1 2 ;x2R d , and thus Z ^ f (x) 2 (1 (x)) 2 I (x)dxC Z ^ f (x) 2 1 +w jxj 1 1 2 2 w jxj 1 1 dx C Z ^ f (x) 2 dx =Cjfj L2(R d ) : Next we aim to establish weak-type statement for the operator T 1 . Note that, with the notation of Lemma 51, let K =K 1;1 , consider first the following elementary estimate, Z 1 0 w (t) 1 G J ;1 f;1 (x;t) 2 dt t 1 2 = 0 @ Z 1 0 w (t) 1 1 t d Z B 0 t jKf (xz)Kf (x)j (dz) ! 2 dt t 1 A 1 2 = 0 @ Z 1 0 w (t) 1 t 2d Z B 0 t Z f (y) [K (xzy)K (xy)]dy (dz) ! 2 dt t 1 A 1 2 Z Z 1 jzj w (t) 1 t 2d Z f (y) [K (xzy)K (xy)]dy 2 dt t ! 1=2 (dz) = Z Z f (y) [K (xzy)K (xy)]dy Z 1 jzj w (t) 1 t 2d dt t ! 1=2 (dz) = Z Z f (y) [K (xzy)K (xy)]dy dm (dz) (6.8) where we denote m (dz) = R 1 jzj ! (t) 1 t 2d dt t 1 2 (dz). Remark 54. By Corollary 10 immediately m (dz) = Z 1 jzj w (t) 1 t 2d dt t !1 2 (dz)Cw (jzj) 1 2 jzj d (dz) =Cw (z) 1 2 (dz): Before proceeding we carry out another basic computation regarding m (dz) to be used in the next Lemma. Remark 55. (i) For > q w 1 2 _ q w 2 2 and > 0, by Lemmas 6 and 10, 82 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES Z 0<jzj jzj m (dz) = Z 0<jzj jzj w (z) 1 2 (dz) = 1 X k=0 Z 2 k1 <jzj2 k jzj w (z) 1 2 (dz) 1 X k=0 2 k w 2 k 1 2 jzj> 2 k1 = 1 X k=0 2 k w 2 k 1 2 w 2 k1 1 C 1 X k=0 2 (k1) w 2 k1 1 2 C Z 0 y w (y) 1 2 dy y C w () 1 2 : (ii) For 0, by a similar computation, Z jzj> jzj m (dz) = Z jzj> jzj w (z) 1 2 (dz) = 1 X k=0 Z 2 k <jzj2 k+1 jzj w (z) 1 2 (dz) 1 X k=0 2 k+1 w 2 k+1 1 2 jzj> 2 k =C 1 X k=0 2 k w 2 k+1 1 2 w 2 k 1 C 1 X k=0 2 k w 2 k+1 1 2 C Z 1 y w (y) 1 2 dy y C w () 1 2 : Now we proceed to show Hörmander condition i.e. weak type estimate for the kernel in 6.8. Lemma 56. Let 2A , w = w be an O-RV function and A, B, C hold. Let (dy) = dy w (jyj) 1 2jyj d , and p2 (1;1), K = K 1;1 defined as in Lemma 51. The following Hörmander condition holds: for any > 0, letjyj< then Z jxj>3 Z j(K (z +x +y)K (x +y)) (K (z +x)K (x))jm (dz)dxC where C is independent of and y. Proof. We follow the splitting in the proof Theorem 1 of [Whe72], i.e. Z jxj>3 Z j(K (z +x +y)K (x +y)) (K (z +x)K (x))jm (dz)dx = Z jxj>3 Z 0<jzj jK (z +x +y)K (x +y) (K (z +x)K (x))jm (dz)dx + Z jxj>3 Z jzj> jK (x +y)K (x)jm (dz)dx + Z jxj>3 Z jzj> jK (z +x +y)K (z +x)jm (dz)dx =A +B +C: 83 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES First we estimate A, applying Lemma 51, jK (z +x)K (x)jjzj Z 1 0 rK (x +rz)dr Cjzj Z 1 0 w (jx +rzj) 1=2 jx +rzj d+1 dr: Now becausejzj< jxj 3 , by Lemma 6, for r2 [0; 1],jzj w (jx+rzj) 1=2 jx+rzj d+1 Cjzj w (jxj) 1=2 jxj d+1 : Similarly,jK (z +x +y)K (x +y)jCjzj w (jxj) 1=2 jxj d+1 . Applying Corollary 10, Remark 54 leads to AC Z 0<jzj< jzjm (dz) ! Z jxj>3 w (jxj) 1=2 jxj d+1 dx ! =Cw () 1=2 Z r>3 w (r) 1=2 r 2 dr Cw () 1=2 w (3) 1=2 1 C: For the estimate of B, sincejyj<< jxj 3 , by Corollary 10, Remark 54, and Lemma 51, BCjyj Z jzj> m (dz) ! Z jxj>3 w (jxj) 1=2 jxj d+1 dx ! Cw () 1=2 w (3) 1=2 1 C: To estimate C we split further, Z jxj>3 Z jzj>;jxzj>2 ::::m (dz)dx + Z jxj>3 Z jzj>;jxzj2 ::::m (dz)dx =C 1 +C 2 : By Corollary 10, Remark 54, and Lemma 51, C 1 = Z jxj>3 Z jzj>;jx+zj>2 jK (z +x +y)K (z +x)jm (dz)dx jyj Z jzj> Z jx+zj>2 Z 1 0 jrK (z +x +ry)jdrdxm (dz) =jyj Z jzj> m (dz) ! Z jxj>2 Z 1 0 jrK (x +ry)jdrdx ! Cw () 1=2 w (2) 1=2 1 C: Finally we estimate C 2 ; by Corollary 10, Remark 54, and Lemma Lemma 51, 84 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES C 2 C Z Z jzj>;jx+zj2 w (jx +z +yj) 1=2 jx +z +yj d + w (jx +zj) 1=2 jx +zj d ! m (dz)dx C Z jxj3 w (jxj) 1=2 jxj d dx ! Z jzj> m (dz) ! Cw (3) 1=2 w () 1=2 C: Lemma 52, Lemma 56 and vector-valued version of Calderón-Zygmund theorem (e.g. Theorem 1.3 of [RRT86]) lead to boundedness in L p R d for p2 (1;1) stated in the following Corollary. Corollary 57. Let 2A , w =w be an O-RV function and A, B, C hold. Let (dy) = dy w (jyj) 1 2jyj d , and p2 (1;1). The following L p boundedness holds, for some C =C (p;)> 0, T 1 J ;1 f Lp(R d ) Cjfj Lp(R d ) ; f2L p R d and Z 1 0 w (t) 1 G f;1 (x;t) 2 dt t 1 2 Lp(R d ) C L ;1 f Lp(R d ) ; f2H ;1 p R d : Proof. The first inequality is an immediate result of Lemma 52, Lemma 56 and vector-valued version of Calderón-Zygmund theorem (see for example, Theorem 1.3 of [RRT86].) For the second estimate, by examining the proof that led to the inequality Z 1 0 w (t) 1 G J ;1 f;1 (x;t) 2 dt t 1 2 Lp(R d ) C (p;)jfj Lp(R d ) ; 8f2L p R d : We observe that all estimates remains valid whenG J ;1 f;1 (x;t) is replaced withG J ;1 f;1 (x;t); > 0 (due to AssumptionA we have always 1< 2d q1 ^ 2d q2 ) without changing the constant C (p;). Indeed we have a stronger result, from Z 1 0 w (t) 1 G J ;1 f;1 (x;t) 2 dt t 1 2 Lp(R d ) C (p;)jfj Lp(R d ) : For f2H ;1 p R d , by substituting f by J ;1 f, Z 1 0 w (t) 1 G f;1 (x;t) 2 dt t 1 2 Lp(R d ) C (p;) J ;1 f Lp(R d ) =C (p;) fL ;1 f Lp(R d ) 85 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES Passing ! 0, Z 1 0 w (t) 1 G f;1 (x;t) 2 dt t 1 2 Lp(R d ) C (p;) L ;1 f Lp(R d ) completing the proof. Remark 58. In Proposition 26, we derive the estimate Z 1 0 w (t) 1 G f;1 (x;t) 2 dt t 1 2 L P (R d ) C (p;)jfj H ;1 p (R d ) : The result in Corollary 57 is hence a stronger results i.e. we can bounde LHS by only L ;1 f Lp(R d ) without jfj Lp(R d ) . Also the result of Proposition 26 only covers the case that (dy) = dy. In Proposition 26, the proof follows general but rather technical results from [KL87]. Here we provide a self-contained proof through the classical Calderón-Zygmund theorem. 6.2.4 Estimate ofG f;2 : Square Function Operators Throughout this section we assume that2A satisfy assumptionA,C andD. Currently we are unable to derive the result for satisfying only A, B and C, but we believe that estimate ofG f;1 should be used for the estimate ofG f;2 . We extend the L p estimate ofG f;1 toG f;2 by following the argument of Dorronsoro in [Dor85]. Recall that AssumptionD stipulates that must be of the form (dy) = a(y)dy ! (y)jyj d wherea (y) is uniformly bounded from below and above. We will prove the result of Corollary 57 for T 2 in place of T 1 . We remind the reader the definition of T 2 , for f2C 1 0 R d , T 2 f (x) = Z 1 0 ! (t) 1 G f;2 (x;t) 2 dt t 1=2 with G f;2 (x;t) = 1 t d Z Bt jf (x +z)f (x)j 2 a (z)dz 1 r ;r> 0: Before proceeding we note the following simple Lemma which is essential in deriving estimates for square functions. Lemma 59. Let 2 A , w = w be an O-RV function and A, B hold. Let 2 0 satisfy sup t>0 Z 1 t U (x) 2 dx 1 2 Z t 0 V (x) 2 dx 1 2 <1: 86 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES Proof. For convenience we denote I (t) = R 1 t U (x) 2 dx 1 2 R t 0 V (x) 2 dx 1 2 . From 2 < p 1 ^ p 2 , by Corollary 10, I (t) = Z 1 t w (x) 1 x 21 dx 1 2 Z t 0 w (x)x 21 dx 1 2 C w (t) 1 t 2 1 2 w (t)t 2 1 2 C: Lemma 60. Let 2A , w =w be an O-RV function and A, C, D hold. Let (dy) = dy w (jyj) 1 2jyj d , and p2 [2;1). Then there exists C =C (p;)> 0 such that jT 2 fj Lp(R d ) C (p;) L ;1 f Lp(R d ) ; 8f2H ;1 p R d : Proof. Now we follow the proof of Theorem 6 of [Dor85] closely and modify it for our case. The key difference is that we must rely on the weighted Hardy’s inequality instead of the classical version. We include here key steps for the sake of completeness. Fix x2R n , t> 0, and a ball B t centered at 0 such that x2B t . By the following standard argument, for the purpose ofL p -boundedness, one may easily replacef (x) with the average value f, inG f;2 . Precisely, denote f Bt (x) = 1 jBtj R jyjt f (x +y)dy . Z 1 0 w (t) 1 G f;2 (x;t) 2 dt t 1 2 = Z 1 0 w (t) 1 1 t d Z jyjt jf (x +y)f (x)j 2 a (y)dy dt t ! 1=2 Z 1 0 w (t) 1 1 t d Z jyjt f (x +y) f Bt (x) 2 a (y)dy dt t ! 1=2 + Z 1 0 w (t) 1 1 t d Z jyjt f (x) f Bt (x) 2 a (y)dy dt t ! 1=2 =I 1 (x) +I 2 (x): Estimate of I 2 : Since a the coefficient in the definition of is bounded from above and below, 87 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES I 2 (x)C Z 1 0 w (t) 1 f (x) f Bt (x) 2 dt t 1=2 C 0 @ Z 1 0 w (t) 1 1 jB t j Z jyjt jf (x +y)f (x)ja (y)dy 2 dt t 1 A 1=2 =C Z 1 0 w (t) 1 G f;1 (x;t) 2 dt t 1 2 Therefore, by Corollary 57, jI 2 j Lp(R d ) C L ;1 f Lp(R d ) : Estimate of I 1 : Changing variable of integration, I 1 (x) = Z 1 0 w (t) 1 1 t d Z jyxjt f (y) f Bt (x) 2 dy dt t ! 1=2 : Before proceeding, we define local non-centered maximal functions as in [Dor85] (p.29) which will be convenient to work with, for r> 0, f2L loc;r R d , define f;r (y;s) := sup Qs ( 1 jQ s j Z Qs f f Qs r 1=r :y2Q s ) where Q s is an open ball with radius s (not necessarily centered at the origin.) Starting with p 2, r = 2; =p 1 ^p 2 . We fix 0< < 2 ^ d p and choose q< 2 such that 1 2 = 1 r = 1 q d : The following observation was provided by Dorronsoro [Dor85] (Theorem 6), fix x2 R d , for a ball Q t containing x, and y2 Q t , denote also Q y;s a ball centered at y with radius s, it follows from Lebesgue differentiation theorem that for a.e. y2Q t ;(we refer to (10) on p.28 of [Dor85] for detailed argument.) f (y) f Qt C Z t 0 f;1 (y;s) ds s C Z t 0 1 jQ t \Q y;s j Z Qy;s f;1 (z;s) Qt (z)dz ! ds s C Z t 0 s s d+ Z Qy;s f;1 (z;s) Qt (z)dz ! ds s C Z t 0 s M ( f;1 (;s) Qt ) (y) ds s (6.9) 88 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES with M g (y) = sup y2Q n jQj 1+=d R Q jgj o where sup is taken over the set of balls containing y. It is well known that M maps L q R d into L r R d =L 2 R d (see for example Theorem 2.1 of [HKK15].) Squaring (6.9) and integrating in y over Q t yields Z Qt f (y) f Qt 2 dy 1 2 C Z t 0 s Z (M ( f;1 (;s) Qt )) 2 (y)dy 1 2 ds s C Z t 0 s Z Qt f;1 (y;s) q dy 1 q ds s which implies, because Q t is an arbitrary ball of radius t containing x, f;2 (x;t)Ct Z t 0 s (M q f;1 (;s)) (x) ds s where M q g = (Mjgj q ) 1=q ; where M is the Hardy-Littlewood maximal operator. Next instead of the standard version of Hardy’s inequality, thanks to Lemma 59, we apply instead the weighted version. Specifically applying Theorem 1 of [Muc72] with U (t) = ! (t) 1 2 t 1 2 ; and V (t) = ! (t) 1 2 t + 1 2 ;t> 0. Z 1 0 w (t) 1 f;2 (x;t) 2 dt t 1=2 C " Z 1 0 w (t) 1 t 2 Z t 0 s M q f;1 (;s) (x) ds s 2 dt t # 1=2 C Z 1 0 w (t) 1 t 2 t t M q f;1 (;t) (x)t 1 2 dt 1=2 =C Z 1 0 w (t) 1 t 1 (M q f;1 (;t) (x)) 2 dt 1=2 : Taking L p norm, applying vector-valued Ferfermen-Stein inequality (recall that q< 2p) i.e. Theorem 1 of [FS71], Z 1 0 w (t) 1 f;2 (;t) 2 dt t 1=2 Lp(R d ) C Z 1 0 w (t) 1 f;1 (;t) 2 dt t 1=2 Lp(R d ) : (6.10) We now estimate RHS, first recall that f;1 (x;t) = sup x2Qt 1 jQ t j Z Qt f f Qt : By standard computation, for a ball Q t of radius t containing x, 89 6.2. DIFFERENCE ESTIMATE BY H ;1=2 P R D CHAPTER 6. L P -NORM ESTIMATES OF DIFFERENCES 1 jQ t j Z Qt f (y) f Qt dy 2 jQ t j Z Qt jf (y)f (x)jdy and hence, f;1 (x;t) 2 jQ t j Z Qt jf (y)f (x)jdy 2 jQ t j Z Qx;2t jf (y)f (x)jdyCG f;1 (x; 2t): Therefore, by (6.10), Corollary 57 and Lemma 6, Z 1 0 w (t) 1 f;2 (;t) 2 dt t 1=2 Lp(R d ) C Z Z 1 0 w (t) 1 f;1 (x;t) 2 dt t p=2 dx ! 1=p C Z Z 1 0 w (t) 1 G f;1 (x; 2t) dt t p=2 dx ! 1=p =C Z Z 1 0 w (t) 1 G f;1 (x;t) dt t p=2 dx ! 1=p C L ;1 f Lp(R d ) concluding the proof. Remark 61. The similar argument can be used to obtained a result for some p< 2 (but not all p2 (1; 2)), however we do not pursue this here. Corollary 57, and Lemma 60 lead to the conclusion of Theorem 48 and Theorem 49. 