Discrete Analogues in Harmonic Analysis

Transcription

1 Discrete Analogues in Harmonic Analysis Lillian Beatrix Pierce A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Mathematics Adviser: Elias M. Stein June 2009

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3 c Copyright by Lillian Beatrix Pierce, All rights reserved.

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5 Abstract iii The study of discrete analogues in harmonic analysis originated in the early 1900 s when M. Riesz noted that his work on the L p boundedness of the Hilbert transform implied the boundedness of a closely related discrete operator. Results for several other simple discrete analogues also follow trivially from their counterparts in the classical setting, but many discrete operators present distinctive difficulties apparently unapproachable from the continuous perspective. It was not until the work of J. Bourgain in the late 1980 s, and in particular the introduction of ideas from the circle method of analytic number theory, that a renewed study of discrete analogues began in earnest. This thesis presents new results for several classes of discrete operators. First, this thesis presents sharp results for families of discrete fractional integral operators along paraboloids. These results are further generalized to operators defined in terms of arbitrary positive definite quadratic forms with integer coefficients. Second, this thesis presents results on the boundedness of twisted discrete singular Radon transforms and related discrete oscillatory integral operators. Third, this thesis presents results for a discrete analogue of fractional integration on the Heisenberg group, as well as generalizations to fractional integral operators associated to symplectic bilinear forms of lower rank. The methods employed include classical analytic methods as well as techniques from number theory, including the circle method, theta functions, and exponential sums.

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7 Contents Abstract Acknowledgements iii x 1 Introduction A brief history of discrete analogues Integration over submanifolds Maximal Radon transforms Bourgain s method Recent progress on discrete analogues Translation invariant fractional integral operators Translation invariant singular Radon transforms Quasi-translation invariant operators Spherical averages The results of the Thesis Discrete fractional integration along paraboloids Discrete fractional integration and quadratic forms Twisted discrete singular Radon transforms and related operators Discrete fractional integration on the Heisenberg group Outline of the Thesis Preliminaries Analytic preliminaries Fourier transforms L p spaces Weak L r spaces Lorentz spaces L p,q Restricted weak-type operators Calderón-Zygmund kernels T T arguments Discrete analogues Conventions Equivalence of discrete operators Simple arguments for discrete analogues The implication method The imitation method

8 vi CONTENTS Discrete fractional integral operators Number theoretic preliminaries Notation Dissections of the unit interval Exponential sum bounds Linear exponential sums Gauss sums Weyl sums Gauss sums in higher dimensions Quadratic forms Notation Comparison of forms Gauss sums of quadratic forms Theta functions Jacobi inversion formula Jacobi inversion formula for a twisted theta function Theta function of a quadratic form Twisted theta function of a quadratic form Sums of squares Average results Asymptotics Representations by quadratic forms Remarks on the singular series Results for the average A Q,k The classical circle method Waring s problem Major and minor arcs Main contribution from the major arcs Bounding the contribution of the minor arcs Parallels to work in discrete analogues Fractional integration on paraboloids Introduction Continuous analogues Statement of the Theorems Outline of Chapter Preliminaries Auxiliary bounds Necessary conditions for I λ Necessary conditions for J λ The operator I λ : 1/2 < λ < Weak-type result for m λ Major and minor arcs The operator I λ : methods for k = 2, A basic decomposition for J λ

9 CONTENTS vii A dyadic decomposition of the multiplier Major/minor decomposition The approximate identity The operator J λ : 1/2 < λ < The minor arcs The remainder term The major arcs Removing the restriction p 2 q The operator J λ : 2/(k + 4) < λ < A major/minor decomposition An interpolation argument Motivation for the double decomposition method An alternative derivative method The double decomposition method for J λ The double decomposition of the major arcs The main term supported on the major arcs: r The main term supported on the major arcs: r = The remainder term supported on the major arcs The contribution of the minor arcs Fractional integration II: Quadratic forms Introduction Statement of the Theorems Outline of Chapter Preliminaries: full rank A reduction argument Auxiliary bounds Necessary conditions for I Q,λ, J Q,λ The operator I Q,λ : full rank The main term on the minor arcs The remainder term The main term on the major arcs The operator J Q,λ : full rank The approximate identity The major arcs: a double decomposition The main term supported on the major arcs The remainder term supported on the major arcs The contribution of the minor arcs Lower rank quadratic forms Preliminaries: lower rank A reduction Auxiliary bounds Necessary conditions for I λ Necessary conditions for J λ The operator I λ : lower rank

10 viii CONTENTS Minor arcs Major arcs Remainder terms The operator J λ : lower rank Approximate identity Major/minor decomposition Major arcs M 1 M 2 on the major arcs M 1 E 2 on the major arcs M 2 E 1 on the major arcs E 1 E 2 on the major arcs Minor arcs Minor arc estimate for all terms on R(λ) = Minor arcs estimates for R(λ) = 2/k Twisted discrete singular Radon transforms Introduction Translation invariant operators Quasi-translation invariant operators Outline of Chapter Proof of Theorem Method of descent Proof of Corollary Proof of Theorem Outline of the inductive major/minor decomposition The inductive procedure The base case The inductive step Bounding the final operator The error term operators Product formulation for T Bounding the Gauss sum operator Bounding T Fractional integration on H k Introduction Continuous analogue Statement of the Theorems Outline of method Outline of Chapter Preliminaries Auxiliary bounds Necessary conditions A decomposition of the operator Bounds on the line R(λ) = 1/k The major arcs operator