90 Chapter 7 Proof of the Main Results In this Chapter, we will prove main results for the elliptic equation (5.7) and the parabolic equation (1.1) i.e. Theorem 2. The existence of the solution is established via the method of apriori estimates. The main estimates are achieved by Calderón-Zygmund theorems for the free term f, and the stochastic term . 7.1 Elliptic Equation Theorem 62. Let p2 (1;1);s2 R: Let 2 A ; and w = w be continuous O-RV function at zero and infinity with p i ;q i ;i = 1; 2; defined in (2.3), (2.4). Assume that Assumptions A, B hold then for f2H ;s p R d , there exists a unique u2H ;s+1 p R d solving (5.7). Moreover, there exists C =C (d;p;) such that jL uj H ;s p (R d ) Cjfj H ;s p (R d ) juj H ;s p (R d ) 1 jfj H ;s p (R d ) : Define for f2 ~ C 1 R d , > 0, 0, K f (x) = Z 1 e t Ef (x +Z t )dt = Z 1 Z e t f (x +y)p (t;y)dydt = Z f (xy) Z 1 e t p (t;y)dtdy = Z f (y) Z 1 e t p (t;xy)dtdy;x2R d : 91 7.1. ELLIPTIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS Let 2A be such that the integrability Assumption (5.4) in Lemma 35 holds and define T f (x) =L K f (x) = Z L f (xy) Z 1 e t p (t;y)dtdy = Z 1 e t Z L p (t;xy)f (y)dydt = Z m (xy)f (y)dy;f2 ~ C 1 R d : (7.1) with m (x) = Z 1 e t Z L p (t;x)dt Note that by Lemma 35, Z jm (x)jdx Z 1 e t dt t <1: We remind the reader the definitions of generalized inverse a (r) = infft> 0 :w (t)rg;r> 0 and a 1 (r) = infft> 0 :a (t)rg;r> 0: Infact, underassumptionsofLemma13,a 1 =w, wewillusebothnotationsinterchangeablybecausea 1 is more intuitive and make proof clearer. We define for convenience that a (0) = 0 and a 1 (0) =w (0) = 0. Lemma 63. Let 2 A ; and w = w be continuous O-RV function and A,B hold. Let 2 A be such that Assumption (5.4) in Lemma 35 holds. Then for each p2 (1;1) there is a constant C = C (d;p;) independent of such that jT " fj Lp(R d ) Cjfj Lp(R d ) ;f2 ~ C 1 R d : (7.2) Proof. (i) First we prove the statement for p = 2. Observe that c m " () = () Z 1 " expf ()ttgdt;2R d : Hence by Lemma 20, c m " () Cw jj 1 1 Z 1 0 exp cw jj 1 1 t dtC 92 7.1. ELLIPTIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS for all 2R d ; and thus jT " fj L2(R d ) Cjfj L2(R d ) ;f2 ~ C 1 R d : (ii) We prove the estimate for p2 (1; 2). Since we already have an L 2 -estimate, according to Theorem 3 of Chapter I in [Ste93], it suffices to show that Z jxj3jsj jm " (xs)m " (x)jdxC; 8s6= 0: i.e. B = Z jxj3jsj Z 1 e t h L p (t;xs)L p (t;x) i dt dxC Spliting B as follows, B Z jxj3jsj Z a 1 (jsj) 0 ::: dx + Z jxj3jsj Z 1 a 1 (jsj) ::: dx =A 1 +A 2 : By Lemma 35, and by Corollary 14 for > 0, A 1 2 Z jxj2jsj Z a 1 (jsj) 0 L p (t;x) dtdx Cjsj Z a 1 (jsj) 0 t 1 a (t) dtdx Cjsj a a 1 (jsj) C: and similarly, A 2 = Z jxj3jsj Z 1 a 1 (jsj) h L p (t;xs)L p (t;x) i dt dx Z 1 a 1 (jsj) jsj Z 1 0 Z L rp (t;xs) dxddt Cjsj Z 1 a 1 (jsj) a (t) 1 t 1 dtCjsja a 1 (jsj) 1 C jsj a (2a 1 (jsj)) a 2a 1 (jsj) a (a 1 (jsj)) C where we used Lemma 6 in the last inequality. 93 7.1. ELLIPTIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS (iii) We prove the estimate for p> 2. We follow a standard duality argument. Let g2C 1 0 R d , Z T f (x)g (x)dx = Z L K f (x)g (x)dx = Z Z Z 1 L f (xy)e t p (t;y)g (x)dtdydx = Z Z Z 1 L f (y)e t p (t;xy)g (x)dtdydx = Z Z Z 1 f (y)e t L p (t;xy)g (x)dtdydx = Z f (y) Z Z 1 e t L p (t;yx)g (x)dtdxdy Thus, by Hölder’s inequality, and (7.2) for q2 (1; 2), Z T f (x)g (x)dx Cjfj Lp(R d ) jgj Lq (R d ) : The estimate holds for p2 (2;1) as well. Next we have the main apriori estimate for the elliptic equation. Corollary 64. Under the same assumptions in Lemma 63. Let f 2 ~ C 1 R d and u2 ~ C 1 R d be the unique solution to (L )u =f inR d : (7.3) Then for each p2 (1;1) there is a constant C =C (d;p;) such that jL uj Lp(R d ) Cjfj Lp(R d ) Moreover, jL gj Lp(R d ) CjL gj Lp(R d ) ;g2 ~ C 1 R d : Proof. Let f2 ~ C 1 R d ;> 0. There is a unique u2 ~ C 1 R d solving (7.3): According to Lemma 39, L u(x) = Z 1 0 e t EL f (x +Z t )dt;x2R d : By (7.1), T " f (x) = Z L f(xy) Z 1 " e t p (t;y)dtdy (7.4) = Z 1 " e t EL f (x +Z t )dt;x2R d : 94 7.1. ELLIPTIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS By Lemma 63, for each p2 (1;1) there is a constant C =C (d;p;) independent of and such that jT " fj Lp(R d ) Cjfj Lp(R d ) : (7.5) Passing to the limit in (7.5) and (7.4) as "! 0; we have jL uj Lp(R d ) Cjfj Lp(R d ) : (7.6) For any g2 ~ C 1 R d , let f = (L )g2 ~ C 1 R d , hence by (7.6), jL gj Lp(R d ) Cjfj Lp(R d ) =Cj(L )gj Lp(R d ) The proof is completed by passing ! 0. Proof. Proof of Theorem 62 Because J s ;s2R is an isomorphism between Bessel potential spaces, it suffices to prove the case s = 0. Existence. Let f2L p R d . There is a sequence f n 2 ~ C 1 (R d ) such that f n !f in L p . For each n; there is unique u n 2 ~ C 1 R d solving (5.7). Hence (L ) (u n u m ) =f n f m : By Corollary 64 and Lemma 39, jL (u n u m )j Lp(R d ) Cjf n f m j Lp(R d ) ! 0; ju n u m j Lp(R d ) 1 jf n f m j Lp(R d ) ! 0; as n;m!1. Hence there is u2 H ;1 p R d so that u n ! u in H ;1 p R d . In particular, u n ! u and L u n !L u in L p R d . Passing n!1, obviously, (L )u =f in L p R d . Uniqueness. Assume u 1 ;u 2 2 H ;1 p R d solve (5.7): Then u = u 1 u 2 2 H ;1 p R d solves (L )u = 0, i.e.8'2 ~ C 1 R d Z ' (L )u = Z u L 'dx = 0 According to Lemma 39, R ufdx = 08f2 ~ C 1 R d . Hence u = 0 a.e. The statement is proved. 95 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS 7.2 Parabolic Equation 7.2.1 Existence, Uniqueness, and Apriori Estimates for Smooth Input Func- tions Let 2A , and Z t =Z t ;t 0; be the Lévy process associated to it. Let P t (dy) be the distribution of Z t ;t> 0, and for a measurable f 0; T t f (x) = Z f (x +y)P t (dy); (t;x)2E: We represent the solution to (1.1) with smooth input functions using the following operators: T t g (x) =e t Z g (x +y)P t (dy); (t;x)2E; g2 ~ C 1 0;p R d ;p> 1; R f (t;x) = Z t 0 e (ts) Z f (s;x +y)P ts (dy)ds; (t;x)2E; f2 ~ C 1 0;p (E);p> 1; and ~ R (t;x) = Z t 0 e (ts) Z U Z (s;x +y;z)P ts (dy)q (ds;dz); (t;x)2E; 2 ~ C 1 2;p (E)\ ~ C 1 p;p (E) if p 2; 2 ~ C 1 p;p (E) if p2 (1; 2): We remind the reader that hereq (ds;dz) is the martingale measure as given in (1.1). We start by deriving basic estimates of the solution. Lemma 65. Let f2 ~ C 1 0;p (E);g2 ~ C 1 0;p R d ; 2 ~ C 1 2;p (E)\ ~ C 1 p;p (E) for p2 [2;1) and 2 ~ C 1 p;p (E) for p2 (1; 2), then the following estimates hold for any multiindex 2N d 0 , (i)P-a.s. D T g Lp(E) 1 p jD gj Lp(R d ) ; g2 ~ C 1 0;p R d ;p> 1; jD R fj Lp(E) jD fj Lp(E) ; f2 ~ C 1 0;p (E);p> 1; and jD R f (t;)j Lp(R d ) Z t 0 jD f (s;)j Lp(R d ) ds;t 0;p> 1 T t g Lp(R d ) e t jgj Lp(R d ) ;t 0;p> 1; 96 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS (ii) For each p 2; D ~ R p Lp(E) C " p 2 E Z T 0 jD (s;)j p L2;p(R d ) ds + jD j p Lp;p(E) # ; 2 ~ C 1 2;p (E)\ ~ C 1 p;p (E); and for each p2 (1; 2); D ~ R p Lp(E) C jD j p Lp;p(E) ; 2 ~ C 1 p;p (E); where =T^ 1 : Moreover, D ~ R (t;) p Lp(R d ) C ( E " Z t 0 jD (s;)j 2 L2;p(R d ) ds p=2 # +E Z t 0 jD (s;)j p Lp;p(R d ) ds ) ; if p 2, and D ~ R (t;) p Lp(R d ) CE Z t 0 jD (s;)j p Lp;p(R d ) ds;t 0; if p2 (1; 2): Proof. (i) We start with estimates of g, obviously D T t g Lp(R d ) e t jD gj Lp(R d ) and D T t g Lp(E) =jD gj Lp(R d ) Z T 0 e pt dt ! 1=p 1=p jD gj Lp(R d ) : Now we estimate f, by Hölder’s inequality for > 0, jD R fj p Lp(E) p Z T 0 Z t 0 e (ts) jD f (s;)j p Lp(R d ) dsdt p jD fj p Lp(E) : By Hölder’s inequality for 0, jD R fj p Lp(E) Z T 0 t p1 Z t 0 jD f (s;)j p Lp(R d ) dsdt T p p jD fj p Lp(E) : 97 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS (ii) Due to similarity, we only prove (ii) for the casep 2: Let 2 ~ C 1 2;p (E)\ ~ C 1 p;p (E), recall that = Un for some U n 2U with (U n )<1: Obviously, for any multiindex , (t;x)2E; D ~ R (t;x) = Z t 0 e (ts) Z U Z D (s;x +y;z)P ts (dy)q (ds;dz) = Z t 0 e (ts) Z U Z D (s;x +y;z)P ts (dy)p (ds;dz) Z t 0 e (ts) Z U Z D (s;x +y;z)P ts (dy) (dz)ds: By Kunita’s inequality (see Theorem 2.11 of [Kun04]), for t> 0; E Z D ~ R (t;x) p dx CE Z Z t 0 e 2(ts) Z Z U jD (s;x +y;z)j 2 (dz) P ts (dy)ds p=2 dx +CE Z Z t 0 e p(ts) jT ts D (s;x;z)j p (dz)dsdx = B (t) +D (t): By Fubini’s theorem and Minkowski’s inequality, D (t)CE Z t 0 e p(ts) jD (s;)j p Lp;p(R d ) ds;t> 0; and B (t) CE " Z t 0 e 2(ts) jD (s;)j 2 L2;p(R d ) ds p=2 # C 1 p=2 E Z t 0 2e 2(ts) jD (s;)j p L2;p(R d ) ds : Now, Z T 0 D (t)dtC E Z T 0 jD (s;)j p Lp;p(R d ) ds and Z T 0 B (t)dtC p 2 E Z T 0 jD (s;)j p L2;p(R d ) ds: The proof for p2 (1; 2) follows from BDG inequality and the estimate of D (t). In the next Lemma we show existence and uniqueness for smooth input functions f;g and . 98 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS Lemma 66. For 2 A , let f 2 ~ C 1 0;p (E);g 2 ~ C 1 0;p R d ; 2 ~ C 1 2;p (E)\ ~ C 1 p;p (E) for p2 [2;1) and 2 ~ C 1 p;p (E) for p2 (1; 2), then there is unique u2 ~ C 1 0;p (E) solving (1.1). Moreover, u (t;x) =T t g (x) +R f (t;x) + ~ R (t;x); (t;x)2E; and u 1 (t;x) =T t g (x); (t;x)2E; solves (1.1) with f = 0; = 0, u 2 =R f solves (1.1) with g = 0; = 0, and u 3 = ~ R solves (1.1) with f = 0;g = 0: Proof. Uniqueness. Letu 1 ;u 2 2 ~ C 1 0;p (E)solve(1.1)andu =u 1 u 2 . Thenusolves(1.1)withf =g = = 0: Let t2 (0;T ], consider smooth function F (s;y) =e (ts) u (ts;y); 0st;y2R d : Note that for (s;y)2 [0;t]R d , F (t;y) = 0;F (0;y) =e t u (t;y); @ s F (s;y) =e (ts) [@ t u (ts;y)u (ts;y)]: By Itô’s formula for F (s;x +Z s ) =e (ts) [u (ts;x +Z s )]; 0st we have e t u (t;x) =E Z t 0 e (ts) [@ t +L ]u (ts;x +Z s )ds = 0 Hence, u (t;x) = 0 for all (t;x)2E andP-a.s. Existence. We prove that u 1 ;u 2 ;u 3 solve their corresponding equations. Let g2 ~ C 1 0;p R d , by Itô’s formula u 1 (t;x) =T t g (x) solves (1.1) with f = = 0. Let f2 ~ C 1 0;p (E), we fix s2 [0;T ];x2R d , and applying Itô’s formula with e r f(s;x +) and Z r ; 0 rts; we have e (ts) f(s;x +Z ts ) = f(s;x) + Z ts 0 Z R0 e r f(s;x +Z r +y)f(s;x +Z r ) q(dr;dy) + Z ts 0 e r (L )f (s;x +Z r )dr 99 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS Taking expectation on both sides, and integrating with respect to s, we obtain by Fubini’s theorem for each (t;x)2E; Z t 0 e (ts) Ef s;x +Z ts ds = Z t 0 f (s;x)ds + Z t 0 Z ts 0 e r E [L ]f (s;x +Z r )drds = Z t 0 f (s;x)ds + Z t 0 Z t s e (rs) E [L ]f s;x +Z rs drds = Z t 0 f (s;x)ds + Z t 0 Z r 0 e (rs) E [L ]f s;x +Z rs dsdr = Z t 0 f (s;x)ds + Z t 0 Z r 0 e (rs) [L ] Z f (s;x +y)P ts (dy)dsdr Since for each (r;x)2E Z r 0 e (rs) EL f (s;x +Z rs )ds =L u 2 (r;x); it follows that for each (t;x)2E; u 2 (t;x) = Z t 0 f (s;x)ds + Z t 0 [L u 2 (r;x)u 2 (r;x)]dr: Let 2 ~ C 1 2;p (E)\ ~ C 1 p;p (E) if p2 [2;1) or 2 ~ C 1 p;p (E) if p2 (1; 2]. Recall that = Un for some U n 2U with (U n )<1. A simple application of Itô’s formula (similar to u 2 ) and Fubini’s theorem show thatP-a.s. u 3 (t;x) = Z t 0 e (ts) Z U Z (s;x +y;z)P ts (dy)q (ds;dz) = Z t 0 Z U (s;x;z)q (ds;dz) + Z t 0 Z t s e (rs) Z U Z [L (s;x +y;z) (s;x +y;z)]P rs (dy)drq (ds;dz) = Z t 0 Z U (s;x;z)q (ds;dz) + Z t 0 [L u 3 (s;x)u 3 (s;x)]ds; (t;x)2E: 100 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS 7.2.2 Estimate of T g In this section, we derive the main apriori estimate of the T g. The solution associated to the initial value function is given explicitly by T g =G t g; g2 ~ C 1 0;p R d ; where G t (x) = exp (t)p (t;x), (dy) = (dy): We prove that there is C =C (;d;p) so that a.s. jL T gj H ;s p (E) Cjgj B ;s+11=p pp (R d ) : (7.7) Since by Proposition 15,J t :H ;s p R d !H ;st p R d andJ t :B ;s pp R d !B ;st pp R d are isomorphisms for any s;t2R, it is enough to derive the estimate for s = 0: Lemma 67. Let 2 A ;w = w be a continuous O-RV function and A, B hold. Let p2 (1;1). Then there is C =C (;d;p) so that a.s. jL T gj Lp(E) Cjgj B ;11=p pp (R d ) ;g2 ~ C 1 0;p R d : (7.8) Proof. We will use an equivalent norm(see Proposition 15.) Let N > 1 be an integer. Recall that there exists a function 2C 1 0 (R d ) such that supp =f : 1 N 6jj6Ng, ()> 0 if N 1 <jj<N and 1 X j=1 (N j ) = 1 if 6= 0: Let ~ () = (N) + () + N 1 ;2R d : Note that supp ~ N 2 jjN 2 and ~ = . Let ' k =F 1 N k ;k 1; and ' 0 2S R d is defined as ' 0 =F 1 " 1 1 X k=1 N k # : Let 0 () =F' 0 (); ~ 0 () =F' 0 () +F' 1 ();2R d ; ~ ' =F 1 ~ ;' =F 1 . Let ~ ' k = 1 X l=1 ' k+l ;k 1; ~ ' 0 =' 0 +' 1 : 101 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS Note that ' k = ~ ' k ' k ;k 0. For j 1; F [L T g (t;)' j ] = w N j 1 ~ N j N j exp n w N j 1 ~ N j N j tt o ~ N j ^ g j (); and F [L T g (t;)' 0 ] = () expf ()ttg ~ 0 () ^ g 0 (); where g j =g' j ;j 0: LetZ j =Z ~ N j ;j 1: Let 2C 1 0 R d ; 0 = 2supp and ~ = ~ , =F 1 . Denoting =F 1 ~ 0 ; we have L T g (t;)' j = w N j 1 H ;j t g j ;j 1; (7.9) L T g (t;)' 0 = H ;0 t g 0 ;t> 0; where for j 1; H ;j t (x) = N jd H ;j w(N j ) 1 t N j x ; (t;x)2E; H ;j t = e w(N j )t (L ~ N j )E ~ ' +Z j t ;t> 0; and H ;0 t (x) =e t L E ( +Z t ); (t;x)2E: By Lemma 5 in [MX19] and Corollary 64 , sup j Z L ~ N j dx<1. Hence by Lemma 19, Z H ;j t dx Z L ~ N j dx Z E ~ ' +Z j t dx Ce ct ;t> 0;j 1; 102 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS and hence, Z H ;j t dxC exp n cw N j 1 t o ;t> 0;j 1: (7.10) and by Lemma 35, Z H ;0 t dxC 1 t ^ 1 ;t> 0: (7.11) It follows by Proposition 15 and (7.9) that jL T g (t)j p Lp(R d ) C 0 @ 1 X j=1 w N j 1 H ;j t g j 2 1 A 1=2 p Lp(R d ) +C Z H ;0 t g 0 p dx: Hence, jL T g (t)j p Lp(R d ) C 1 X j=0 w N j 1 H ;j t g j p Lp(R d ) if p2 (1; 2]; and, by Minkowski’s inequality, jL T g (t)j p Lp(R d ) C 0 @ 1 X j=1 Z w N j 1 H ;j t g j p dx 2=p 1 A p=2 +C Z H ;0 t g 0 p dx if p> 2. Now, by (7.10), Z w N j 1 H ;j t g j p dx Z H ;j t dx p Z w N j 1 g j p dx Cw N j p exp n cw N j 1 t o jg j j p Lp(R d ) if j 1; and, by (7.11), Z H ;0 t g 0 p dxC 1 t ^ 1 p Z jg 0 j p dx: Therefore for p2 (1; 2]; Z 1 0 jL T g (t)j p Lp(R d ) dtC 1 X j=0 w N j (11=p) jg j j Lp(R d ) p ; and (7.8) follows by Proposition 15. 103 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS Let p> 2: In this case, Z 1 0 jL T g (t)j p Lp(R d ) dtC[G +jg 0 j p Lp(R d ) ]; where G = Z 1 0 0 @ 1 X j=1 exp n cw N j 1 t o k 2 j 1 A p=2 dt with c> 0 and k j =w N j 1 jg j j Lp(R d ) ;j 1: Now, let B = j :a (t)N j where a (t) = inffr :w (r)tg;t> 0. Then 1 X j=1 e cw(N j ) 1 t k 2 j = X j2B ::: + X j= 2B ::: =D (t) +E (t);t> 0: Let 0< p 2 <p 1 ^p 2 , by Hölder’s inequality, D (t) C 1 X j=1 fj:a(t)N j g a (t) N j a (t) N j k 2 j C 0 @ 1 X j=1 fj:a(t)N j g a (t) N j p p2 1 A 1 2 p 0 @ 1 X j=1 fj:j:a(t)N j g a (t) N j p 2 k p j 1 A 2 p = CD p2 p 1 D 2 p 2 : Denoting 0 =p= (p 2); we have for t> 0; D 1 (t) = 1 X j=1 fj: a(t) N j 1g a (t) N j 0 C Z 1 0 f a(t) N x 1g a (t) N x 0 dxC Z 1 0 y 0dy y <1: Applying Corollary 14 with 1> p 2 1 p1 _ 1 p2 , and arbitrary > 1; Z 1 0 D p=2 2 dtC 1 X j=1 Z 1 0 fj: a(t) N j 1g a (t) N j p 2 k p j dt C 1 X j=1 Z w(N j ) 0 a (t) N j p 2 dtk p j C 1 X j=1 w N j a w N j N j ! p 2 k p j : 104 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS Hence Z 1 0 D p=2 2 dtC 1 X j=1 w N j k p j : Now, we estimate the second term E (t);t> 0. By Hölder’s inequality, for t> 0, E (t) = X a(t)>N j e cw(N j ) 1 t w N j 2 jg j j 2 Lp 0 @ X a(t)N j e cw(N j ) 1 t 1 A p2 p 0 @ X a(t)N j e cw(N j ) 1 t k p j 1 A 2 p : Changing the variable of integration; we have, by Lemma 6, for some l> 0, X a(t)N j e cw(N j ) 1 t X a(t)N j exp c w (a (t)) w (N j ) X a(t)N j exp ( c a (t) N j l ) = X N j a(t)1 exp n c N j a (t) l o C Z N x a(t)1 exp n c (N x a (t)) l o dx =C Z 1 1 exp cy l dy y : Hence, Z 1 0 E (t) p=2 dt C Z 1 0 X w(N j ) 1 t1 e cw(N j ) 1 t k p j dt C X j w N j k p j . The estimate (7.8) is proved. 7.2.3 Estimates of R f; ~ R : verification of Hörmander Conditions In this section, we derive main apriori estimates for R f; ~ R . First we show that for each p> 1 there is C > 0 so that jL R fj H ;s p (E) Cjfj H ;s p (E) ; f2 ~ C 1 0;p (E): (7.12) By applying J s , taking expectation and using standard density argument, it is sufficient to prove (7.12) for s = 0 and f2 ~ C (E) i.e. jL R fj Lp(E) Cjfj Lp(E) f2 ~ C (E): (7.13) 105 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS We will apply the classical Calderón-Zygmund theorem e.g. Theorem 3 of Chapter I in ([Ste93]). For this purpose, we routinely justify that ((t;x); (s;y)) = w (jtsj) +jxyj; (t;x); (s;y)2 RR d is a quasi-distance inR d+1 in the sense of Section 2.4 of Chapter I in [Ste93]. Indeed, (i) ((t;x); (s;y)) = 0 if (t;x) = (s;y) (ii) ((t;x); (s;y)) = ((t;x); (s;y)) (iii) By Lemma 6, for some C > 0, w (jtsj)w (jtrj +jrsj)C (w (jtrj) +w (jrsj));r2R: Wecommenton a simple calculation, which will beused in thenext Lemma, letk> 0, then by integration by-part, Z 1 1 e kt t 1 dtC e k k : (7.14) In the next Lemma we show that a version a Hörmander condition, (7.15), holds and thus a Calderón- Zygmund theorem applies; as a result, (7.13) holds. Lemma 68. Let 2 A ;w = w be a continuous O-RV function and A, B hold. Let 1 > q 1 _q 2 , 0< 2 1 if 2 (1; 2); 1 1 if 2 (0; 1); 1 2 if 2 [1; 2): Let 2A sign ; and assume that Z jyj1 jyj 1 d f jj R + Z jyj>1 jyj 2 d f jj R M;R> 0: Let > 0 and K (t;x) =e t L p (t;x) [;1] (t);t> 0;x2R d ; where (dy) = (dy). There exist C 0 > 1 and C so that I = Z Q C 0 (0) c (t;x)jK (ts;xy)K (t;x)jdxdtCM (7.15) for alljsjw ();jyj;> 0, where Q C0 (0) = (w (C 0 );w (C 0 ))fx :jxj<C 0 g: 106 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS Proof. We will follow splitting idea in Corollary 5.18 of [KK17] (see also Corollary 4.11 of [KKK19].) First, we assume that s6= 0. By Lemma 6 we choose C 0 > 3 such that w (C 0 )> 3w ();> 0. We split I = Z 2jsj 1 Z ::: + Z 1 2jsj Z ::: =I 1 +I 2 Since w (C 0 ) > 3w (); > 0; it follows by Lemma 35, Corollary 14 and Lemma 6 for a, denoting k 0 =C 0 1; I 1 C Z 3jsj 0 Z jxj>k0a(jsj) e t L p (t;x) dxdt CM 1 a (jsj) 2 Z 3jsj 0 t 1 a (t) 2 dtCM a (3jsj) 2 a (jsj) 2 CM: Now, I 2 Z 1 2jsj Z Q c C 0 (0) e (ts) [;1] (ts) L p (ts;xy)L p (ts;x) dxdt + Z 1 2jsj Z Q c C 0 (0) e (ts) j [;1] (ts)L p (ts;x) [;1] (t)L p (t;x)jdxdt + Z 1 2jsj Z Q c C 0 (0) e (ts) e t j [;1] (t)L p (t;x)jdxdt = I 2;1 +I 2;2 +I 2;3 : We split the estimate ofI 2;1 into two cases. Case 1. Assumejyja (2jsj): Then, by Lemma 35 , Corollary 14 , and Lemma 6, I 2;1 CMjyj Z 1 2jsj (ts) 1 a (jtsj) 1 dt CMjyja (2jsjs) 1 CMjyja (jsj) 1 CM jyj a (2jsj) a (3jsj) a (jsj) CM: Case 2. Assumejyj>a (2jsj), i.e. jyj>a (2jsj) and a 1 ()a 1 (jyj) 2jsj. We split I 2;1 = Z 2jsj+a 1 (jyj) 2jsj Z ::: + Z 1 2jsj+a 1 (jyj) Z ::: =I 2;1;1 +I 2;1;2 : 107 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS If 2jsj t 2jsj +a 1 (jyj), then 0 t 3a 1 () = 3w () < w (C 0 ). Hencejxj C 0 a (2jsj) +jyj and jxyj (C 0 1) =k 0 k 0 2 (a (2jsj) +jyj) a (2jsj) +jyj if (t;x) = 2Q C0 (0): and by Lemma 6, a 3jsj +a 1 (jyj) a (2jsj) +jyj a 5 2 a 1 (jyj) a (2jsj) +jyj a 5 2 a 1 (jyj) a (a 1 (jyj)) a a 1 (jyj) jyj C: (7.16) By Lemma 35, Corollary 14 and (7.16), I 2;1;1 C Z 2jsj+a 1 (jyj) 2jsj Z jxj>a(2jsj)+jyj L p (ts;x) dtdx CM (a (2jsj) +jyj) 2 Z 2jsj+a 1 (jyj) 2jsj (ts) 1 a (jtsj) 2 dt CM (a (2jsj) +jyj) 2 Z 3jsj+a 1 (jyj) 0 t 1 a (t) 2 dt CM a 3jsj +a 1 (jyj) 2 (a (2jsj) +jyj) 2 CM: By Lemma 35, Corollary 14, I 2;1;2 Z 1 2jsj+a 1 (jyj) Z R d L p (ts;xy)L p (ts;x) dx dt CMjyj Z 1 2jsj+a 1 (jyj) (ts) 1 a (jtsj) 1 dr CMjyja 2jsj +a 1 (jyj)s 1 CMjyja a 1 (jyj) +jsj 1 CMjyjjyj 1 CM: Hence,I 2;1 =I 2;1;1 +I 2;1;2 C. 108 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS By Lemma 35, I 2;2 Z 1 2jsj Z Q c C 0 (0) L p (ts;x)L p (t;x) dxdt + Z "+jsj "_jsj Z L p (t;x) dxdt CM: Finally, by Lemma 35 and (7.14), I 2;3 = Z 1 2jsj Z Q c C 0 (0) e (ts) e t j [;1] (t)L p (t;x)jdxdt CM 1e s Z 1 2jsj e t t 1 dtCM 1e s Z 1 1 e 2jsjt t 1 dt CM 1e s e 2jsj 2jsj CM: The proof is complete. Remark 69. With notation of the proof above, for the case s = 0, we may assume without loss of generality thatjyj> 0, and we only need to estimateI 2;1 =I 2;1;1 +I 2;1;2 withjyj> 0 (Case 2.). The estimate ofI 2;1;1 remains unchanged. To estimateI 2;1;2 , we replace 2jsj +a 1 (jyj) with 2jsj +a 1 (jyj) + with > 0, then using Fatou’s lemma we pass ! 0. Remark 70. Although we write Lemma 68 with general , in this manuscript we only need the result for =. Corollary 71. Under the same assumptions as Lemma 68. Let p 2 (1;1), then there exists C > 0 independent of such that jK fj Lp(R d+1 ) Cjfj Lp(R d+1 ) ; f2C 1 0 R d+1 (7.17) here the convolution is in both t and x variables. Consequently, jL R fj Lp(E) Cjfj Lp(E) ; f2 ~ C (E): In particular,jL R fj Lp(E) Cjfj Lp(E) ; f2 ~ C (E): 109 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS Proof. (i) We start by deriving the L 2 estimate, F (K f) (t;) = () Z expf () (ts)(ts)g ^ f (s;) [;1] (ts)ds Therefore, by Lemma 20 and Hölder’s inequality, jF (K f) (t;)jCw jj 1 1 Z t 1 exp cw jj 1 1 (ts) ^ f (s;) ds C Z t 1 w jj 1 1 exp cw jj 1 1 (ts) ^ f (s;) 2 ds 1=2 Therefore, by Fubini’s theorem, Z Z jF (K f) (t;)j 2 dtd C Z Z Z t 1 w jj 1 1 exp cw jj 1 1 (ts) ^ f (s;) 2 dsdtd C Z Z ^ f (s;) 2 dsd: and L 2 estimate follows from Plancherel’s theorem i.e.jK fj L2(R d+1 ) Cjfj L2(R d+1 ) : (ii) By Lemma 68, the estimate for p = 2, and Theorem 3 of Chapter I in [Ste93], the desired estimate for p2 (1; 2) is proved. (iii) We now prove the estimate for p2 (1; 2) by duality argument. First note that L p (t;x) =L p (t;x);t> 0;x2R d and let ~ K " (t;x) =K " (t;x) =e t L p (t;x) [";1) (t); (t;x)2R d+1 ; and ~ K " g (s;y) = Z e (st) ~ K " (st;yx)g (t;x)dtdx; (s;y)2R d+1 : 110 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS Let 1=p+1=q = 1;h;g2C 1 0 R d+1 . Then, denoting ~ g (t;x) =g (t;x); (t;x)2R d+1 ; we have (by Fubini’s theorem and changing the variable of integration) Z K h (t;x)g (t;x)dtdx = Z Z Z Z e (ts) K " (ts;xy)h (s;y)dsdyg (t;x)dtdx = Z Z Z Z e (st) ~ K " (st;yx)g (t;x)dtdxh (s;y)dsdy = Z Z ~ K " ~ g (s;y)h (s;y)dsdy; and (7.17) holds for ~ K " ~ g and q2 (1; 2). Therefore, by Hölder’s inequality, Z K h (t;x)g (t;x)dtdx ~ K " ~ g (s;y) Lq (R d+1 ) jhj Lp(R d+1 ) Cjgj Lq (R d+1 ) jhj Lp(R d+1 ) and (7.17) holds for p> 2 as well, that is for all p2 (1;1). (iv) Now consider h = [0;T] f for f2 ~ C (E), then by a standard density argument, jK hj Lp(R d+1 ) Cjhj Lp(R d+1 ) =Cjfj Lp(E) : Recall the definition K (t;x) =e t L p (t;x) [;1] (t), hence, for (t;x)2E, K h (t;x) = Z t 0 Z e (ts) L p (ts;xy) [;1] (ts)f (s;y)dsdy Clearly,jK hL R fj Lp(E) ! 0. Hence, passing ! 