11 CONTENTS ix The shifted ball reduction The first approximation step The second approximation step The Gauss sum operator S The integral operator I Concluding the major arcs The minor arcs operator A comparison to other methods of estimating the exponential sum The error term operators The term E (2) The term E (1) : a simple estimate The term E (1) : a sharper bound Balancing the contributions of the major and minor arcs Choosing the parameters β, β 0, β Balancing the contributions from the major and minor arcs Concluding remarks Fractional integration on H k II: Lower rank Introduction Fractional integration on H 1 : interior results Higher dimensions General symplectic bilinear forms The operator TΩ λ A simple reduction A basic decomposition The major arcs The minor arcs operator Error term operators Concluding arguments Bibliography 211

12 x Acknowledgements CONTENTS To Eli Stein I owe my greatest thanks. It is because of the good fortune that ten years ago he taught the first mathematics course I took as an undergraduate at Princeton, soon followed by many other appealing courses, that I have studied mathematics. In the years since, he has taught me much of the analysis that I know, and his beautiful lectures and uniquely comprehensible mathematical writings continue to serve as my model for how to speak and write about math. I am grateful that for my thesis topic he suggested the area of discrete analogues in harmonic analysis, which has allowed me to explore an unusual intersection between analytic number theory and harmonic analysis. I have benefited greatly from his suggestions of new and interesting problems, his encouragement, and our many intricate discussions. I always leave Fine 802 with a simpler way to think about something, or a new perception of what really matters in the paper I am reading; I hope some day I ll be able to arrive at such clarity so easily in my own office! I have also benefited from many conversations with Peter Sarnak, who has always been very generous in discussing my questions about number theory, as has been Roger Heath- Brown. Charlie Fefferman also elucidated several points for me, for which I am very grateful. During a visit to the University of Wisconsin at Madison in May 2008, I had several stimulating conversations with Alexandru Ionescu and Stephen Wainger, whose work on discrete analogues has also been an inspiration. I am also grateful to Michael Christ for providing a copy of his unpublished manuscript from 1988 on fractional integration on the Heisenberg group. Finally, many thanks go to Stephen Wainger for serving as an official reader of this thesis, and to Alice Chang and Robert Gunning for serving on the final public oral committee. My officemates Melanie Matchett Wood and Polam Yung have been wonderfully pleasant and knowledgeable companions for these years. It has always been a privilege to work in the Math Department: the resources of the library and the generosity and flexibility of the department have seemed at times miraculous, and I thank all of the staff for their constant efforts to make everything run smoothly. During my graduate studies at Princeton I have been generously funded by the National Science Foundation, the Princeton University Centennial Fellowship, and the 250th Anniversary Fund for Innovation in Undergraduate Education, and I am grateful for the opportunities these fellowships have afforded me. And to my husband, Tobias Overath, I am grateful for his constancy and perseverance, which allowed us to best the Atlantic Ocean as we pursued our individual scientific explorations. Through his simultaneous doctoral studies in neuroscience, I gained perspective on the special ways the study of mathematics is different from other sciences, as well as the way in which all of us are searching for exactly the same thing: the truth.

13 Chapter 1 Introduction 1.1 A brief history of discrete analogues The study of discrete analogues in harmonic analysis stretches companionably back to the early history of singular integrals. In 1928, M. Riesz [34] proved that the Hilbert transform, Hf(x) = p.v. 1 π R f(x t) dt, t is bounded on L p (R), for all 1 < p <. In this classic paper, Riesz further remarked that the boundedness of the Hilbert transform implies the boundedness of its discrete analogue, Hf(n) = m Z m 0 f(n m) m, on L p (Z), for all 1 < p <. Riesz s observation was generalized in 1952 to more general singular integral operators by Calderón and Zygmund [7], who observed that the boundedness on L p (R k ) for 1 < p < of the singular integral operator T f(x) = p.v. f(x y)k(y)dy, R k where K belongs to the broad class of singular kernels now known as Calderón-Zygmund kernels, implies the L p (Z k ) boundedness of its discrete analogue T f(n) = m Zk m 0 f(n m)k(m). In a survey of the fundamental operators in harmonic analysis, the maximal function of Hardy and Littlewood would appear prominently, and it does here too: the boundedness on L p (R k ) for 1 < p of the classical maximal function 1 Mf(x) = sup f(x y) dy r>0 V (B r ) y <r

14 2 CHAPTER 1. INTRODUCTION implies the L p (Z k ) boundedness of its discrete analogue, Mf(n) = sup r>0 1 N(B r ) m Z k m <r f(n m). (1.1) Here V (B r ) denotes the volume of the open ball B r of radius r in R k, while N(B r ) denotes the number of integer lattice points lying within B r. In fact, in both the case of Calderón- Zygmund operators and the maximal function, the discrete result may be obtained not only by means of the continuous result, but also by following the method of proof used in the continuous setting (see Section 2.3). Another classical family of operators in harmonic analysis are the fractional integral operators, defined for 0 < λ < 1 by f(x y) I λ f(x) = R y kλ dy. (1.2) k For these operators, the Hardy-Littlewood-Sobolev theorem states that for 1 < p < q < with 1/q = 1/p (1 λ), I λ is a bounded operator from L p (R k ) to L q (R k ). The simplest discrete analogue of this operator is defined by I λ f(n) = m Zk m 0 f(n m) m kλ, (1.3) where n Z k and 0 < λ < 1. As a consequence of the boundedness of I λ, it follows that I λ maps L p (Z k ) to L q (Z k ) boundedly for all 1/q 1/p (1 λ) with 1 < p < q <, and moreover, this result is sharp (see Proposition 2.4). Let us consider a variation on the operator I λ, defining instead I P λ f(n) = m Z k 1 m 0 f(n P (m)) m k1λ, (1.4) where n Z k2, 0 < λ < 1, and P is a one-to-one mapping from Z k1 to Z k2. For this operator, the simple trick of comparing it to its continuous analogue, or imitating a method of proof used in the continuous case, is not so successful. In fact, such a simple comparison argument yields only the same result as for I λ (see Proposition 2.5): that Iλ P maps Lp (Z k2 ) to L q (Z k2 ) boundedly for 1/q 1/p (1 λ) with 1 < p < q <. That this is not in general sharp may already be seen in the one-dimensional case k 1 = k 2 = 1 with P (m) = m s, s 2. Then the operator I P λ f(n) = m=1 f(n m s ) m λ (1.5) is expected to be bounded from L p (Z) to L q (Z) whenever 1/q 1/p (1 λ)/s and 1/q < λ, 1/p > 1 λ. That these conditions are necessary can be shown by simple examples; in fact, note that the first condition arises naturally, since (with equality) it is the homogeneity condition required by the continuous analogue of Iλ P. Proving that these conditions are also sufficient is much more difficult. This is known only for s = 2, due to work of Stein, Wainger,