0, since C is independent of we have jL R fj Lp(E) Cjfj Lp(E) : Let2A ;w =w be a continuous O-RV function andA,B hold. Now we will show that forp2 [2;1), there is C so that for all 2 ~ C 1 2;p (E)\ ~ C 1 p;p (E), L ~ R H ;s p (E) C h jj B ;s+11=p p;pp (E) +jj H ;s+1=2 2;p (E) i ; (7.18) 111 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS and for p2 (1; 2), there is C so that for all 2 ~ C 1 p;p (E), L ~ R H ;s p (E) Cjj B ;s+11=p p;pp (E) : (7.19) Due to Proposition 15, it is enough to consider the case s = 0. Let "> 0; G ;" s;t (x) = exp ( (ts))p (ts;x) [";1] (ts); 0<s<t;x2R d ; where (dy) = (dy). In addition, we may replace L with L ; = sym due to Corollary 64 (i.e. jL fj Lp(R d ) ujL fj Lp(R d ) ;f2 ~ C 1 R d ) Denote Q (t;x) = Z t 0 Z U ~ " (s;x;z)q (ds;dz); (t;x)2E; ~ " (s;x;z) = Z L G ;" s;t (xy) (s;y;z)dy; (s;x)2E: If 2p<1, then by Kunita’s inequality, E Z T 0 jQ (t;)j p Lp(R d ) dt CE 8 < : Z T 0 Z t 0 Z U ~ " (s;;z) 2 (dz)ds 1=2 p Lp(R d ) dt 9 = ; + CE ( Z T 0 Z t 0 Z U ~ " (s;;z) p Lp(R d ) (dz)dsdt ) =C (EI 1 +EI 2 ): If 1<p< 2, then by BDG inequality, E Z T 0 jQ (t;)j p Lp(R d ) dtCEI 2 : Estimate of EI 2 . Let B t g (x) =e t Eg (x +Z t ); (t;x)2E;g2 ~ C 1 0;p R d . Then I 2 = Z T 0 Z t 0 Z U ~ " (s;;z) p Lp(R d ) (dz)dsdt = Z T 0 Z t 0 Z U L G ;" s;t (s;;z) p Lp(R d ) (dz)dsdt Z T 0 Z t 0 Z U L B ts (s;;z) p Lp(R d ) (dz)dsdt = Z U Z T 0 Z T s L B ts (s;;z) p Lp(R d ) dtds (dz) 112 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS It follows from Proposition 15 and Lemma 67 that for p> 1; EI 2 E Z U Z T 0 Z T s L B ts (s;;z) p Lp(R d ) dtds (dz) CE Z U Z T 0 1 X j=0 w N j (11=p) j (s;;z)' j j Lp(R d ) p ds (dz) = Cjj p B ;11=p p;pp (E) : Hence (7.19) holds for p2 (1; 2). We prove that for p 2; EI 1 Cjj H ;1=2 2;p (E) ; 2 ~ C 1 2;p (E); (7.20) Denote K " (t;x) =e t L ;1=2 p (t;x) [";1] (t);t> 0;x2R d , EI 1 = E Z T 0 Z t 0 Z U ~ " (s;;z) 2 (dz)ds 1=2 p Lp(R d ) dt = E " Z t 0 Z U Z L G ;" s;t (xy) (s;y;z)dy 2 (dz)ds # 1=2 p Lp(E) = E " Z t 0 Z U Z L ;1=2 G ;" s;t (xy)L ;1=2 (s;y;z)dy 2 (dz)ds # 1=2 p Lp(E) = E " Z t 0 Z U Z K " (ts;xy)L ;1=2 (s;y;z)dy 2 (dz)ds # 1=2 p Lp(E) : It is enough to show that Z T 0 Z Z t 0 jK " (ts;) (s;) (x)j 2 V2 ds p 2 dxdt (7.21) C Z T 0 Z j (t;x)j p V2 dxdt; 2 ~ C 1 2;p (E): where V 2 = L 2 (U;U; ), C is independent of "; and - by setting to be L ;1=2 later. This will be shown by verifying Hörmander condition (7.22) in Lemma 72 below. In the following statement we show that a version a stochastic Hörmander condition holds which implies (7.21) and thus (7.20)- see Theorem 3.1, and Theorem 2.5 of [KK17]. 113 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS Lemma 72. Let 2A sym : Let 2A ;w =w be a continuous O-RV function and A, B hold. Assume 1 >q w 1 _q w 2 ; 0< 2 1 if 2 (1; 2); 1 1 if 2 (0; 1); 1 2 if 2 [1; 2); and Z jyj1 jyj 1 de R + Z jyj>1 jyj 2 de R M;R> 0: Let > 0 and K (t;x) =e t L ; 1 2 p (t;x) [;1] (t);t> 0;x2R d ; where (dy) = (dy). Then there exists C 0 > 0 and N > 0 such that for alljsjw ();jyj; > 0, we have I = Z Z Q C 0 (0) cjK " (ts;xy)K " (t;x)jdx 2 dtN; (7.22) where Q C0 (0) = (w (C 0 );w (C 0 ))fx :jxj<C 0 g: Proof. We will follow splitting idea in Corollary 5.18 of [KK17] (see also Corollary 4.11 of [KKK19].) The proof is similar to Lemma 68 - essentially following the same splitting. Due to similarity, we skip some details. We will assume that s6= 0, the case of s = 0 is handled as in Remark 69. By Lemma 6 we choose C 0 > 3 such that w (C 0 )> 3w ();> 0. We split I = Z 2jsj 1 Z ::: 2 dt + Z 1 2jsj Z ::: 2 dt =I 1 +I 2 : Sincew (C 0 )> 3w ();> 0, it follows by Lemma 36, Lemma 6 and Corollary 14 with k 0 =C 0 1 and 2 (0; 2 2 ), jI 1 j C Z 3jsj 0 " Z jxj>k0a(jsj) L ; 1 2 p (t;x) dx # 2 dt C Z 3jsj 0 t 1 2 a (t) (k 0 a (jsj)) 2 dtCa (jsj) 2 Z 3jsj 0 a (t) 2 dt t Ca (jsj) 2 a (3jsj) 2 C: Now, 114 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS jI 2 j 3 Z 1 2jsj Z Q c C 0 (0) L ; 1 2 p (ts;xy)L ; 1 2 p (ts;x) dx 2 dt + 3 Z 1 2jsj Z Q c C 0 (0) j [";1] (ts)L ; 1 2 p (ts;x) [";1] (t)L ; 1 2 p (t;x)jdx 2 dt + 3 Z 1 2jsj Z Q c C 0 (0) e (ts) e t j [;1] (t)L ; 1 2 p (t;x)jdx 2 dt = I 2;1 +I 2;2 +I 2;3 : We split the estimate ofI 2;1 into two cases. Case 1. Assumejyja (2jsj). Then by Lemma 36, Lemma 6 and Corollary 14 I 2;1 C Z 1 2jsj jyj 2 (ts)a (ts) 2 dt Cjyj 2 Z 1 2jsj 1 a (ts) 2 (ts) 1 dt Cjyj 2 Z 1 jsj a (t) 2 dt t Cjyj 2 a (jsj) 2 C: Case 2. Assumejyj>a (2jsj) i.e. jyj>a (2jsj) and a 1 ()a 1 (jyj) 2jsj. We split I 2;1 = Z 2jsj+a 1 (jyj) 2jsj Z ::: 2 + Z 1 2jsj+a 1 (jyj) Z ::: 2 =I 2;1;1 +I 2;1;2 : Hence, with 2 (0; 2 2 ), by Lemma 36, Lemma 6 and Corollary 14, I 2;1;1 C Z 2jsj+a 1 (jyj) 2jsj " Z jxj>a(2jsj)+jyj L ; 1 2 p (ts;x) dx # 2 dt C (a (2jsj) +jyj) 2 Z 2jsj+a 1 (jyj) 2jsj a (ts) 2 dt (ts) C (a (2jsj) +jyj) 2 a 3jsj +a 1 (jyj) 2 C: 115 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS Then by Lemma 36 and Corollary 14, I 2;1;2 C Z 1 2jsj+a 1 (jyj) Z R d L ; 1 2 p (ts;xy)L ; 1 2 p (ts;x) dx 2 dt C Z 1 2jsj+a 1 (jyj) jyj 2 a (ts) 2 dt ts Cjyj 2 a 2jsj +a 1 (jyj)s 2 Cjyj 2 a jsj +a 1 (jyj) 2 C: Hence,I 2;1 C. Since I 2;2 Z 1 2jsj Z Q c C 0 (0) L ; 1 2 p (ts;x)L ; 1 2 p (t;x) dx 2 dt + Z "+jsj "_jsj Z L ; 1 2 p (t;x) dx 2 dt = I 2;2;1 +I 2;2;2 ; it follows by Lemma 36 that I 2;2;1 Z 1 2jsj Z Q c C 0 (0) L ; 1 2 p (ts;x)L ; 1 2 p (t;x) dx 2 dt C: Clearly,I 2;2;2 C as well, and thusI 2;2 C. Finally, by Lemma 36 and (7.14), I 2;3 = Z 1 2jsj Z Q c C 0 (0) e (ts) e t j [;1] (t)L ; 1 2 p (t;x)jdx 2 dt C 1e s 2 Z 1 2jsj e 2t t 1 dtC 1e s 2 Z 1 1 e 4jsjt t 1 dt C 1e s 2 e 4jsj 4jsj C: The proof is complete. Corollary 73. Under the same assumptions as Lemma 72. Let p 2 [2;1), then there exists C > 0 independent of ; such that 116 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS Z T 0 Z Z t 0 jK " (ts;) (s;) (x)j 2 V2 ds p 2 dxdt (7.23) C Z T 0 Z j (t;x)j p V2 dxdt; 2 ~ C 1 2;p (E): where V 2 =L 2 (U;U; ), C is independent of "; and . Consequently, (7.20) holds. Proof. For p = 2, by Plancherel’s theorem and Fubini’s theorem, denoting ^ =F, using Lemma 20, Z T 0 Z Z t 0 jK " (ts;) (s;) (x)j 2 V2 dsdxdt = Z T 0 Z Z t 0 L ; 1 2 G ; s;t (s;) (x) 2 V2 dsdxdt = Z T 0 Z Z t" 0 Z U jexpf2 ( ()) (ts)gjj ()j ^ (s;;z) 2 (dz)dsddt C Z T 0 Z Z U ^ (s;;z) 2 (dz)dds =C Z T 0 Z Z U j (s;x;z)j 2 (dz)dxds Hence (7.23) follows for p = 2. By Lemma 72, the estimate for p2 [2;1) follows from Theorem 3.1 of [KK17] (see also Theorem 2.5 in [KK17].) Now we apply (7.23) to L ;1=2 . In fact, by extending norm equivalence (2.1) to separable Hilbert space-valued functions. Z T 0 Z Z t 0 K " (ts;)L ;1=2 (s;) (x) 2 V2 ds p 2 dxdt C Z T 0 Z L ;1=2 (t;x) p V2 dxdt Cjj p H ;1=2 2;p (E) : and (7.20) follows. By passing ! 0, we derive (7.18). Remark 74. The estimates in Lemmas 68 and 72 are independent of > 0. Now, Lemmas 65- 72 and their Corollaries, specifically (7.7), (7.12), (7.18), and (7.19) allow us to derive all the estimates of u and L u claimed in Theorem 2 for the solution u of (1.1) with smooth input functions. We now prove the main theorem for general input functions. 117 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS 7.2.4 Proof of the Main Theorem We finish the proof of Theorem 2 in a standard way. Since by Proposition 15, J t : H ;s p R d ;l 2 ! H ;st p R d ;l 2 , J t :H ;s p R d !H ;st p R d and J t :B ;s pp R d !B ;st pp R d is an isomorphism for any s;t2R, it is enough to derive the statement for s = 0: Let f2L p (E);g2B ;11=p pp R d ; and 2 B ;11=p p;pp (E)\H ;1=2 2;p (E) if p 2; 2 B ;11=p p;pp (E) if p2 (1; 2): According to Lemma 30, there are sequences f n 2 ~ C 1 0;p (E);g n 2 ~ C 1 0;p R d ; n 2 ~ C 1 2;p (E)\ ~ C 1 p;p (E) if p 2, and n 2 ~ C 1 p;p (E) if p2 (1; 2), such that f n !f inL p (E);g n !g inB ;11=p pp R d ; and n ! inB ;11=p p;pp (E)\H ;1=2 2;p (E) if p 2; n ! inB ;11=p p;pp (E) if p2 (1; 2): For each n; there is unique u n 2 ~ C 1 0;p (E) solving (1.1). Hence for u n;m =u n u m; we have @ t u n;m = (L )u n;m +f n f m + Z U ( n m )q (dt;dz); u n;m (0;x) = g n (x)g m (x);x2R d : By estimates in Theorem 2 for smooth inputs (see Remark 74), jL u n;m j Lp(E) C[jf n f m j Lp(E) +jg n g m j B ;11=p pp (R d ) +j n m j B ;11=p p;pp (E) +j n m j H ;1=2 2;p (E) ]; if p 2 and jL u n;m j Lp(E) C[jf n f m j Lp(E) +jg n g m j B ;11=p pp (R d ) +j n m j B ;11=p p;pp (E) : 118 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS if p2 (1; 2). By Lemma 65, ju n;m j Lp(E) C[ jf n f m j Lp(E) + 1=p jg n g m j Lp(R d ) + 1=p j n m j Lp;p(E) + 1=2 j n m j L2;p(E) ] if p 2, and ju n;m j Lp(E) C[ jf n f m j Lp(E) + 1=p jg n g m j Lp(R d ) + 1=p j n m j Lp;p(E) ] ifp2 (1; 2). Hence there isu2H ;1 p (E) so thatu n !u inH ;1 p (E). In particular,u n !u andL u n !L u inL p (E). Moreover, by Lemma 65, sup tT ju n (t)u (t)j Lp(R d ) ! 0; (7.24) and u isL p R d -valued continuous. By Lemma 93 (see Appendix) and Remark 24, sup tT Z t 0 Z U n q (ds;dz) Z t 0 Z U q (ds;dz) Lp(R d ) ! 0 (7.25) as n!1 in probability. Hence (see (7.24)-(7.25)) we can pass to the limit in the equation u n (t) =g n + Z t 0 [L u n (s)u n (s) +f n (s)]ds + Z t 0 Z U n q (ds;dz); 0tT: (7.26) Obviously, (7.26) holds foru;g andf; . We proved the existence part of Theorem 2. The solution estimates follows easily from passing n!1 as well. Uniqueness. Assume u 1 ;u 2 2H ;1 p (E) solve (1.1): Thenu =u 1 u 2 2H ;1 p (E) solves (1.1) with f = 0;g = 0; = 0. Now, let '2 ~ C 1 (E), and ~ ' (t;x) =' (Tt;x); (t;x)2E. By Lemma 66, there is unique ~ k2 ~ C 1 0;p (E) solving (1.1) with f = ~ ';g = 0; = 0 and instead of . Let k (t;x) = ~ k (Tt;x); (t;x)2 E. Then 119 7.2. PARABOLIC EQUATION CHAPTER 7. PROOF OF THE MAIN RESULTS @ t k +L kk +' = 0 in E and k (T ) =k (T;) = 0. Integrating by parts, Z E 'u = Z E u @ t kL k +k = Z E k (@ t uL u +u) = 0: Hence R E u' dtdx = 08'2 ~ C 1 (E). Hence u = 0 a.e. Theorem 2 is proved. 120 Chapter 8 Time Regularity In this chapter, we investigate time regularity of the following deterministic equation. du (t;x) = [L u (t;x)u (t;x) +f (t;x)]dt u (0;x) =g (x); (t;x)2E; (8.1) with 0: Proposition 75. Let p2 (1;1);s2R. Let 2A ; w =w be continuous O-RV function and A;B hold. For each f 2 H ;s p (E);g2 B ;s+11=p pp R d , we denote u f ;u g 2 H ;s+1 p (E) the unique solution of (8.1) with g = 0 and f = 0 respectively. Then the following estimates hold. For 2 1 p ; 1 i , there is C > 0 independent of T such that for any t;t 0 2 [0;T ], L ;1 (u f (t)u f (t 0 )) H ;s p (R d ) Cjtt 0 j ( 1 p ) jfj H ;s p (E) : Moreover, for 1 2 h 0; 1 p and 2 2 [0; 1], there exists C > 0 independent of T such that for any t 0 ;t2 [0;T ] , L ;1 (u g (t)u g (t 0 )) H ;s p (R d ) C jtt 0 j +jtt 0 j 1 +jtt 0 j 2 2 jgj B ;s+11=p pp (R d ) : Proof. Without loss of generality, we assume that f2 ~ C 1 (E);g2 ~ C 1 R d . By Lemma 66, u f (t) = Z t 0 G ts f (s)ds;t 0 u g (t) = G t g;t 0 121 CHAPTER 8. TIME REGULARITY where G t (x) = e t p (t;x) = e t G t (x);t > 0;x2R d ;and p is the probability density associated to Z t and (dy) = (dy): G t is interpreted as the Dirac delta function if t = 0: Clearly, by Proposition 15, it suffices to consider the case of scale s = 0. Throughout this proof the constant C change from line to line but remains independent of T. Without loss of generality we assume that t 0 t. 1. Estimate of u f (t): Following closely the proof of Proposition 2 of [MP14], for r;l 0, u f (l +r)u f (l) = Z l+r 0 e (l+rs) G l+rs f (s)ds Z l 0 e (ls) G ls f (s)ds = Z l+r l e (l+rs) G l+rs f (s)ds + e r 1 Z l 0 e (ls) G l+rs f (s)ds + Z l 0 e (ls) [G l+rs G ls ]f (s)ds =A 1 (l;r) +A 2 (l;r) +A 3 (l;r) and thus by taking L ;1 on both sides, L ;1 u f (l +r)L ;1 u f (l) =L ;1 A 1 (l;r) +L ;1 A 2 (l;r) +L ;1 A 3 (l;r): Using the fact that G t+s =G t G s ;t;s> 0; A 3 (l;r) = Z l 0 e (ls) [G l+rs G ls ]f (s)ds =G r u f (l)u f (l): Examining the solution of (1.1) for the initial value function, for h2 ~ C 1 R d , r> 0, G r hh = Z r 0 L G s hds: For 2 (0; 1], we have, by Lemma 36, jG r hhj Lp(R d ) Z r 0 jL G s hj Lp(R d ) ds Z r 0 L ;1 G s L ; h Lp(R d ) ds C Z r 0 s 1 dsjL ; hj Lp(R d ) =Cr jL ; hj Lp(R d ) : (8.2) Let 2 1 p ; 1 i , applying (8.2) to h =L ;1 u f (l), 122 CHAPTER 8. TIME REGULARITY L ;1 A 3 (l;r) Lp(R d ) Cr jL u f (l)j Lp(R d ) : We now estimate A 1 . By Lemma 35, and Hölder’s inequality with 1=p + 1=q = 1; L ;1 A 1 (l;r) Lp(R d ) = Z r 0 e s L ;1 G s f (l +rs)ds Lp(R d ) = Z r 0 e s L ;1 G s f (l +rs)ds Lp(R d ) Z r 0 e s L ;1 G s L1(R d ) jf (l +rs)j Lp(R d ) ds C Z r 0 s (1)q ds 1 q Z r 0 jf (l +rs)j p Lp(R d ) ds 1 p =Cr 1 p Z r 0 jf (l +rs)j p Lp(R d ) ds 1 p : We now deal with A 2 , with 2 [0; 1], by Lemma 35 and Hölder’s inequality, for any 2 [0; 1] we have L ;1 A 2 (l;r) Lp(R d ) = e r 1 Z l 0 e (ls) L ;1 G l+rs f (s)ds Lp(R d ) C e r 1 Z l 0 e (ls) (l +rs) 1 jf (s)j Lp(R d ) ds C e r 1 r 1 " Z l 0 e (ls) ds #1 q " Z l 0 jf (s)j p Lp(R d ) ds #1 p Cr +1 1 q jfj Lp(E) : We have by letting = 1, then + 1 = L ;1 A 2 (l;r) L P (R d ) Cr 1 p jfj L P (E) : To finish the estimate of u f (t), we repeat the argument in Proposition 2 of [MP14] with slight modifi- cations. The details are included for completeness. Precisely, applying Lemma 7.4 in [Kry99], combining estimates for A 1 ;A 2 ;A 3 , fixing > 1 p such that >, L ;1 (u f (t)u f (t 0 )) p Lp(R d ) C (tt 0 ) p1 Z tt 0 0 dr r 1+p Z tr t 0 X i=1;2;3 L ;1 A i (r;l) p Lp(R d ) dl =B 1 +B 2 +B 3 123 CHAPTER 8. TIME REGULARITY Estimating term by term, B 1 C (tt 0 ) p1 Z tt 0 0 dr r 1+p Z tr t 0 r p1 Z r 0 jf (l +rs)j p Lp(R d ) dsdl C (tt 0 ) p1 Z tt 0 0 dr r 2+()p Z tr t 0 Z r 0 jf (l +rs)j p Lp(R d ) dsdl =C (tt 0 ) p1 Z tt 0 0 dr r 2+()p Z r 0 Z ts t 0 +rs jf (l)j p Lp(R d ) dlds C (tt 0 ) p1 Z tt 0 0 rdr r 2+()p jfj p Lp(E) =C (tt 0 ) p1 jfj p Lp(E) : We now estimate B 2 , B 2 C (tt 0 ) p1 Z tt 0 0 dr r 1+p Z tr t 0 r p dljfj p Lp(E) =C (tt 0 ) p1 Z tt 0 0 dr r 1+()p (tt 0 r)jfj p Lp(E) C (tt 0 ) p1 jfj p Lp(E) : Finally, using the estimate in Theorem 2, B 3 C (tt 0 ) p1 Z tt 0 0 dr r 1+p Z tr t 0 r p jL u f (l)j p Lp(R d ) dl C (tt 0 ) p1 Z tt 0 0 dr r 1+()p jL u f j p Lp(E) C (tt 0 ) p1 Z tt 0 0 dr r 1+()p jfj p Lp(E) C (tt 0 ) p1 jfj p Lp(E) : Summarizing for all 2 1 p ; 1 i there is C independent of T, L ;1 (u f (t)u f (t 0 )) Lp(R d ) C (tt 0 ) ( 1 p ) jfj Lp(E) : 2. Estimate of u g (t) =G t g; 0tT. We have for t 0 ;t2 (0;T ]; 124 CHAPTER 8. TIME REGULARITY L ;1 u g (t)L ;1 u g (t 0 ) Lp(R d ) = G t L ;1 gG t 0L ;1 g Lp(R d ) e t 1 X j=0 G t L ;1 g ~ ' j ' j e t 0 G t 0L ;1 g ~ ' j ' j Lp(R d ) 1 X j=0 e t G t ~ ' j e t 0 G t 0 ~ ' j L1(R d ) L ;1 g' j Lp(R d ) : We note that e t G t ~ ' j e t 0 G t 0 ~ ' j (x) =e t N jd E ~ ' N j x +Z ~ N j wjt e t 0 N jd E ~ ' N j x +Z ~ N j wjt 0 ;j 1 with w j =w N j 1 ;j 1; and e t G t ~ ' 0 e t 0 G t 0 ~ ' 0 (x) =e t E ~ ' 0 (x +Z t )e s E ~ ' 0 (x +Z t 0): Hence, for j 1, e t G t ~ ' j e t 0 G t 0 ~ ' j L1(R d ) = e t E ~ ' +Z ~ N j wjt e t 0 E ~ ' +Z ~ N j wjt 0 L1(R d ) e t E ~ ' +Z ~ N j wjt E ~ ' +Z ~ N j wjt 0 L1(R d ) + e t e t 0 E ~ ' +Z ~ N j wjt 0 L1(R d ) =I j 1 +I j 2 : By Lemma 19, following a computation in Proposition 6 of [MX19], 125 CHAPTER 8. TIME REGULARITY I j 1 =e t E Z wjt wjt 0 L ~ N j ~ ' +Z ~ N j r dr L1(R d ) Ce t Z wjt wjt 0 e cr dr =Ce t e cwjt 0 h 1e cwj(tt 0 ) i Cw j (tt 0 ) ;2 [0; 1] ; j 1; and I j 2 Ce cwjt 0 e t e t 0 C (tt 0 ) ;2 [0; 1] ; j 1 and corresponding terms for j = 0 I 0 1 Z t t 0 EL ~ ' 0 (x +Z r )dr L1(R d ) C (tt 0 ); I 0 2 C (tt 0 ) ;2 [0; 1]: Hence, for 1 ; 2 2 [0; 1], L ;1 u g (t)L ;1 u g (t 0 ) Lp(R d ) C (tt 0 ) 1 1 X j=1 w 1 j L ;1 g' j Lp(R d ) +C (tt 0 ) 2 2 1 X j=0 L ;1 g' j Lp(R d ) +C (tt 0 ) L ;1 g' 0 Lp(R d ) =A +B +D: For 2 1 p ; 1 i , 1 ; 2 2 [0; 1], 1 < 1 p ; by Hölder’s inequality and Proposition 15, AC (tt 0 ) 1 1 X j=1 w 1+ 1 p j w 1 p j L ;1 g' j Lp(R d ) C (tt 0 ) 1 0 @ 1 X j=1 w (1+ 1 p )q j 1 A 1 q 0 @ 1 X j=1 w ( 1 p )p j L ;1 g' j p Lp(R d ) 1 A 1 p C (tt 0 ) 1 jgj B ;1 1 p pp (R d ) : Similarly for B, by Hölder’s inequality, 126 CHAPTER 8. TIME REGULARITY BC (tt 0 ) 2 2 1 X j=0 L ;1 g' j Lp(R d ) C (tt 0 ) 2 2 0 @ 1 X j=0 w ( 1 p )q j 1 A 1 q 0 @ 1 X j=0 w ( 1 p )p j L ;1 g' j p Lp(R d ) 1 A 1 p C (tt 0 ) 2 2 jgj B 1 1 p pp (R d ) : Now, obviously, DCjtt 0 j L ;1 g Lp(R d ) Cjtt 0 jjgj B ;1 1 p pp (R d ) : Summarizing for 2 1 p ; 1 i , 1 2 h 0; 1 p and 2 2 [0; 1], L ;1 (u (t)u (t 0 )) Lp(R d ) C (tt 0 ) + (tt 0 ) 1 + (tt 0 ) 2 2 jgj B 1 1 p pp (R d ) : 127 Chapter 9 Stochastic Variable Coefficient We establish the existence of H ;1 p (E)solution of the following equation with possibly non-trivial adapted coefficient depending on the space variable x. du (t;x) = L 2 u (t;x)u (t;x) +f (t;x) dt + Z R d 0 (1 + (t;x;z)) [u (t;x +z)u (t;x)] + (t;x;z)q (dt;dz); (9.1) u (0;x) = 0; (t;x)2E: Remark 76. Although we only pursue here the case where is sufficiently small, we believe that the estimates from Chapter 6 are essential for more general coefficients. We mention for clarity that B ;11=p p;pp R d ;H ;1=2 2;p R d and their variations are defined with respect to Banach spaces L p R d 0 ; and L 2 R d 0 ; respectively. For = 0, using Theorem 1.1 and Itô-Wentzel formula foru 0 (t;x +Z t ) whereu 0 is the solution to (1.1), for any f2L p (E) and 2B ;11=p p;pp (E)\H ;1=2 2;p (E), there exists a unique solution u2H ;1 p (E) of (9.1.) For details, we refer to the proof of Theorem 81 of the next section. We skip the proof for the time being due to similarity. We will now use the method of continuation by parameter to establish the result for a non-zero (see for example; proof of Theorem 1 in [MP14].) LetH ;1 p (E) be the space of all functions u2H ;1 p (E), such that for each t2 [0;T ] and P-a.s. u (t) = R t 0 F (s)ds + R t 0 R R d 0 G (s;;z)q (ds;dz); 0tT withF2L p (E) andG2B ;11=p p;pp (E)\H ;1=2 2;p (E). It is a Banach space with respect to the norm. juj H ;1 p (E) =juj H ;1 p (E) +jFj Lp(E) +jGj B ;11=p p;pp (E) +jGj H ;1=2 2;p (E) : 128 CHAPTER 9. STOCHASTIC VARIABLE COEFFICIENT Let f 2 L p (E) and 2 B ;11=p p;pp (E)\H ;1=2 2;p (E), we say that u2H ;1 p (E) is a solution to (9.1) if L 2 u2L p (E) and u (t;x) = Z t 0 L 2 u (s;x)u (s;x) +f (s;x) ds + Z t 0 Z R d 0 (1 + (s;x;z)) [u (s;x +z)u (s;x)] + (t;x;z)q (dt;dz) P a.s. in L p R d . Theorem 49 and Lemma 45 yield the following summary. Corollary 77. Let 2A ; w =w be an O-RV function and A, B hold, p 2. Then there exists C > 0 such that Z jf ( +y)f ()j p (dy) 1 p Lp(R d ) Cjfj B ;1=p pp (R d ) ;f2B ;1=p pp R d : If Assumptions A, C and D hold. Then there exists C > 0 such that Z jf ( +y)f ()j 2 (dy) 1 2 Lp(R d ) Cjfj H ;1=2 p (R d ) ;f2H ;1=2 p R d : From Theorem 2, the following estimate holds for the solution u of (1.1), when g = 0, juj H ;1 p (E) C 1 jfj H ;1 p (E) +jj B ;11=p p;pp (E) +jj H ;1=2 2;p (E) : We will need the following Assumption P for the estimate of pointwise multiplication i.e. the term u. Let p 2, we assume that = (t;x;y) satisfies the following Assumption , Assumption P(). For t2 [0;T ], and y2R d 0 , j (t;;y)fj B ;11=p pp (R d ) jfj B ;11=p pp (R d ) ; f2B ;11=p pp R d and for t2 [0;T ], j (t)fj H ;1=2 2;p (R d ) jfj H ;1=2 2;p (R d ) ; f2H ;1=2 2;p R d : Remark 78. We believe that a more concrete assumption can be provided, but it will require an extensive investigation of boundeness of pointwise multiplication in B ;11=p pp R d and H ;1=2 2;p R d which is beyond the scope of this manuscript. We provide here some examples of that satisfies AssumptionP () with some > 0. 129 CHAPTER 9. STOCHASTIC VARIABLE COEFFICIENT Let (t;x;y) = (t;y) be uniformly bounded In the classical setting, that is =dy=jyj d+ , 2 (0; 1), we also have following examples Let (t;x;y) = (x)besuchthat P jnjm jD n j L1(R d ) <1forsomesufficientlylargem(seeTheorem 4.2.2 of [Tri92] and Proposition 6.9 of [Wal03]) InR, assume that 1 1 p < 1 p , let characteristic function of half space (x) = fx>0g (x) (see [Tri77] and Corollary 6.10 of [Wal03]) Now we will establish well-posedness of (9.1). Lemma 79. Let 2 A ; w = w be an O-RV function and A, C, D hold, p 2. Let f 2 L p (E) and 2B ;11=p p;pp (E)\H ;1=2 2;p (E). Assume that satisfies P() for some sufficiently small, then there exists a unique solution u2H ;1 p (E) to (9.1). Proof. Suppose that u2H ;1 p (E) is a solution to (9.1), by Theorem 2, there exists C 1 > 0 such that juj H ;1 p (E) C 1 jfj H ;1 p (E) +jj B ;11=p p;pp (E) +jj H ;1=2 2;p (E) +j uj B ;11=p p;pp (E) +j uj H ;1=2 2;p (E) (9.2) where u (t;x;z) =u (t;x +z)u (t;x). Note that we have substitute u (t) byu (t) because it is clear from the equation that u is a càdlàgL p R d valued process and thus u (t;x) =u (t;x) a.s. (in t;x and !. ) By Corollary 77, Assumption P() where > 0 will be specified later, Proposition 15 (see also Remark 24), j uj B ;11=p p;pp (E) +j uj H ;1=2 2;p (E) Cjuj H ;1 p (E) : (9.3) Thus from (9.2) for sufficiently small (depending on C, C 1 ), juj H ;1 p (E) 2C 1 jfj H ;1 p (E) +jj B ;11=p pp (E) +jj H ;1=2 p (E) (9.4) The same estimate holds if we replace (t;x;z) by (t;x;z) for any 2 [0; 1]: We use the method of continuation by parameter to show existence of a solution. The method should be standard. In particular, we adopt ideas from proof of Theorem 1 in [MP14]. Define for x;z2R d and t> 0, Au (t;x) :=L 2 u (t;x)u (t;x) M u (t;x;z) := [u (t;x +z)u (t;x)] (1 + (t;x;z)); 2 [0; 1] 130 CHAPTER 9. STOCHASTIC VARIABLE COEFFICIENT LetV ;1 p be Banach space of all pairs l = (f; ), f2L p (E) and 2B ;11=p p;pp (E)\H ;1=2 2;p (E) with the norm jlj ;1 =jfj Lp(E) +jj B ;11=p p;pp (E) +jj H ;1=2 2;p (E) : Consider the mapping T :H ;1 p (E)!V ;1 p defined by u (t;x) = Z t 0 F (s;x)ds + Z t 0 Z G (s;x;z)q (ds;dz)! (FAu;GM u) Similar to (9.3), using Corollary 77 and Assumption P(), for some constant C not depending on , jT uj V ;1 p jFj Lp(E) +jAuj Lp(E) +jGj B ;11=p p;pp (E) +jM uj B ;11=p p;pp (E) +jGj H ;1=2 2;p (E) +jM uj H ;1=2 2;p (E) Cjuj H ;1 p (E) : (9.5) We now show the reverse inequality of (9.5), u (t;x) = Z t 0 F (s;x)ds + Z t 0 Z G (s;x;z)q (ds;dz) = Z t 0 (F (s;x)Au +Au)ds + Z t 0 Z (G (s;x;z)M u +M u)q (ds;dz): By apriori estimate (9.4), juj H ;1 p (E) C jFAuj Lp(E) +jGM uj B ;11=p p;pp (E) +jGM uj H ;1=2 2;p (E) : Therefore, juj H ;1 p (E) =juj H ;1 p (E) +jFj L P (E) +jGj B ;11=p p;pp (E) +jGj H ;1=2 2;p (E) C(jFAuj Lp(E) +jGM uj B ;11=p p;pp (E) +jGM uj H ;1=2 2;p (E) +jAuj Lp(E) +jM uj B ;11=p p;pp (E) +jM uj H ;1=2 2;p (E) ) C jFAuj Lp(E) +jGM uj B ;11=p p;pp (E) +jGM uj H ;1=2 2;p (E) =CjT uj V ;1 p (E) with C independent of . We already know thatT 0 is an onto map. Therefore, applying Theorem 5.2 of [GT01],T 1 must also be an onto map, establishing the existence of theH ;1 p solution of (9.1) . The uniqueness follows from (9.4). 