15 CHAPTER 1. INTRODUCTION 3 and others [42], [43], [31], [23]; the full result currently remains out of reach for s 3, due to its relation to a major unsolved conjecture in number theory (the Hypothesis K of Hardy and Littlewood; see [42]). This is one indicator of the distinctive complications that can arise in the discrete case but are not at all apparent in the continuous case. 1.2 Integration over submanifolds If the boundedness of discrete analogues always arose simply from comparison to the continuous case, there would be no further story to tell. But as we have already seen for operators of the form Iλ P, this is not the case. In fact, many difficult classical problems in number theory are discrete versions of comparatively easy continuous problems: for example, the infinite sum defining the Riemann zeta function may be regarded as analogous to the corresponding simple integral; counting the number of integer lattice points lying on a sphere of a fixed radius in R k (Waring s problem) is the discrete analogue of computing the surface area of the sphere. In the setting of harmonic analysis, one particular type of problem provides particularly challenging discrete analogues: integration over submanifolds. This naturally points to the study of Radon transforms, which are for the most part well understood in the continuous case, but still pose interesting problems in the discrete case Maximal Radon transforms Perhaps the simplest Radon transform to consider is a translation invariant maximal Radon transform. Given a portion of a k 0 -dimensional submanifold S in R k that may be written as the image of a smooth mapping t γ(t) as t ranges over the unit ball in R k0, the corresponding maximal function is defined by 1 M γ f(x) = sup f(x γ(t))dt 0<r 1 r k0. Under the geometric assumption that S is of finite type at γ(0), i.e. that S has a finite order of contact with any hyperplane in R k, it is known that for every 1 < p, M γ is bounded on L p (R k ). Most relevant to the discrete setting is the case in which γ(t) is given by a polynomial function, γ(t) = P (t) = (P 1 (t),..., P k (t)), where P j (t) are real-valued polynomials of t R k0. Then the corresponding maximal function M P is defined by 1 M P f(x) = sup f(x γ(t))dt r>0 r k0 ; M P is again bounded on L p (R k ) for 1 < p. (For a full discussion, see Theorem 1 in Chapter XI of [37], whose notation we follow.) A discrete analogue of M P may be defined by 1 M P f(n) = sup f(n P (m)), r>0 N(B r ) t r t r m Z k 0 m r

16 4 CHAPTER 1. INTRODUCTION where n Z k and P : Z k0 Z k is a polynomial. This maximal operator was considered by Bourgain [4], [5], [6], who proved (among other results) that in the one-dimensional case k = 1, M P is a bounded operator from L p (Z) to itself, for all 1 < p. (Bourgain was initially interested in this in an ergodic context, but for this presentation we will simply consider it as a maximal function.) The key contribution of Bourgain s work was the introduction of ideas from the circle method, a technique from analytic number theory. As this has influenced the field of discrete analogues significantly, and will play a fundamental role in the work of this thesis, it is worth outlining the key points of his original treatment Bourgain s method Crucial to obtaining the L p bound was an estimate of the operator M P on L 2. For each r > 0, let A r f(n) = 1 r f(n P (m)) r + 1, m=0 so that M P = sup r>0 A r. For the L 2 bound Bourgain used the discrete Fourier transform in order to define for each A r its Fourier multiplier, m r (θ) = 1 r + 1 r e 2πiP (m)θ, m=0 where θ [0, 1]. Bourgain showed that each m r (θ) may be closely approximated in the L norm by another function k r (θ) supported on a subset of [0, 1] called, in the terminology of the classical circle method, the major arcs. The major arcs may be described (very roughly) as consisting of those θ [0, 1] that are sufficiently close to a rational number a/q (in reduced form) with denominator q sufficiently small. (We will describe the concept and function of major arcs more precisely in Section 2.11). The meaning of sufficiently close and sufficiently small is chosen to depend on r, so that a different set of major arcs is defined for each r > 0. To make the approximation k r of m r for each fixed r, Bourgain showed that on each interval belonging to the collection of major arcs, say where θ is approximated by a/q, the multiplier m r (θ) may be well-approximated by an integral version of m r, call it ν r, evaluated at the small error θ a/q, times a particular type of finite exponential sum S evaluated at a/q. (Precisely, the exponential sum is a Gauss sum, if P has degree 2, or a Weyl sum, if deg(p ) 3.) Bourgain then defined the function k r (θ) approximating the multiplier m r (θ) to be a smoothed sum of the contribution of these terms ν r (θ a/q)s(a/q) on the collection of all major arcs for that fixed r. The error between k r (θ) and m r (θ) is supported on the complement of the major arcs in [0, 1], also known as the minor arcs. This error can be shown to be small, so the problem is reduced to showing that the operator with multiplier k r (θ) is bounded on L 2, or more properly, the supremum over all r > 0 of such operators is bounded on L 2. This is achieved by combining three main facts: first, the normalized Gauss (or Weyl) sum 1 q S(a/q) is bounded with decay in q; second, all the major arcs corresponding to distinct rationals a/q, a /q are disjoint (for each fixed r); third, the factors ν r are multipliers corresponding to convolution operators whose supremum over r > 0 is shown to be bounded on L 2 in the continuous setting.

17 CHAPTER 1. INTRODUCTION 5 While this outline is purposefully vague, the key points to remember are as follows: first, the main contribution to the multiplier m r (θ) comes from θ in a collection of major arcs, while the contribution to m r (θ) from θ in the minor arcs is bounded as an error term. Second, on each major arc, say where θ is approximated by a/q, the arithmetic information encapsulated in the rational a/q is separated out and used in a finite exponential sum for which number theoretic bounds are known, while the continuous information, or error term θ a/q, is used to compare the discrete multiplier to a multiplier, often of an operator in the continuous setting, for which the relevant bound is already known or may be shown more easily. We will return to these ideas in Section 2.11, in which we describe the circle method as it arises in number theory, and highlight the key points that will be used to study discrete analogues in this thesis. 1.3 Recent progress on discrete analogues We now turn to recent progress on discrete analogues. While there are a range of approaches to discrete analogues, including interesting results using sampling theory and Carleson s theorem, we will focus here on results that involve number theoretic methods, and in particular on those that are most closely related to the work of this thesis Translation invariant fractional integral operators Recall the fractional integral operators Iλ P along curves defined in (1.4). Obtaining sharp bounds for such operators in the specific case where P (m) = m 2 has been the focus of work by Stein and Wainger in [42], [43]. Define operators I λ and J λ for 0 < λ < 1 by where n Z, and I λ f(n) = m Z m 0 J λ f(n, t) = m Z m 0 f(n m 2 ) m λ, f(n m, t m 2 ) m λ, where (n, t) Z 2. Stein and Wainger s work, in combination with a result from Ionescu and Wainger [23] (and independent work by Oberlin [31]), resulted in the following theorems: Theorem A (Stein and Wainger; Oberlin; Ionescu and Wainger). For 0 < λ < 1, I λ : L p (Z) L q (Z) boundedly if and only if p, q satisfy (i) 1/q 1/p 1 2 (1 λ), (ii) 1/q < λ, 1/p > 1 λ. Theorem B (Stein and Wainger; Oberlin; Ionescu and Wainger). For 0 < λ < 1, J λ : L p (Z 2 ) L q (Z 2 ) boundedly if and only if p, q satisfy (i) 1/q 1/p 1 3 (1 λ), (ii) 1/q < λ, 1/p > 1 λ.

18 6 CHAPTER 1. INTRODUCTION These results ultimately arose from a combination of methods. In the original paper [42], Stein and Wainger derived the boundedness of I λ, J λ for λ > 1/2 and certain p, q from a weak-type L r, bound for the Fourier multipliers associated to the operators, proved using a circle method decomposition of the spectral variables. In later work presented in [43], Stein and Wainger then extended their result to values λ < 1/2 by an intricate decomposition and interpolation argument, again using a circle method decomposition of the spectral variables. In this work they utilized both L 2 L 2 bounds on the line R(λ) = 1 and also L 1 L bounds for certain R(λ) < 0; these last bound are particularly interesting, as they arise from bounds for the Fourier coefficients of the multipliers. We extend these methods in Chapters 3 and 4 to treat a wide range of operators related to I λ, J λ, in particular considering higher dimensional analogues involving not only the quadratic form m 2 but also arbitrary positive definite quadratic forms Q(m). Oberlin s independent work [31] on the operators I λ, J λ had a distinctly different flavor, as it relied not on the circle method, but on a lemma of Christ [9], somewhat combinatorial in nature. With his approach, Oberlin was able to prove interior results, i.e. those cases with strict inequality in the conditions (i) in the above theorems. We use a method inspired by Oberlin s idea to treat a discrete analogue of fractional integration on the Heisenberg group H 1 in Chapter Translation invariant singular Radon transforms A class of Radon transforms that has elicited significant interest in the continuous case is the class of singular Radon transforms, given by T γ f(x) = p.v. f(x γ(t))k(t)dt, t 1 where γ is a smooth mapping of the unit ball in R k0 to R k that is of finite type at the origin, and K is a type of Calderón-Zygmund kernel in R k0. A special case of this class of operator occurs when γ(t) = P (t) is a polynomial function, in which case we define T P f(x) = p.v. f(x P (t))k(t)dt. R k 0 It is known that both T γ and T P extend to bounded operators on L p (R k ), for 1 < p < (see 4.5 of Chapter XI in [37], as well as [39]). A discrete analogue of T P is given by T P f(n) = f(n P (m))k(m), m Z k 1 m 0 where K is a Calderón-Zygmund kernel and P = (P 1,..., P k2 ) is a polynomial mapping R k1 R k2 such that P (Z k1 ) Z k2. The result of Ionescu and Wainger [23] mentioned in the context of the operators I λ, J λ is a general theorem about the discrete analogue T P : Theorem C (Ionescu and Wainger). The operator T P is bounded on L p (Z k2 ) for every 1 < p <, with T P f Lp (Z k 2 ) A p f Lp (Z k 2 ). The constant A p depends only on p, the dimension k 1, and the degree of the polynomial P.