131 Chapter 10 Quasilinear Equation We will show the existence and uniqueness of the solution to the following quasilinear equation. Let ;2A, we will now consider the following equation u (t;x) = Z t 0 L + u (s;x) +D (s;x;u (s))ds + Z t 0 Z R d 0 [u (s;x +y)u (s;x) +Q (s;x;y;u (s))]q (ds;dy); (t;x)2E (10.1) where for u2 H ;1 p R d , D (t;x;u) is an adaptedR (F)B R d measurable, L p R d -valued function and Q (t;x;y;u) is an adaptedR (F)B R d B R d 0 measurable, B ;11=p p;pp R d \H ;1=2 2;p R d - valued function. We mention for clarity that B ;11=p p;pp R d ;H ;1=2 2;p R d and their variations are defined with respect to Banach spaces L p R d 0 ; and L 2 R d 0 ; respectively. We only consider the case where the initial value function is 0 since its generalization does not offer significant interest. Following ideas and conventions in [MP13], we denote for convenience for an adapted function 2 B ;11=p p;pp (E)\H ;1=2 2;p (E), (t;x;y) = (t;xy;y) (I) (t;x) = Z R d 0 [ (t;x;y) (t;xy;y)] (dy) = Z R d 0 [ (t;x;y) (t;x;y)] (dy) for (t;x)2E;y2R d 0 assuming that 132 CHAPTER 10. QUASILINEAR EQUATION I (t;x) = lim !0 Z jyj> [ (t;x;y) (t;x;y)] (dy); (t;x)2E inL p (E) (I is interpreted as this limit.) Define also D 0 (t;x) =D (t;x; 0) Q 0 (t;x;y) =Q (t;x;y; 0) andA p (E) = n 2B ;11=p p;pp (E)\H ;1=2 2;p (E) : 2B ;11=p p;pp (E)\H ;1=2 2;p (E) and I2L p (E) o : When there is no chance of confusion for a function f on ER d 0 we denote f (t;y) = f (t;;y), and f (t) =f (t;;). The following growth and continuity assumptions on D and Q will be used. Assumption L(p;;). For any u2H ;1 p R d , D (t;x;u) is an adapted L p -valued function and Q (t;x;y;u) is an adapted B ;11=p p;pp R d \H ;1=2 2;p R d - valued function such that (i) For every > 0, there exists C > 0 such that for any u 1 ;u 2 2H ;1 p R d ,t2 [0;T ]; andPa.s. jD (t;u 1 )D (t;u 2 )j Lp(R d ) +jIQ (t;u 1 )IQ (t;u 2 )j Lp(R d ) +jQ (t;u 1 ) Q (t;u 2 )j B ;11=p p;pp (R d ) +jQ (t;u 1 ) Q (t;u 2 )j H ;1=2 2;p (R d ) jL u 1 L u 2 j Lp(R d ) +C ju 1 u 2 j Lp(R d ) : (ii) D 0 ;IQ 0 2L p (E) and Q 0 ; Q 0 2B ;11=p p;pp (E)\H ;1=2 2;p (E): When the context is clear we simply say that D;Q satisfy Assumption L instead of L(p;;): The following statement is our main result for (10.1). Theorem 80. Let 2 A ;w = w be a continuous O-RV function and A, B hold, p2 (1;1);2 A. Assume that D;Q satisfy L(p;;). Then there exists a unique u2H ;1 p (E) solving (10.1). Moreover, there is C =C (d;p;;T ) such that for p2 [2;1), t2 [0;T ], E sup 0st ju (s)j p Lp(R d ) + Z t 0 jL u (s)j p Lp(R d ) ds CE Z t 0 h D 0 (s)IQ 0 (s) p Lp(R d ) + Q 0 (s) p B ;11=p p;pp (R d ) + Q 0 (s) p H ;1=2 2;p (R d ) i ds and for p2 (1; 2), 133 10.1. PROOF OF THE MAIN RESULTS CHAPTER 10. QUASILINEAR EQUATION E sup 0st ju (s)j p Lp(R d ) + Z t 0 jL u (s)j p Lp(R d ) ds CE Z t 0 h D 0 (s)IQ 0 (s) p Lp(R d ) + Q 0 (s) p B ;11=p p;pp (R d ) i ds: 10.1 Proof of the Main Results Due to similarity we only prove the result for the case p 2. The proof of the main theorem follows several steps starting from a linear equation, a partial-quasi linear equation, and then a general quasilinear equation as stated in the main theorem. We start by proving existence, uniqueness, estimates of the following linear equation. Theorem 81. Let 2A ;w =w be a continuous O-RV function and A, B hold, p2 [2;1);2A. Let f;I2L p (E) and; 2B ;11=p p;pp (E)\H ;1=2 2;p (E), then there exists a unique solutionu2H ;1 p (E) solving u (t;x) = Z t 0 L + u (s;x) +f (s;x)ds + Z t 0 Z R d 0 [u (s;x +y)u (s;x) + (s;x;y)]q (ds;dy) (10.2) (t;x) 2 E Moreover, there exists C =C (d;p;;T )> 0 such that E Z t 0 jL u (s)j p Lp(R d ) ds CE Z t 0 jfI (s)j p Lp(R d ) +j (s)j p B ;11=p p;pp (R d ) +j (s)j p H ;1=2 2;p (R d ) ds (10.3) and E sup 0st ju (s)j p Lp(R d ) CE Z t 0 h jfI (s)j p Lp(R d ) +j (s)j p Lp;p(R d ) +j (s)j p L2;p(R d ) i ds: (10.4) Proof. We prove this Theorem similar to Proposition 22 of [MP13]. Some details are omitted because it is a straightforward modification of Proposition 22 of [MP13]. We consider first the case of smooth input functions. 134 10.1. PROOF OF THE MAIN RESULTS CHAPTER 10. QUASILINEAR EQUATION Smooth Inputs: Assuming first that f2 ~ C 1 0;p (E); and ; 2 ~ C 1 2;p (E)\ ~ C 1 p;p (E) and that there is I2 ~ C 1 0;p (E) such that for every multiindex 2N d 0 , jD x I D x Ij Lp(E) +E " Z T 0 sup x2R d jD x I D x Ij p # ! 0; as ! 0: LetZ t =Z ; t = R t 0 R jyj> (y)yq (ds;dy)+ R t 0 R (1 (y))yp (ds;dy);t 0 wherep is a Poisson point measure withEp (ds;dy) = (dy)ds, q (ds;dy) =p (ds;dy) (dy)ds and Z t =Z ;0 t . Let us define (t;x;y) = t;xZ t y;y f (t;x) = f (t;xZ t )I (t;xZ t ): Clearly, f (t;xZ t );I (t;xZ t )2 ~ C 1 0;p (E), t;xZ t y;y 2 ~ C 1 2;p (E)\ ~ C 1 p;p (E). By Theorem 2 (see also Lemmas 66 and 65) for smooth inputs, there exists a unique u2 ~ C 1 0;p (E) solving the equation d u (t;x) = L u (t;x) u (t;x) + f (t;x) dt + Z (t;x;y)q (dt;dy) u (0;x) = 0: Moreover, we have the estimates, E Z t 0 jL u (s)j p Lp(R d ) ds CE Z t 0 jfI (s)j p Lp(R d ) +j (s)j p B ;11=p p;pp (R d ) +j (s)j p H ;1=2 2;p (R d ) ds (10.5) and for 0tT, E sup 0st j u (s)j p Lp(R d ) CE Z t 0 h jfI (s)j p Lp(R d ) +j (s)j p Lp;p(R d ) +j (s)j p L2;p(R d ) i ds: (10.6) 135 10.1. PROOF OF THE MAIN RESULTS CHAPTER 10. QUASILINEAR EQUATION Applying Itô-Wentzell formula (Proposition 1 of [Mik83]) to u (t;x +Z ; t ), u (t;x +Z ; t ) = u (0;x) + Z t 0 r u s;x +Z ; s dZ ; s + Z t 0 (L ) u (s;x +Z ; s ) + f (s;x +Z ; s )ds + Z t 0 Z s;x +Z ; s ;y q (ds;dy) + X 0st u (s;x +Z ; s ) u s;x +Z ; s r u s;x +Z ; s Z ; s + X 0st u (s;x +Z ; s ) u s;x +Z ; s : (10.7) For convenience we denote the following, and will examine them carefully, we only deal with the case 2 (1; 2), A = X 0st u (s;x +Z ; s ) u s;x +Z ; s r u s;x +Z ; s Z ; s = Z t 0 Z jyj> u s;x +Z ; s +y u s;x +Z ; s r u s;x +Z ; s y p (ds;dy) B = Z t 0 r u s;x +Z ; s dZ ; s = Z t 0 Z jyj> r u s;x +Z ; s yq (ds;dy) C = X 0st u (s;x +Z ; s ) u s;x +Z ; s = X 0st u (s;x +Z ; s ) u (s;x +Z ; s ) u s;x +Z ; s + u s;x +Z ; s = Z t 0 Z jyj> (s;x +Z ; s ;y) s;x +Z ; s ;y p (ds;dy) = Z t 0 Z jyj> s;x +Z ; s +y;y s;x +Z ; s ;y p (ds;dy) = Z t 0 Z jyj> s;x +Z ; s +y;y s;x +Z ; s ;y q (ds;dy) + Z t 0 I x +Z ; s ds Summarizing from above computations, A = Z t 0 Z jyj> u s;x +Z ; s +y u s;x +Z ; s r u s;x +Z ; s y p (ds;dy) B = Z t 0 Z jyj> r u s;x +Z ; s yq (ds;dy) C = Z t 0 Z jyj> s;x +Z ; s +y;y s;x +Z ; s ;y q (ds;dy) + Z t 0 I x +Z ; s ds Denoting u (t;x) = u (t;x +Z t ), plugging A;B;C into (10.7), and taking ! 0. We establish that 136 10.1. PROOF OF THE MAIN RESULTS CHAPTER 10. QUASILINEAR EQUATION u (t;x) = Z t 0 L + u (s;x) +f (s;x)ds + Z t 0 Z [u (s;x +y)u (s;x) + (s;x;y)]q (ds;dy): Reading this derivation backward, clearly u2 ~ C 1 0;p (E) is a unique solution of (10.2) . Estimates (10.3) and (10.4) for smooth inputs follows from (10.5) and (10.6). General Inputs. Using estimates (10.3), (10.4) for smooth input functions, and Lemma 30 , passing to the limit in a standard way leads to existence of the solution for general inputs f2L p (E) , and 2A p (E). We refer to Proposition 22 of [MP13] for additional details. For uniqueness, suppose u 1 ;u 2 are bothH ;1 p (E) solutions. Then u = u 1 u 2 solves (10.2) with f = 0 and = 0 i.e. u (t;x) = Z t 0 L + u (s;x)ds + Z t 0 Z R d 0 [u (s;x +y)u (s;x)]q (ds;dy): For each t, taking convolution of both sides with the standard mollifier ! = d ! (x=);x2R d ;! 0;!2 C 1 0 R d ; R ! = 1. We see that u (t) = u (t)! 2 ~ C 1 0;p (E) is the classical a.s. solution of (10.2) with zero input functions. Hence,u (t;x) = 0 a.s. by uniqueness for smooth inputs. Since> 0 is arbitrary u (t;x) = 0 a.s. Passing to the limit, clearly (10.3), (10.4) holds for general inputs. We will use the method of continuation by parameter to show the existence of the solution to the (1.1). The rest of arguments in these section can be considered routine, and should be well-known in any textbooks concerning the extension of linear equations to quasilinear equations. Nonetheless, we provide the argument for the sake of completeness. Theorem 82. Let 2A ;w =w be a continuous O-RV function and A, B hold, p2 [2;1) and 2A. Suppose that D;Q satisfy L and u2 H ;1 p (E) is a solution of equation of (10.1). Then there exists C = C (d;p;;T )> 0 such that for each t2 [0;T ], E sup 0st ju (s)j p Lp(R d ) + Z t 0 jL uj p Lp(R d ) ds CE Z t 0 h D 0 (s)IQ 0 (s) p Lp(R d ) + Q 0 (s) p B ;11=p p;pp (R d ) + Q 0 (s) p H ;1=2 2;p (R d ) i ds: Proof. If u2H ;1 p (E) solves (10.1), by the estimates (10.3) and (10.4), 137 10.1. PROOF OF THE MAIN RESULTS CHAPTER 10. QUASILINEAR EQUATION E sup 0st ju (s)j p Lp(R d ) +E Z t 0 jL uj p Lp(R d ) ds E Z t 0 [jD (s;u) (IQ) (s;u)j p Lp(R d ) + jQ (s;u)j p B ;11=p p;pp (R d ) +jQ (s;u)j p H ;1=2 2;p (R d ) ]ds: Using the Assumption L. For t2 [0;T ], > 0, there is C > 0 independent of t2 [0;T ] such that E sup 0st ju (s)j p Lp(R d ) +E Z t 0 jL u (s)j p Lp(R d ) ds E Z t 0 jL u (s)j p Lp(R d ) ds +C E Z t 0 ju (s)j p Lp(R d ) ds +CH (t) where H (t) =E Z t 0 h D 0 (s)IQ 0 (s) p Lp(R d ) + Q 0 (s) p B ;11=p p;pp (R d ) + Q 0 (s) p H ;1=2 2;p (R d ) i ds: Since can be made sufficiently small, E sup 0st ju (s)j p Lp(R d ) + (1)E Z t 0 jL uj p Lp(R d ) ds CE Z t 0 ju (s)j p Lp(R d ) ds +CH (t): By Gronwall’s inequality, E sup 0st ju (s)j p Lp(R d ) CH (t): and E Z t 0 jL u (s)j p Lp(R d ) ds CH (t) as needed. Lemma 83. Let 2 A ;w = w be a continuous O-RV function and A, B hold, p2 [2;1) and 2 A: Suppose that D satisfies L and I2L p (E), ; 2B ;11=p p;pp (E)\H ;1=2 2;p (E). Then the exists a unique solution u2H ;1 p (E) solution to the following partial quasilinear equation u (t;x) = Z t 0 L + u (s;x) +D (s;x;u (s))ds + Z t 0 Z R d 0 [u (s;x +y)u (s;x) + (s;x;y)]q (ds;dy) (10.8) (t;x) 2 E: 138 10.1. PROOF OF THE MAIN RESULTS CHAPTER 10. QUASILINEAR EQUATION Moreover, there exists C =C (d;p;;T )> 0 such that for each t2 [0;T ], E sup 0st ju (s)j p Lp(R d ) + Z t 0 jL uj p Lp(R d ) ds CE Z t 0 D 0 (s)I (s) p Lp(R d ) +j (s)j p B ;11=p p;pp (R d ) +j (s)j p H ;1=2 2;p (R d ) ds: (10.9) Proof. (10.9) immediately follows from Theorem 82. Existence: For 2 [0; 1], consider the equation, du (t;x) = L + u (t;x)dt + Z R d 0 [u (t;x +y)u (t;x) + (t;x;y)]q (dt;dy) + D (t;x; 0)dt + (1)D (t;x;u (t))dt +f (t;x)dt (10.10) u (0;x) = 0: Assume that for some = 0 2 [0; 1] (10.10) has a unique H ;1 p (E)-solution for any f 2 L p (E). For other , we rewrite (10.10) with f = 0 as follows, du (t;x) = L + u (t;x)dt + Z R d 0 [u (t;x +y)u (t;x) + (t;x;y)]q (dt;dy) + 0 D (t;x; 0)dt + (1 0 )D (t;x;u (t))dt + ( 0 ) [D (t;x; 0)D (t;x;u (t))]dt (10.11) u (0;x) = 0: We solve (10.11) by iteration, starting with u 0 = 0, and for n 1 define du n+1 (t;x) = L + u n+1 (t;x)dt + Z R d 0 u n+1 (t;x +y)u n+1 (t;x) + (t;x;y) q (dt;dy) + 0 D (t;x; 0)dt + (1 0 )D t;x;u n+1 (t) dt + ( 0 ) [D (t;x; 0)D (t;x;u n (t))]dt u n+1 (0;x) = 0 Now from Assumption L, D (t;x; 0)D (t;y;u n )2L p (E), provided that u n 2H ;1 p (E), indeed, jD (t;x; 0)D (t;y;u n (t))j Lp(R d ) Cju n (t)j H ;1 p (R d ) : 139 10.1. PROOF OF THE MAIN RESULTS CHAPTER 10. QUASILINEAR EQUATION Hence, u n+1 is well-defined as aH ;1 p (E)solution to (10.11) by the Assumption imposed on 0 . Now consider the difference equation for u n+1 =u n+1 u n , d u n+1 (t;x) = L + u n+1 (t;x)dt + Z R d 0 u n+1 (t;x +y) u n+1 (t;x) q (dt;dy) + (1 0 ) D t;x; u n+1 (t) +u n (t) D (t;x;u n (t)) dt ( 0 ) D t;x; u n (t) +u n1 (t) D t;x;u n1 (t) dt u n+1 (0;x) = 0: Hence, by Theorem 82 and Assumption L, E " sup 0tT u n+1 (t) p Lp(R d ) + Z T 0 L u n+1 (t) p Lp(R d ) dt # Cj 0 j p E " sup 0tT j u n (t)j p Lp(R d ) + Z T 0 jL u n (t)j p Lp(R d ) dt # : (10.12) DenoteI =E h sup 0tT u 1 (t) p Lp(R d ) + R T 0 L u 1 (t) p Lp(R d ) dt i <1 and letj 0 j< (2C) 1 p , RHS of (10.12) is bounded by (1=2) n I and hence (u n ) 1 n=1 is a Cauchy sequence and there is a H ;1 p R d valued process u such that E " sup 0tT ju n (t)u (t)j p Lp(R d ) + Z T 0 jL u n (t)L u (t)j p Lp(R d ) dt # ! 