19 CHAPTER 1. INTRODUCTION 7 Such operators were also considered earlier by Arkhipov and Oskolkov [1], who proved an L 2 result in dimension k 1 = k 2 = 1, and in [40], in which Stein and Wainger proved that T P is bounded on L p in the limited region 3/2 < p < 3, in all dimensions. In fact, Ionescu and Wainger further prove a more general result than Theorem C for a related discrete singular Radon transform acting on functions F of R k2 by T P F (x) = F (x P (m))k(m), m Z k 1 m 0 where P is now allowed to be a polynomial map from R k1 to R k2 with real coefficients. The more general result is that T P extends to a bounded operator on L p (R k2 ) with T P F Lp (R k 2 ) A p F L p (R k 2 ) for every 1 < p <, with the same restrictions on the dependence of the constant A p as in Theorem C. In fact, as was observed by E. M. Stein, the results for these two operators are equivalent in the case that P has integral coefficients (see Section 2.2.2). Theorem C is quite a deep result and requires substantial technical work. A key component of the proof is again a decomposition of the spectral variable according to Diophantine approximations of the coefficients of the polynomial P, in the style of the circle method. However, unlike previous arguments, in which the decomposition was based simply on the sizes of the denominators in the rational approximations, the method of Ionescu and Wainger crucially exploits almost orthogonality properties among the denominators as well. 1 We will extend the result of Ionescu and Wainger in Chapter 5 to the case of twisted discrete singular Radon transforms: these are operators of the type T P, but including a twisting factor e 2πiQ(m) in the kernel Quasi-translation invariant operators The most general formulation of a singular Radon transform is, roughly speaking, an operator of the form Rf(x) = p.v. f(γ t (x))k(t)dt, R k 0 where K is a Calderón-Zygmund kernel and γ t ( ) is a family of diffeomorphisms of R k that depend smoothly on t R k0. One must make various assumptions about this family, including that γ 0 is the identity mapping, and that the family of varieties {γ t (x) : t R k0 } x R k have certain curvature properties generalizing the property of finite type; these curvature properties may be stated either in terms of iterates of the mappings γ t or commutation properties of associated vector fields. (The general theory of such Radon transforms may be found in [10], which also points to earlier literature in the field.) Non-translation invariant singular Radon transforms of the form R present new difficulties, even in the continuous case, as the lack of translation invariance severely hampers the use of the Fourier transform. One may easily define a discrete analogue of R, but the inapplicability of Fourier transform methods seems particularly problematic in the discrete case. In fact, 1 In this sense, it could be considered somewhat akin to the so-called double Kloosterman refinement of the circle method [26], [19], which accomplishes cancellation involving the denominators of rational approximations.

20 8 CHAPTER 1. INTRODUCTION recall that the circle method techniques, which open up approaches to many discrete operators, rely on a decomposition of the spectral variable, which would not be possible for a fully nontranslation invariant operator. Thus we turn instead to quasi-translation invariant operators, i.e. those that are translation invariant in one or more variables, which allow one to make use of a partial Fourier transform. That quasi-translation invariant operators are still sufficiently general to be of interest is made clear by the fact that singular integrals on the Heisenberg group are quasi-translation invariant in this sense. Currently, there has not been progress in the fully non-translation invariant discrete case. However, recently substantial progress has been made on a class of quasi-translation invariant operators introduced by Stein and Wainger in [41]: Theorem D (Stein and Wainger). Let R be the discrete singular Radon transform defined by Rf(n, n ) = f(n m, n P (n, m))k(m), m Zk (1.6) m 0 where (n, n ) Z k Z l, m Z k, the mapping P : Z k Z k Z l is the restriction of a polynomial on R k R k, and K is a Calderón-Zygmund kernel. Then Rf L 2 (Z k Z l ) A f L 2 (Z k Z l ), where the constant A depends only on the dimensions k, l and the degree of the polynomial P. While the operator R is not translation invariant in all dimensions (if P is nontrivial), it is translation invariant with respect to the second variable n. Taking the Fourier transform with respect to n leads one to consider a related oscillatory operator: m Zk n m 0 If(n) = e 2πiQ(n,m) K(n m)f(m), (1.7) where n Z k and Q is a real-valued polynomial on R k R k. The main result of [41] is that the operator I is bounded on L 2 (Z k ), with a bound dependent only on the degree of Q, and independent of its coefficients. By Parseval s theorem, R is then also bounded on L 2 (Z k ). The method of proof is quite involved, utilizing an inductive decomposition of the operator I based on the Diophantine properties of the coefficients of the polynomial Q, à la the circle method, working inductively from the highest order terms to the lowest order terms. The goal is to reduce the operator to a tensor product of a discrete Gauss sum operator with nice arithmetic properties, and a discrete operator that may be closely approximated by its continuous analogue, which is known to be bounded on L 2 by the work of Ricci and Stein [32]. We will extend these results in two directions. First, in Chapter 5, we prove by a series of simple observations that the operator I is not only bounded on L 2 (Z k ) but on L p (Z k ) for all 1 < p <. Note that this does not have any implications for the operator R, since only the L 2 properties of these two operators are related. Second, in Chapters 6 and 7, we develop a method inspired both by the approach of Stein and Wainger to bounding the operator I on L 2, and also their later work [43] on the fractional integral operator J λ, to treat a discrete analogue of fractional integration on the Heisenberg group H k. Very recently, Ionescu, Magyar, Stein and Wainger [22] have made further progress, proving L p bounds for operators of the form R:

21 CHAPTER 1. INTRODUCTION 9 Theorem E (Ionescu, Magyar, Stein and Wainger). If the polynomial P is of degree at most 2, the discrete singular Radon transform R defined in (1.6) may be extended to a bounded operator on L p (Z k Z l ), for all 1 < p <, with Rf L p (Z k Z l ) A f L p (Z k Z l ), where the constant A depends only on p and the dimension k. Furthermore, they prove a result for a quasi-translation invariant discrete maximal Radon transform: Theorem F (Ionescu, Magyar, Stein and Wainger). Let M be the discrete maximal Radon transform defined by Mf(n, n ) = sup 1 f(n m, n P (n, m)) r>0 N(B r ), m <r where P : Z k Z k Z l is a polynomial. Then in the case where P has at most degree 2, M extends to a bounded operator on L p (Z k Z l ), for all 1 < p, with Mf L p (Z k Z l ) A f L p (Z k Z l ), where the constant A depends only on p and the dimension k. This last result extends the original work of Bourgain in the translation invariant, onedimensional case, and similarly has applications to pointwise and L p (1 < p < ) ergodic theorems. Both these results use the restriction on the degree of P intrinsically in the method of proof, as the operators R and M are related to group translation invariant Radon transforms on discrete nilpotent groups of step 2, to which Fourier transform methods are then applied. Higher degree polynomials would require nilpotent Lie groups of higher step, on which Fourier transform methods may not be applicable; the writers of [22] hope to return to this problem in future work. Finally, note that for discrete singular Radon transforms, both in the translation invariant and non-translation invariant cases, as well as for their more classical continuous analogues, an appropriate L 1 theory remains unknown Spherical averages Lastly, although we will not address these operators further in this thesis, we turn to the very interesting subject of discrete maximal spherical averages, defined by A f(n) = sup r>0 1 N(r) m Z k m =r f(n m), where N(r) = #{m : m = r} is the number of integer lattice points lying on the sphere of radius r in R k and the supremum is further restricted to those r such that N(r) 0. Such operators have been studied by Magyar [27], Magyar, Stein and Wainger [28], and Ionescu [21], resulting in the following optimal result:

22 10 CHAPTER 1. INTRODUCTION Theorem G (Magyar, Stein and Wainger; Ionescu). The discrete spherical maximal operator A extends to a bounded operator on L p (Z k ) if and only if k 5 and p > p k = k/(k 2), or k 4 and p =. Moreover, for k 5, A satisfies a restricted weak-type estimate at the endpoint, namely it extends to a bounded operator from L p k,1 (Z k ) to L p k, (Z k ). This theorem again uses Fourier transform methods and a circle method decomposition, as well as an interesting method of transference via a sampling theorem. This result is also noteworthy because it is an example of a discrete result that differs from its continuous analogue not only in the method of proof, but also in the result itself. The continuous analogue of A is defined by A f(x) = sup A λ f(x), 0<λ< where A λ f = f dσ λ, with dσ λ the normalized invariant measure on the sphere of radius λ in R k. In contrast, the result for this operator in the continuous setting is that A f L p (R k ) A f L p (R k ) if p > k/(k 1) and k 2. (See [36], [39] for k 3, [3] for k = 2.) Thus while classical operators in the continuous setting play a valuable role in motivating the study of discrete analogues, and often in predicting the optimal results for the discrete operators, the analogy is not always complete. This is yet another reason why discrete analogues are both intriguing and challenging. 1.4 The results of the Thesis We now turn to the main results of this thesis. These pertain to three classes of operators: discrete fractional integral operators along paraboloids, twisted discrete singular Radon transforms and related oscillatory integral operators, and discrete fractional integral operators on the Heisenberg group. Here we state the theorems without elaboration; detailed discussions may be found in the relevant chapters, which are outlined in Section Discrete fractional integration along paraboloids Define operators I λ and J λ for 0 < λ < 1 by where n Z, and I λ f(n) = m Zk m 0 J λ f(n, t) = m Zk m 0 f(n m 2 ) m kλ, f(n m, t m 2 ) m kλ, where n Z k, t Z. These operators were introduced in the one-dimensional case k = 1 by Stein and Wainger [42], [43] (see Theorems A and B in Section 1.3.1); we study these operators in all dimensions k 2. In the case of the operator I λ, we prove the sharp results for all dimensions k 4:

23 CHAPTER 1. INTRODUCTION 11 Theorem 1.1. If k 4 and 0 < λ < 1, I λ is a bounded operator from l p (Z) to l q (Z) if and only if λ > 1 2/k and p, q satisfy (i) 1/q 1/p k 2 (1 λ) (ii) 1/q < 1 k 2 (1 λ), 1/p > k 2 (1 λ). For k = 2, 3 the same result holds, under the additional restriction that 1/2 < λ < 1. For k = 2, 3, we prove nearly sharp results for the remaining region 1 2/k < λ 1/2 : Corollary If k = 2, 3, I λ is a bounded operator from l p (Z) to l q (Z) if 1 2/k < λ 1/2 and p, q satisfy (i) 1/q < 1/p k 2 (1 λ) (ii) 1/q < 1 k 2 (1 λ), 1/p > k 2 (1 λ). In the case of the operator J λ, we prove sharp results in all dimensions k 2: Theorem 1.2. For k 2 and 0 < λ < 1, the operator J λ is a bounded operator from l p (Z k+1 ) to l q (Z k+1 ) if and only if p, q satisfy (i) 1/q 1/p k k+2 (1 λ) (ii) 1/q < λ, 1/p > 1 λ Discrete fractional integration and quadratic forms We furthermore consider operators modeled on I λ, J λ, but with the quadratic form m 2 replaced by more general quadratic forms. Given Q 1, Q 2 positive definite quadratic forms in k variables with integer coefficients, define the operators where n Z, and I Q1,Q 2,λf(n) = m Zk m 0 J Q1,Q 2,λf(n, t) = m Zk m 0 where n Z k, t Z. We prove the following results: 2 f(n Q 1 (m)) Q 2 (m) kλ/2, f(n m, t Q 1 (m)) Q 2 (m) kλ/2, Theorem 1.3. For k 4 and 0 < λ < 1, I Q1,Q 2,λ is a bounded operator from l p (Z) to l q (Z) if and only if λ > 1 2/k and p, q satisfy (i) 1/q 1/p k 2 (1 λ) (ii) 1/q < 1 k 2 (1 λ), 1/p > k 2 (1 λ). 2 Note that if one obtains Theorems 1.3 and 1.4 for any single quadratic form Q 2, the results then automatically hold for all quadratic forms Q 2 ; in particular it would be sufficient to consider only the quadratic form 2 in the role of Q 2. Yet the method of proof will require the ability to choose Q 2 specifically, and to reflect this flexibility we state the theorems as above; see Section for a full discussion.

24 12 CHAPTER 1. INTRODUCTION Specifically, for such p, q there exists a constant A Q1,Q 2 such that I Q1,Q 2,λf l q (Z) A Q1,Q 2 f l p (Z), where the constant A Q1,Q 2 depends on k, Q 1, Q 2, p, q, λ. For k = 2, 3 the same result holds, with the additional restriction that 1/2 < λ < 1. Once again, in the cases k = 2, 3, we obtain nearly sharp results for the remaining region 1 2/k < λ 1/2: Corollary If k = 2, 3, I Q1,Q 2,λ is a bounded operator from l p (Z) to l q (Z) if 1 2/k < λ 1/2 and p, q satisfy (i) 1/q < 1/p k 2 (1 λ) (ii) 1/q < 1 k 2 (1 λ), 1/p > k 2 (1 λ). For the operator J Q1,Q 2,λ we obtain sharp results in all dimensions k 2: Theorem 1.4. For any k 2 and 0 < λ < 1, the operator J Q1,Q 2,λ is a bounded operator from l p (Z k+1 ) to l q (Z k+1 ) if and only if p, q satisfy (i) 1/q 1/p k k+2 (1 λ) (ii) 1/q < λ, 1/p > 1 λ. Specifically, for such p, q there exists a constant A Q1,Q 2 such that J Q1,Q 2,λf l q (Z k+1 ) A Q1,Q 2 f l p (Z k+1 ), where the constant A Q1,Q 2 depends on k, Q 1, Q 2, p, q, λ Lower rank quadratic forms One might further ask how the operators I Q1,Q 2,λ, J Q1,Q 2,λ behave if Q 1 is not a full rank quadratic form, but a form in k 1 variables, with 1 k 1 k. We consider such operators in the following theorems: Theorem 1.5. Let Q 1 be a positive definite quadratic form in k 1 variables with integer coefficients, with 1 k 1 k. Let Q be a positive definite quadratic form in k variables with integer coefficients. For k 2 and 0 < λ < 1, I Q1, Q,λ is a bounded operator from lp (Z) to l q (Z) if λ > max(1 k 1 /k, 1 2/k) and if additionally λ > 1 k 1 /2k and p, q satisfy (i) 1/q 1/p k 2 (1 λ) (ii) 1/q < 1 k 2 (1 λ), 1/p > k 2 (1 λ). Specifically, for such p, q there exists a constant A Q1, Q such that I Q1, Q,λ f l q (Z) A Q1, Q f l p (Z), where the constant A Q1, Q depends on k, Q 1, Q, p, q, λ. For k = 2, 3 the same result holds, with the additional restriction that 1/2 < λ < 1.

25 CHAPTER 1. INTRODUCTION 13 Only the restriction λ > 1 k 1 /2k makes this result nonsharp, if k 1 < k. For operators of the form J Q1, Q,λ we prove: Theorem 1.6. Let Q 1 be a positive definite quadratic form in k 1 variables with integer coefficients, with 1 k 1 k. Let Q be a positive definite quadratic form in k variables with integer coefficients. For any k 2 the operator J Q1, Q,λ is a bounded operator from lp (Z k+1 ) to l q (Z k+1 ) if λ k,k1 < λ < 1, where and if p, q satisfy (i) 1/q 1/p k k+2 (1 λ) (ii) 1/q < λ, 1/p > 1 λ. λ k,k1 = 4k 2k 1 k(k 1 + 4), Specifically, for such p, q there exists a constant A Q1, Q such that J Q1, Q,λ f l q (Z k+1 ) A Q1, Q f l p (Z k+1 ), where the constant A Q1, Q depends on k, Q 1, Q, p, q, λ. This is sharp in the full rank case, when k 1 = k, in which case we recover Theorem 1.4. In the remaining region 0 < λ λ k,k1 we obtain a weaker result: Corollary Under the conditions of Theorem 1.6, J Q1, Q,λ is a bounded operator from l p (Z k+1 ) to l q (Z k+1 ) if 0 < λ λ k,k1 and p, q satisfy ) (i) 1/q < 1/p 1 + λ (ii) 1/q < λ, 1/p > 1 λ. ( kλk,k1 +2 λ k,k1 (k+2) Twisted discrete singular Radon transforms and related operators The next family of operators we consider relates to the discrete singular Radon transforms studied by Ionescu and Wainger (Theorem C) and Stein and Wainger (Theorem D). We first generalize the result of Ionescu and Wainger to the twisted case, proving: Theorem 1.7. Let T P,Q be the operator T P,Q f(n) = m Z k 1 m 0 f(n P (m))k(m)e 2πiQ(m), where P : Z k1 Z k2 is the restriction of a polynomial mapping R k1 R k2, Q : R k1 R is a polynomial, and K is a Calderón-Zygmund kernel. Then T P,Q is bounded on l p (Z k ) for all 1 < p <, with a bound dependent only on the degrees of P and Q, and independent of their coefficients.