0 as n!1. Obviously, by passing to the limit, u is a solution to (10.10) with f = 0. It follows that we have a solution for any such thatj 0 j < (2C) 1 p (we assumed that we have one for 0 .) For 0 = 1, the H ;1 p (E) solution exists by Theorem 81 , therefore in finite number of steps starting with = 1, there exists the solution for = 0. Uniqueness. Let u 1 and u 2 beH ;1 p (E)solution to (10.8), then u =u 2 u 1 is a solution to u (t;x) = Z t 0 L + u (s;x) +D (s;x;u 2 (s))D (s;x;u 1 (s))ds + Z t 0 Z R d 0 [u (s;x +y)u (s;x)]q (ds;dy): By Assumption L, for > 0;s2 [0;T ], jD (s;u 2 (s))D (s;u 1 (s))j Lp(R d ) Cju (s)j Lp(R d ) +jL u (s)j Lp(R d ) 140 10.1. PROOF OF THE MAIN RESULTS CHAPTER 10. QUASILINEAR EQUATION and by (10.9), E sup 0st ju (s)j p Lp(R d ) + Z t 0 jL u (s)j p Lp(R d ) ds E C Z t 0 ju (s)j p Lp(R d ) + Z t 0 jL u (s)j p Lp(R d ) ds Thus by choosing sufficiently small, Gronwall’s inequality implies u = 0 a.s. We now prove the main result Theorem 80. Proof. Proof of theorem 80. The solution estimate is proved in Theorem 82. Existence: According to Lemma 83, we construct a sequence of iteration- starting with u 0 = 0 and define u n+1 (t;x) = L + u n+1 (t;x) +D t;x;u n+1 (t) dt + Z R d 0 u n+1 (t;x +y)u n+1 (t;x) +Q (t;x;y;u n (t)) q (dt;dy) u n+1 (0;x) = 0: Then u n+1 =u n+1 u n isH ;1 p (E) solution to u n+1 (t;x) = L + u n+1 (t;x) +D t;x; u n+1 +u n D (t;x;u n ) dt + Z R d 0 u n+1 (t;x +y) u n+1 (t;x) +Q (t;x;y;u n )Q t;x;y;u n1 q (dt;dy) u (t;x) = 0: By the estimate from Lemma 83, with Q n (s;x;y) =Q (s;x;y;u n )Q s;x;y;u n1 ; E sup 0st u n+1 (s) p Lp(R d ) + Z t 0 L u n+1 (s) p Lp(R d ) ds CE Z t 0 I Q n (s) p Lp(R d ) + Q n (s) p B ;11=p p;pp (R d ) + Q n (s) p H ;1=2 2;p (R d ) ds : The integrand of RHS is bounded by Cj u n (s)j p H ;1 p (R d ) by Assumption L. Therefore, 141 10.1. PROOF OF THE MAIN RESULTS CHAPTER 10. QUASILINEAR EQUATION E sup 0st u n+1 (s) p Lp(R d ) + Z t 0 L u n+1 (s) p Lp(R d ) ds CE Z t 0 j u n (s)j p H ;1 p (R d ) ds CE Z t 0 sup 0sr j u n (s)j p H ;1 p (R d ) dr (Ct) n n! E sup 0st u 1 (s) p H ;1 p (R d ) where u 1 =u 1 is a solution to u 1 (t;x) = L + u 1 (t;x) +D t;x;u 1 + Z R d 0 u 1 (t;x +y)u 1 (t;x) +Q 0 (t;x;y) q (dt;dy) u 1 (0;x) = 0: By Lemma 83, the following estimate holds for u 1 , E sup 0st u 1 (s) p Lp(R d ) + Z t 0 L u 1 p Lp(R d ) ds CE Z t 0 h D 0 (s)IQ 0 (s) p Lp(R d ) + Q 0 (s) p B ;11=p p;pp (R d ) + Q 0 (s) p H ;1=2 2;p (R d ) i ds < 1: Therefore, there is a H ;1 p R d valued process u (t) such that E " sup 0tT ju n (t)u (t)j p Lp(R d ) + Z T 0 jL u n (t)L u (t)j Lp(R d ) dt # ! 0 as n!1 which solves (10.1). Uniqueness: Let u 1 and u 2 beH ;1 p (E) solution to 10.1, then u =u 2 u 1 is a solution to u (t;x) = Z t 0 L + u (s;x) +D (s;x;u 2 (s))D (s;x;u 1 (s))ds + Z t 0 Z R d 0 [u (s;x +y)u (s;x) +Q (s;x;y;u 2 (s))Q (s;x;y;u 1 (s))]q (ds;dy): 142 10.2. EXAMPLES CHAPTER 10. QUASILINEAR EQUATION By Lemma 82, and Assumption L, for any > 0, with D 2;1 (s;x) = D (s;x;u 2 (s))D (s;x;u 1 (s)), Q 21 (s;x;y) =Q (s;x;y;u 2 (s))Q (s;x;y;u 1 (s)), E sup 0st ju (s)j p Lp(R d ) + Z t 0 jL uj p Lp(R d ) ds CE Z t 0 [jD 21 (s)j p Lp(R d ) ds +jIQ 21 (s)j p B ;11=p p;pp (R d ) + jQ 21 (s)j p H ;1=2 2;p (R d ) +jQ 21 (s)j p H ;1=2 2;p (R d ) ]ds CE Z t 0 ju (s)j p Lp(R d ) +E Z t 0 jL u (s)j p Lp(R d ) ds Thus by choosing sufficiently small, Gronwall’s inequality implies u = 0 a.s. Completing the proof of Theorem 80. 10.2 Examples Assumption L is proposed to facilitate the standard argument for quasilinear equations. In this section, we confirm that some standard cases are covered. First, we need some preliminary computations. We will define a broad definition of L p continuity for ; 2 A, p2 (1;1). We say that is L p continuous with respect to if for any > 0 there exists C ()> 0 such that jL fj Lp(R d ) jL fj Lp(R d ) +C ()jfj Lp(R d ) ; f2H ;1 p R d : Example 84. Let2A ;w =w be a continuous O-RV function andA, B hold,p2 (1;1). Assume that dd=q p w 1 ^ dd=q p w 2 < 1< d+1d=q q w 1 ^ d+1d=q q w 2 for someq 1. Let2A 0 with 0 2 (0; 1) and R jzj1 w (z) (dz)< 1: Then is L p - continuous with respect to . Proof. Applying Lemma 42, f (x +z)f (x) =C Z L ;1 f (xy)k ;1 (y;z)dy Fix a> 0, integrating with respect to (dz) on both sides and applying Minkowski’s inequality, 143 10.2. EXAMPLES CHAPTER 10. QUASILINEAR EQUATION Z jzj<a [f ( +z)f ()] (dz) Lp(R d ) = C Z jzj<a Z L f (y)k ;1 (y;z)dy (dz) Lp(R d ) CjL fj Lp(R d ) Z jzj<a Z k ;1 (y;z) dy (dz) ! CjL fj Lp(R d ) Z jzj<a w (z) (dz) ! On the other hand, Z jzja [f ( +z)f ()] (dz) Lp(R d ) 2jfj Lp(R d ) Z jzja (dz): Therefore, jL fj Lp(R d ) Z jzj<a ! (z) (dz) ! jL fj Lp(R d ) + 2 Z jzja (dz) ! jfj Lp(R d ) : Since R jzj<a w (z) (dz) can be arbitrarily small, the statement follows. Remark 85. As suggested by Example 84, from Lemma 42, one can construct various examples by adjusting and q. Lemma 86. Let 2 A ;w = w be a continuous O-RV function and A, B hold, p 2 (1;1); 2 0; 1 q ! 1 ^ 1 q ! 2 : For any f2C 1 0 R d , let Q 2 l f (x) = Z jzj1 [f (x + 2lz) 2f (x +lz) +f (x)]dz;l2R;x2R d ; then there exists C =C (;)> 0 such that Q 2 l f Lp(R d ) Cw (l) 2 L ;2 f Lp(R d ) : Proof. Using Lemma (42) with 2 0; 1 q1 ^ 1 q2 and q = 1. We may write for z2R d , z f (x) =f (x +z)f (x) =c Z L ; f (xw) ; (w;z)dw Hence, 144 10.2. EXAMPLES CHAPTER 10. QUASILINEAR EQUATION f (x + 2lz) 2f (x +lz) +f (x) = lz lz f (x) = z Z L ; f (xw) ; (w;lz)dw = Z Z L ;2 f (xvw) ; (w;lz) ; (v;lz)dwdv: Therefore, by Minkowski’s inequality and Lemma 6, for some 2 > 0; Q 2 l f Lp(R d ) C L ;2 f Lp(R d ) Z jzj1 Z Z ; (w;lz)k ; (v;lz) dwdvdz C L ;2 f Z jzj1 w (ljzj) 2 dzC L ;2 f Lp(R d ) w (l) 2 Z jzj1 jzj 22 dz Cw (l) 2 L ;2 f Lp(R d ) : Lemma 87. Let 2 A ;w = w be a continuous O-RV function and A, B hold, p2 (1;1). Then for f;g2C 1 0 R d , there exists C =C ()> 0, 0<< 1 2 such that jL (fg)j Lp(R d ) jfL gj Lp(R d ) +jL fgj Lp(R d ) +C L ; 1 2 + f Lp(R d ) L ; 1 2 g L1(R d ) +jfj Lp(R d ) jgj L1(R d ) Proof. It follows a straightforward computation that L fg (x) = fL g +gL f + Z [f (x +y)f (x)] [g (x +y)g (x)] (dy) We estimate L p norm of the last term, Z [f ( +y)f ()] [g ( +y)g ()] (dy) Lp(R d ) Z jyj1 ::: (dy) Lp(R d ) + Z jyj>1 ::: (dy) Lp(R d ) By Lemma (42) for some 0<< 1 2 , and Lemma 6, 145 10.2. EXAMPLES CHAPTER 10. QUASILINEAR EQUATION Z jyj1 ::: (dy) Lp(R d ) Z jyj1 Z L ; 1 2 + f (z) ; 1 2 + (z;y)dz Z L ; 1 2 g (w) ; 1 2 (w;y)dw (dy) Lp(R d ) L ; 1 2 + f Lp(R d ) L ; 1 2 g L1(R d ) Z jyj1 w (jyj) 1+ (dy) C L ; 1 2 + f Lp(R d ) L ; 1 2 g L1(R d ) : Now trivially, Z jyj>1 ::: (dy) Lp(R d ) Cjfj Lp(R d ) jgj L1(R d ) Z jyj>1 (dy) Cjfj Lp(R d ) jgj L1(R d ): Example 88. Let 2A ;w = w be a continuous O-RV function and A, B hold, p2 [2;1) and 2A. Suppose that 2A is L p -continuous with respect to and that for (t;x)2E;y2R d 0 ;u2H ;1 p R d , D (t;x;u) = ~ D t;x;u (x);L u (x) ; ~ D :ERR!R Q (t;x;y;u) =g 1 (t;x;y)u (x) +g 0 (t;x;y); g 1 ;g 0 :ER d 0 !R are adapted and satisfy L.1 There exists C > 0 such thatPa.s. and (t;x)2E, z 1 ;z 2 ; z 1 ; z 2 2R, ~ D (t;x;z 1 ; z 1 ) ~ D (t;x;z 2 ; z 2 ) C [jz 1 z 2 j +j z 1 z 2 j] . L.2 There exists C > 0 such thatPa.s. and t2 [0;T ], Z R d 0 jg 1 (t;y)j L1(R d ) (dy)<C Z R d 0 jL g 1 (t;y)j p L1(R d ) + L ; 1 2 g 1 (t;y) p L1(R d ) +jg 1 (t;y)j p L1(R d ) (dy) < C 146 10.2. EXAMPLES CHAPTER 10. QUASILINEAR EQUATION Z R d 0 jL g 1 (t;y)j 2 L1(R d ) + L ; 1 2 g 1 (t;y) 2 L1(R d ) +jg 1 (t;y)j 2 L1(R d ) (dy)<C and D 0 ;IQ 0 2L p (E) and Q 0 ; Q 0 2B ;11=p p;pp (E)\H ;1=2 2;p (E): Then D;Q satisfy Assumption L. Note that in this case Q 0 =g 0 : Proof. In particular, we will show that for > 0, there exists C > 0 such that for u 1 ;u 2 2 H ;1 p R d , t2 [0;T ] andPa.s. jD (t;u 1 )D (t;u 2 )j Lp(R d ) +jIQ (t;u 1 )IQ (t;u 2 )j Lp(R d ) +jQ (t;u 1 ) Q (t;u 2 )j B ;11=p p;pp (R d ) +jQ (t;u 1 ) Q (t;u 2 )j H ;1=2 2;p (R d ) jL u 1 L u 2 j Lp(R d ) +C ju 1 u 2 j Lp(R d ) where we remind the reader for convenience that f (t;x;y) =f (t;xy;y); (t;x)2E;y2R d 0 (If) (t;x) = Z t 0 Z R d 0 [f (r;x;y)f (r;xy;y)]dr (dy); (t;x)2E: (i) Estimate ofjD (t;u 1 )D (t;u 2 )j Lp(R d ) . By Assumption L.1 and L p continuity of with respect to , for every > 0, jD (t;u 1 )D (t;u 2 )j Lp(R d ) C ju 1 u 2 j Lp(R d ) + L (u 1 u 2 ) Lp(R d ) C ju 1 u 2 j Lp(R d ) +jL (u 1 u 2 )j Lp(R d ) : (ii) Estimate ofjIQ (t;u 1 )IQ (t;u 2 )j Lp(R d ) . IQ (t;x;y;u 1 )IQ (t;x;y;u 2 ) = Z R d 0 g 1 (t;x;y) (u 1 (x)u 2 (x)) +g 1 (t;xy;y) (u 1 (xy)u 2 (xy)) (dy) = Z R d 0 A 1 (t;x;y) +A 2 (t;x;y) (dy): By Minkowski’s inequality, A 1 and A 2 are estimated similarly, by Assumption L.2, 147 10.2. EXAMPLES CHAPTER 10. QUASILINEAR EQUATION Z R d 0 A 1 (t;y) (dy) Lp(R d ) Z R d 0 jg 1 (r;y)j L1(R d ) ju 1 u 2 j Lp(R d ) (dy) = Z R d 0 jg 1 (r;y)j L1(R d ) (dy)ju 1 u 2 j Lp(R d ) Cju 1 u 2 j Lp(R d ) : (ii) Estimate ofjQ (t;u 1 ) Q (t;u 2 )j B ;11=p p;pp (R d ) . For convenience we write u =u 1 u 2 , jQ (t;u 1 ) Q (t;u 2 )j B ;11=p p;pp (R d ) = j (g 1 (t)u)j p B ;11=p p;pp (R d ) = B (t): To estimate B (t), we use characterization of Besov space. Indeed by Proposition 26, for some 2 (0; 1) sufficiently small which will be specified later, B (t) Cf Z R d 0 jg 1 (t;y;y)u (y)j p Lp(R d ) (dy) + Z R d 0 Z 0 Q 2 l g 1 (t;y;y)u (y) p Lp(R d ) dl lw (l) p1 (dy) + Z R d 0 Z 1 Q 2 l g 1 (s;y;y)u (y) p Lp(R d ) dl lw (l) p1 (dy)g = CfB 1 (t) +B 2 (t) +B 3 (t)g We note that R 1 0 w (l) dl l <1 by Corollary 10. Estimate B 1 (t): By Assumption L.2, B 1 (t)Cjuj p Lp(R d ) : Estimate B 2 (t): Applying Corollary 86 and 87 for some 1 2 0; 1 2 , 148 10.2. EXAMPLES CHAPTER 10. QUASILINEAR EQUATION Q 2 l g 1 (t;y;y)u (y) Lp(R d ) Cw (l)jL (g 1 (t;y;y)u (y))j Lp(R d ) = Cw (l)jL (g 1 (t;y)u)j Lp(R d ) Cw (l)fjuL g 1 (t;y)j Lp(R d ) +jg 1 (t;y)L uj Lp(R d ) + + L ; 1 2 +1 u Lp(R d ) L ; 1 2 g 1 (t;y) L1(R d ) +juj Lp(R d ) jg 1 (t;y)j L1(R d ) g = C (b 1 +b 2 +b 3 +b 4 ): Using Assumption L.2, Z R d 0 Z 0 b p 1 dl lw (l) p1 (dy) = Z R d 0 Z 0 juL g 1 (t;y)j Lp(R d ) w (l) p dl lw (l) p1 (dy) juj p Lp(R d ) Z R d 0 jL g 1 (t;y)j p L1(R d ) (dy) ! Z 0 w (l) dl l ! Cjuj p Lp(R d ) : Similarly using Assumption L.2, for > 0 sufficiently small, Z R d 0 Z 0 b p 2 dl lw (l) p1 (dy) jL uj p Lp(R d ) Z R d 0 jg 1 (t;y)j p L1(R d ) (dy) ! Z 0 w (l) dl l ! jL uj p Lp(R d ) : Using Assumption L.2, and Lemma 38, Z R d 0 Z 0 b p 3 dl lw (l) p1 (dy) L ; 1 2 +1 u p Lp(R d ) Z R d 0 L ; 1 2 g 1 (t;y) p L1(R d ) (dy) ! jL uj p Lp(R d ) +C Z t 0 juj p Lp(R d ) : By similar calculation , Z R d 0 Z 0 b p 4 dl lw (l) p1 (dy)Cjuj p Lp(R d ) : 149 10.2. EXAMPLES CHAPTER 10. QUASILINEAR EQUATION Estimate B 3 (t): Q 2 l g 1 (s;y;y)u (y) Lp(R d ) 3jg 1 (s;;y)j L1(R d ) juj Lp(R d ) : By Assumption L.2 and from Corollary 10 R 1 dl lw(l) p1 <1 , B 3 (t)Cjuj p Lp(R d ) : CombiningtheestimatesforB 1 ;B 2 ;B 3 wederivethedesiredestimateforjQ (t;u 1 ) Q (t;u 2 )j B ;11=p p;pp (R d ) . (iii) Estimate ofjQ (t;u 1 ) Q (t;u 2 )j H ;1=2 2;p (R d ) , denote again u =u 1 u 2 . jQ (t;u 1 ) Q (t;u 2 )j H ;1=2 2;p (R d ) = j (g 1 (t)u)j H ;1=2 2;p (R d ) = J 1=2 f (t) L2;p(R d ) where we denote f (t;x;y) =g 1 (t;xy;y)u (xy) for all (t;x)2E and y2R d 0 . For (t;x)2 E, we let(f k (t;x)) 1 k=1 be l 2 - representation of f (t;x)2 V 2 = L 2 R d 0 ; i.e. f (t;x) = P f k (t;x)e k wherefe k g 1 k=1 is a orthonormal basis ofV 2 =L 2 R d 0 ; : Letf k g 1 k=1 be iid random variables with k = 1 or1 with probability 1 2 . By the standard characterization of L p e.g. Lemma 90 we derive, J 1=2 f (t) p L2;p(R d ) = Z R d 0 J 1=2 f (t;y) 2 (dy) !1 2 p Lp(R d ) = 1 X k=1 J 1=2 f k (t) 2 !1 2 p Lp(R d ) CE 1 X k=0 k J 1=2 f k (t) p Lp(R d ) = CE 1 X k=0 k f k (t) p H ;1=2 p (R d ) : 150 10.2. EXAMPLES CHAPTER 10. QUASILINEAR EQUATION Apply characterization of H ;1=2 p R d in Proposition 26 to (t;x) = P 1 k=0 k f k (t;x), E 1 X k=0 k f k (t) p H ;1=2 p (R d ) C 0 B B @ E 1 X k=0 k f k (t) p Lp(R d ) +E 0 @ Z 1 0 1 X k=0 Q 2 l k f k (t) 2 dl lw (l) 1 A 1=2 p Lp(R d ) 1 C C A = C (F 1 (t) +F 2 (t)) (10.13) Estimate F 1 (t): by Minkowski’s inequality, F 1 (t) =E 1 X k=0 k f k (t) p Lp(R d ) C Z 1 X k=0 f k (t) 2 ! p=2 dx =C Z jf (t)j p V2 dx =C Z Z R d 0 jg 1 (t;xy;y)u (xy)j 2 (dy) ! p=2 dx C Z R d 0 Z jg 1 (t;xy;y)u (xy)j p dx 2=p (dy) ! p=2 =Cjuj p Lp(R d ) Z R d 0 jg 1 (t;;y)j 2 1 (dy) ! p=2 : Estimate F 2 (t): apply Minkowski’s inequality, F 2 (t) = E 0 @ Z 1 0 1 X k=0 k Q 2 l f k (t) 2 dl lw (l) 1 A 1=2 p Lp(R d ) C Z 1 0 " Z E 1 X k=0 k Q 2 l f k (t;x) p dx # 2=p dl lw (l) p=2 C Z 1 0 2 4 Z 1 X k=0 Q 2 l f k (t;x) 2 p=2 dx 3 5 2=p dl lw (l) p=2 = C Z 1 0 2 4 Z " Z R d 0 Q 2 l f (t;x) 2 (dy) # p=2 dx 3 5 2=p dl lw (l) p=2 Applying Minkowski’s inequality again, RHS is bounded by 151 10.2. EXAMPLES CHAPTER 10. QUASILINEAR EQUATION C Z 1 0 Z R d 0 Z Q 2 l f (t;x) p dx 2=p (dy) dl lw (l) p=2 C 2 4 Z 0 Z R d 0 Q 2 l f (t) 2 Lp(R d ) (dy) dl lw (l) p=2 + Z 1 Z R d 0 Q 2 l f (t) 2 Lp(R d ) (dy) dl lw (l) p=2 3 5 = C(F 21 (t) +F 22 (t)): It is easily verified that F 22 (t)Cjuj Lp(R d ) . By Corollary 86 and Corollary 87, for some 1 2 0; 1 2 , Q 2 l f (t) Lp(R d ) Cw (l)fjuL g 1 (t;y)j Lp(R d ) +jg 1 (t;y)L uj Lp(R d ) + L ; 1 2 +1 u Lp(R d ) L ; 1 2 g (t;y) L1(R d ) +juj Lp(R d ) jg (t;y)j L1(R d ) g To estimate F 21 , we split F 21 (t) =F 211 (t) +F 212 (t) +F 213 (t) +F 214 (t) corresponding to each term on RHS. Using Assumption L.2, Corollary 38, and R 1 0 w (l) dl l <1, it can be verified by choosing sufficiently small that F 21 (t)jL uj p Lp(R d ) +Cjuj p Lp(R d ) : To avoid redundant repetition, we only show the estimate for F 211 as an example, F 211 (t) = " Z R d 0 Z 0 juL g (t;y)j Lp(R d ) w (l) 2 dl lw (l) (dy) # p 2 " Z R d 0 Z 0 juL g (t;y)j Lp(R d ) w (l) 2 dl lw (l) (dy) # p 2 juj p Lp(R d ) Z R d 0 jL g (t;y)j 2 L1(R d ) (dy) ! p 2 Z 0 w (l) dl l ! p 2 Cjuj p Lp(R d ) : CombiningtheestimatesforF 1 andF 2 ,wederivethedesiredestimateforjQ (t;u 1 ) Q (t;u 2 )j H ;1=2 2;p (R d ) . Next we present an example in which Q is nonlinear. We denoteD =L ; (dy) =dy=jyj d+ the standard fractional Laplacian of order 2 (0; 2). We now let (dy) =dy=jyj d+ ;2 (0; 2): 152 10.2. EXAMPLES CHAPTER 10. QUASILINEAR EQUATION Example 89. (Locally Lipschitz) Let p2 [2;1), F :R!R, F (0) = 0 and Q (t;x;y;u) = 1^jyj 2 F (u (x)); (t;x)2E;y2R d 0 ;u2H ;1 p R d : We assume that F :R!R, such thatjrFj L1(R) <K <1 d=p << d=p+1 for some 2 (0; 1) (see Proposition 44) Proof. By Theorem 3 (Product Rule) and Theorem 5 (Chain Rule) of [Gat02] with = (1 1=p), we estimate the highest order term in H ;11=p p R d of the difference. jD F (u 1 )D F (u 2 )j Lp(R d ) Cf Z 1 0 jD rF (su 1 + (1s)u 2 )j Lp(R d ) dsj(u 1 u 2 )j L1(R d ) +jD (u 1 u 2 )j Lp(R d ) Z 1 0 jrF (su 1 + (1s)u 2 )j L1(R d ) dsg C Z 1 0 jsD u 1 + (1s)D u 2 j Lp(R d ) dsju 1 u 2 j L1(R d ) +CjD (u 1 u 2 )j Lp(R d ) C jD (u 1 u 2 )j Lp(R d ) +jD u 2 j Lp(R d ) ju 1 u 2 j L1(R d ) +CjD (u 1 u 2 )j Lp(R d ) : By L p - continuity of D with respect to D and by embedding (Proposition 44), it follows that Q is locally Lipschitz i.e. it is Lipschitz in the sense of Assumption L if we only consider u2 H ;1 p R d with juj H ;1 p (R d ) <K <1;K > 0: 153 Bibliography [AA77] Aljančić, S., Arandjelović, D.: O-regularly varying functions. Publ. Inst. Math. (Beograd) 22(36), 5-22 (1977) [BGR14] Bogdan, K., Grzywny, T., Ryznar, M.: Density and tails of unimodal convolution semigroups. J. Funct. 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For p 1 set = 0 @ 1 X j=0 1 X k=0 k a kj ! 2 1 A p=2 : Then there are constants 0<c 1 <c 2 so that c 1 jjajj p Ec 2 jjajj p : Proof. Case1. Let p 2. Since k are independent standard normal, by Minkowski’s inequality, E 0 @ 1 X j=0 " E 1 X k=0 k a kj p !# 2=p 1 A p=2 C 0 @ 1 X j=0 1 X k=0 a 2 kj 1 A p=2 : On the other hand, by Hölder’s inequality, E 0 @ E 1 X j=0 1 X k=0 k a kj ! 2 1 A p=2 c 0 @ 1 X j;k=0 a 2 kj 1 A p=2 : 159 BIBLIOGRAPHY Appendix A : Gaussian Moments Case 2. Let p2 [1; 2). Then, by Hölder’s inequality, E 2 6 4 0 @ 1 X j=0 1 X k=0 k a kj ! 2 1 A p=2 3 7 5 0 @ E 1 X j=0 1 X k=0 k a kj ! 2 1 A p=2 = 0 @ 1 X j;k=0 a 2 kj 1 A p=2 : On the other hand, by Hölder’s and Minkowski’s inequality (recall k are independent standard normal), E 2 6 4E 0 @ 1 X j=0 1 X k=0 k a kj 2 1 A 1=2 3 7 5 p 0 @ 1 X j;k=0 (Ej k a kj j) 2 1 A p=2 c 0 @ 1 X j;k=0 a 2 kj 1 A p=2 : 160 BIBLIOGRAPHY Appendix B : Moment Estimate of Lévy Process Appendix B: Moment Estimate of Lévy Process We will need the following Lévy process moment estimate. Lemma 91. Let 2A : Assume Z jzj1 jzj 1 (dz) + Z jzj>1 jzj 2 (dz)M; (B.1) where 1 ; 2 2 (0; 1] if 2 (0; 1); 1 ; 2 2 (1; 2] if 2 (1; 2); 1 2 (1; 2] and 2 2 [0; 1) if = 1. Let t be the Lévy process associated to , that is Ee i2t = expf ()tg;t 0 There is a constant C =C (M) such that E [j t j 2 ]C (1 +t);t 0: Proof. Recall t = Z t 0 Z (y)yq(ds;dy) + Z t 0 Z (1 (y))yp(ds;dy);t 0; p(ds;dy) is Poisson point measure with Ep (ds;dy) = (dy)ds;q (ds;dy) =p (ds;dy) (dy)ds: Now, t = t + ~ t with t = Z t 0 Z jyj1 (y)yq(ds;dy) + Z t 0 Z jyj1 (1 (y))yp(ds;dy); ~ t = Z t 0 Z jyj>1 (y)yq(ds;dy) + Z t 0 Z jyj>1 (1 (y))yp(ds;dy);t 0: Case 1: 2 (0; 1). In this case (B.1) holds with 1 ; 2 2 (0; 1]. Then for any t> 0; E t t Z jyj1 jyj 1 (dy)Ct; and ~ t 2 = X st h ~ s 2 ~ s 2 i Z t 0 Z jyj>1 jyj 2 p (ds;dy) 161 BIBLIOGRAPHY Appendix B : Moment Estimate of Lévy Process implies thatE ~ t 2 Ct: Case 2: 2 (1; 2). In this case, 1 ; 2 2 (1; 2]. Then E h t 2 i = Z jyj1 jyj 2 (dy)tCt; and, by BDG inequality, E h ~ t 2 i CE 2 6 4 0 @ X st ~ s 2 1 A 2=2 3 7 5 CE 2 4 X st ~ s 2 3 5 =Ct Z jyj>1 jyj 2 d: Case 3: = 1. In this case, 1 2 (1; 2] and 2 2 (0; 1). Similarly as above, we find that E h t 2 i = t Z jyj1 jyj 2 (dy)Ct; E h ~ t 2 i Ct: The statement is proved. 162 BIBLIOGRAPHY Appendix C : A Non-Degeneracy Estimate Appendix C: A Non-Degeneracy Estimate We present a sufficient condition for 2 A in Example 4 for Assumption B to hold. For the sake of completeness we give a summary from the proof in [MX19] (Corollary 6 and Lemma 9 of [MX19]) and refer to the original source for full details. Lemma 92. (Corollary 6 of [MX19]) Let 2A , () = Z 1 0 Z S d1 (rz) (r;dz)d (r); 2B R d 0 ; where = = x2R d :jxj>r ; (r;dz);r > 0 is a measurable family of measures on S d1 with (r;S d1 ) = 1;r> 0. Assume that w =w = 1 is an O-RV function at zero and infinity, inf j ^ j=1 Z S d1 ^ z 2 (r;dz)c 0 > 0; r> 0 andjfs2 [0; 1] :r i (s)< 1gj> 0;i = 1; 2: Then assumptionB holds. That is inf R2(0;1);j ^ j=1 Z jyj1 ^ y 2 ~ R (dy)> 0 Proof. For ^ = 1;R>0; with c 0 > 0, Z jyj1 ^ y 2 ~ R (dy)c 0 Z jyj1 jyj 2 ~ R (dy) = 2c 0 Z 1 0 s 2 w (R) w (Rs) 1 ds s : By Fatou’s Lemma, lim inf R!0 Z jyj1 jyj 2 ~ R (dy) 2 Z 1 0 s 2 1 r 1 (s) 1 ds s =c 1 > 0; lim inf R!1 Z jyj1 jyj 2 ~ R (dy) 2 Z 1 0 s 2 1 r 2 (s) 1 ds s =c 2 > 0; ifjfs2 [0; 1] :r i (s)< 1gj> 0;i = 1; 2; completing the proof. 163 BIBLIOGRAPHY Appendix D : Stochastic Integral Appendix D: Stochastic Integral We discuss here the definition of stochastic integrals with respect to a martingale measure. Let ( ;F;P) be a complete probability space with a filtration of algebras on F = (F t ;t 0) satisfying the usual conditions. Let (U;U; ) be a measurable space with finite measure , R d 0 = R d nf0g. Let p (dt;dz) be Fadapted point measures on ([0;1)U;B ([0;1)) U) with compensator (dz)dt: We denote the martingale measure q (dt;dz) =p (dt;dz) (dz)dt. We prove the following based on Lemma 12 from [MP13]. Lemma 93. Let s2R, 2H s 2;p (E)\H s p;p (E) for p2 [2;1) and 2H s p;p (E) for p2 [1; 2). There is a unique càdlàg H s p R d valued process M (t) = Z t 0 Z (r;x;z)q (dr;dz); 0tT;x2R d ; such that for every '2S R d , hM (t);'i = Z t 0 Z Z J s (r;;z)J s 'dx q (dr;dz); 0tT: (D.1) Moreover, there is a constant C independent of such that E sup tT Z t 0 Z (r;;z)q (dr;dz) H s p (R d ) (D.2) C X j=2;p jj H s j;p (E) ;p 2; E sup tT Z t 0 Z (r;;z)q (dr;dz) H s p (R d ) Cjj H s p;p (E) ;p2 (1; 2): Proof. According to Proposition 15, it is enough to consider the case s = 0. Let n be a sequence defined in Lemma 28 that approximates . Note first that by Lemma 28, for all x, E Z T 0 sup x Z jD n (r;x;z)j p (dz)dr<1: 164 BIBLIOGRAPHY Appendix D : Stochastic Integral Recall for each n, we have n = n Un for some U n 2U with (U n ) <1. Consequently, we define for each x2R d andP-a.s. for all (t;x)2E; M n (t;x) = Z t 0 Z n (r;x;z)q (dr;dz) = Z t 0 Z n (r;x;z)p (dr;dz) Z t 0 Z n (r;x;z) (dz)dr: Obviously, M n (t;x) is càdlàg in t and infinitely differentiable in x. Obviously,M n (t) =M n (t;) isL p R d - valued càdlàg and, by Kunita’s inequality (Theorem 2.11 of [Kun04]), there is a constant C independent of n such that E sup tT jM n (t)j p Lp(R d ) CE Z " Z T 0 Z j n (r;x;z)j 2 dr (dz) # p 2 dx (D.3) +CE Z Z T 0 Z j n (r;x;z)j p dr (dz)dx; CE X j=2;p j n j p Lj;p(E) ; 2p<1: By BDG inequality, for 1<p< 2, E sup tT jM n (t;x)j p CE 2 4 Z T 0 Z j n (r;x;z)j 2 p(dr;dz) ! p 2 3 5 CE " Z T 0 Z j n (r;x;z)j p p (dr;dz) # =CE " Z T 0 Z j n (r;x;z)j p dr (dz) # Hence, by integrating in x, E sup tT jM n (t)j p Lp(R d ) CEj n j p Lp;p(E) : (D.4) In addition, by Fubini theorem,P-a.s. for 0tT;'2S R d ; hM n (t);'i = Z M n (t;x)' (x)dx (D.5) = Z t 0 Z Z n (r;x;z)' (x)dx q (dr;dz): 165 BIBLIOGRAPHY Appendix D : Stochastic Integral By Lemma 28, E X j=2;p j n j p Lj;p(R d ) ! 0; 2p<1; Ej n j p Lp;p(E) ! 0;p2 (1; 2): Similarly, for each p2 [2;1), E sup tT jM n (t)M m (t)j p Lp(R d ) CE X j=2;p j n m j p Lj;p(E) ! 0; and for each p2 (1; 2) E sup tT jM n (t)M m (t)j p Lp(R d ) CEj n m j p Lp;p(E) ! 0; as n;m!1. Therefore there is an adapted càdlàg L p R d -valued process M (t) so that E sup tT jM n (t)M (t)j p Lp(R d ) ! 0 as n!1: Passing to the limit as n!1 in (D.3), (D.4) and (D.5) we derive (D.2), (D.1). Henceforth, we define R t 0 R (r;x;z)q (dr;dz) to be M (t) in this lemma. 166

Abstract (if available)

Abstract Parabolic integro-differential nondegenerate Cauchy problem is considered in the scale of Lₚ-spaces of functions whose regularity is defined by a Lévy measure with O-regularly varying radial profile. The existence of the solution is proved by deriving apriori estimates. The uniqueness of the solution is proved by Itô’s formula. Some probability density estimates of the associated Lévy process are used as well. We also study some quasilinear and stochastic variable coefficient models.

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Creator Phonsom, Chukiat (author)

Core Title On stochastic integro-differential equations

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Publication Date 04/26/2020

Defense Date 12/16/2019

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Levy processes,non-local parabolic integro-differential equations,OAI-PMH Harvest,O-regularly varying functions

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Mikulevicius, Remigijus (committee chair), Ghanem, Roger (committee member), Lototsky, Sergey (committee member)

Creator Email chukiatp@gmail.com,phonsom@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-289469

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Legacy Identifier etd-PhonsomChu-8352.pdf

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