26 14 CHAPTER 1. INTRODUCTION Secondly, we prove that the oscillatory integral operator I defined in (1.7), closely related to the quasi-translation invariant Radon transforms considered by Stein and Wainger in Theorem D, is bounded on l p for all 1 < p < : Theorem 1.8. Let T be the operator m Zk n m 0 T f(n) = e 2πiQ(n,m) K(n m)f(m), where n Z k, Q is a real-valued polynomial on R k R k, and K is a Calderón-Zygmund kernel. Then T is bounded on l p (Z k ) for all 1 < p <, with a bound dependent only on the degree of Q, and independent of its coefficients. Finally, we note a simple corollary that extends the l 2 result of Stein and Wainger for quasi-translation invariant Radon transforms to the twisted case: Corollary Let R be the operator Rf(n, n ) = m Zk m 0 f(n m, n P (n, m))k(m)e 2πiQ(n,m), where (n, n ) Z k Z l, m Z k, the mapping P : Z k Z k Z l is the restriction of a polynomial on R k R k, Q : R k R k R is a polynomial, and K is a Calderón-Zygmund kernel. Then R is bounded on l 2 (Z k ), with a bound dependent only on the degrees of P and Q, and independent of their coefficients Discrete fractional integration on the Heisenberg group The third family of operators we consider pertains to fractional integration on the Heisenberg group. Define for any 0 < λ < 1 the operator T λ acting on functions f of Z 2k Z by T λ f(n, t) = m Z 2k m 0 f(n m, t ω(n, m)) m 2kλ, (1.8) where n = (n 1, n 2 ) Z k Z k represents n 1 + in 2 C k and ω(n, m) = 2(n 2 m 1 n 1 m 2 ) denotes the action of the symplectic bilinear form associated to the Heisenberg group. We prove: Theorem 1.9. For λ k < λ < 1, T λ maps l p (Z 2k+1 ) to l q (Z 2k+1 ), where if p, q satisfy (i) 1/q 1/p k(1 λ) k+1 (ii) 1/q < λ, 1/p > 1 λ. λ k = 2k + 1(k + 1) (2k + 1) k 2, This theorem is sharp in the region λ k < λ < 1 in which it holds; note that for all k 1, λ k < 1/2. We obtain a weaker bound in the remaining region 0 < λ λ k.

27 CHAPTER 1. INTRODUCTION 15 Corollary For 0 < λ λ k, T λ maps l p (Z 2k+1 ) to l q (Z 2k+1 ) if p, q satisfy (i) 1/q < 1/p 1 + ( kλ k+1 λ k (k+1) )λ (ii) 1/q < λ, 1/p > 1 λ. In the case k = 1 we obtain a nearly sharp result for T λ for all 0 < λ < 1: Theorem For 0 < λ < 1, T λ maps l p (Z 3 ) to l q (Z 3 ) if p, q satisfy (i) 1/q < 1/p 1 λ 2 (ii) 1/q < λ, 1/p > 1 λ. This improves upon Corollary in the range 0 < λ λ k ; with equality in condition (i), it would be the best possible result for T λ in the case k = Lower rank symplectic bilinear forms We furthermore consider operators of the type T λ defined in 1.8, but with ω replaced by a more general symplectic bilinear form Ω, possibly of lower rank: T λ Ωf(n, t) = m Z 2k m 0 f(n m, t Ω(n, m)) m 2kλ. We prove a result for a specific class of symplectic bilinear forms Ω, defined precisely in Theorem 7.2 in Chapter 7: Theorem Suppose Ω is a symplectic bilinear form of rank 2k 1 acting on Z 2k, where 1 k 1 k, such that Ω is reducible to a canonical symplectic form via a lattice preserving orthogonal linear transformation. Then for λ k,k1 < λ < 1, the operator TΩ λ is bounded from l p (Z 2k+1 ) to l q (Z 2k+1 ), where 2k1 + 1(k + 1) (k + k 1 + 1) λ k,k1 =, kk 1 if p, q satisfy (i) 1/q 1/p k(1 λ) k+1 (ii) 1/q < λ, 1/p > 1 λ. Precisely, there exists a constant A Ω depending on Ω, p, q, λ such that T λ Ωf lq (Z 2k+1 ) A Ω f lp (Z 2k+1 ). In the full rank case, where k 1 = k, this recovers Theorem 1.9. We furthermore obtain a weaker bound in the remaining region 0 < λ λ k,k1 : Corollary For Ω and λ k,k1 as in Theorem 1.11, TΩ λ is bounded from lp (Z 2k+1 ) to l q (Z 2k+1 ) if 0 < λ λ k,k1 and p, q satisfy ) (i) 1/q < 1/p 1 + λ (ii) 1/q < λ, 1/p > 1 λ. ( kλk,k1 +1 λ k,k1 (k+1)