Geometric Aspects of Harmonic Analysis

Description

This volume originated in talks given in Cortona at the conference "Geometric aspects of harmonic analysis" held in honor of the 70th birthday of Fulvio Ricci. It presents timely syntheses of several major fields of mathematics as well as original research articles contributed by some of the finest mathematicians working in these areas. The subjects dealt with are topics of current interest in closely interrelated areas of Fourier analysis, singular integral operators, oscillatory integral operators, partial differential equations, multilinear harmonic analysis, and several complex variables. The work is addressed to researchers in the field. Preface At the Occasion of Fulvio's Conference Contents An Extension Problem and Hardy Type Inequalities for the Grushin Operator 1 Introduction and Main Results 2 Preliminaries on the Grushin Operator 3 An Extension Problem for the Grushin Operator 4 Fractional Powers of the Grushin Operator 5 Trace Hardy and Hardy Inequalities 6 An Isometry Property of the Solution Operator Associated to the Extension Problem 7 Hardy–Littlewood–Sobolev Inequality for Gs References Sharp Local Smoothing Estimates for Fourier Integral Operators 1 Basic Definitions and Examples of Fourier Integral Operators 1.1 Motivating Examples 1.2 Distributions Defined by Oscillatory Integrals 1.2.1 Distributions 1.2.2 Homogeneous Oscillatory Integrals 1.3 Local Fourier Integral Operators 1.4 Wave Front Sets and Equivalence of Phase 1.4.1 The Singular Support 1.4.2 Tempered Distributions and the Fourier Transform 1.4.3 The Wave Front Set 1.4.4 Non-degeneracy and Lagrangian Manifolds 1.4.5 Equivalence of Phase 1.5 Global Theory 1.5.1 Distributions on Manifolds 1.5.2 Global Homogeneous Oscillatory Integrals 1.5.3 Global FIOs and Canonical Relations 1.5.4 Global Versus Local Theory 2 Local Smoothing Estimates 2.1 Fixed-Time Estimates for FIOs 2.2 Local Smoothing Estimates for FIOs 2.3 The Local Smoothing Conjecture 2.3.1 The Euclidean Wave Semigroup 2.3.2 Wave Equations on Manifolds 2.3.3 General FIOs 2.4 Positive Results 2.5 Formulating a Local Smoothing Conjecture for General FIOs 2.6 The Geometric Conditions in Terms of the Canonical Relation 2.6.1 Lp Estimates and Canonical Graphs 2.6.2 Local Smoothing Estimates and Cinematic Curvature Condition 3 Local Smoothing and Maximal Estimates 3.1 Bochner–Riesz Estimates 3.2 Maximal Bochner–Riesz Estimates 3.3 Circular Maximal Function Estimates 3.4 Variable Coefficient Circular Maximal Function Estimates 3.5 Maximal Bounds for Half-Wave Propagators 4 Local Smoothing and Oscillatory Integral Estimates 4.1 Lp Estimates for Oscillatory Integrals Satisfying the Carleson–Sjölin Condition 4.2 Necessary Conditions in Conjecture 24: Sharpness of Theorem 34 in Odd Dimensions 4.3 Maximal Oscillatory Integral Estimates 5 Wolff's Approach to Local Smoothing Estimates 5.1 Preliminary Observations 5.2 Basic Dyadic Decomposition 5.3 Angular Decomposition 5.4 Decoupling into Localised Pieces 5.5 Bounding the Localised Pieces 5.5.1 L∞-Bounds 5.5.2 L2-Bounds 6 Variable-Coefficient Wolff-Type Inequalities 6.1 Preliminaries 6.2 Inductive Setup 6.3 Key Ingredients of The Proof 6.4 Approximation by Constant Coefficient Operators 6.5 Application of Constant-Coefficient Decoupling 6.6 Parabolic Rescaling 6.6.1 A Prototypical Example 6.6.2 The General Case 6.7 Applying the Induction Hypothesis Appendix: Historical Background on the Local Smoothing Conjecture The Euclidean Wave Equation Fourier Integral Operators Figure and Table for the Euclidean Wave Equation for n=2 Figure and Table for the Euclidean Wave Equation for n≥3 Figure and Table for Fourier Integrals References On the Hardy–Littlewood Maximal Functions in High Dimensions: Continuous and Discrete Perspective 1 Introduction 1.1 Statement of the Results: Continuous Perspective 1.2 Statement of the Results: Discrete Perspective 1.3 Notation 2 A Review of the Current State of the Art 2.1 Dimension-Free Estimates for Semigroups 2.2 The Case of the Euclidean Balls SteinMax,SteinStro 2.3 The L2 Result for General Symmetric Bodies via Fourier Transform Methods B1 2.4 Interlude: The Isotropic Conjecture 2.5 The Lp Results for p(3/2, ∞] and Fractional Integration Method 2.6 The Lp Result for p(1, ∞], the Case of q-Balls 2.7 Weak Type (1,1) Considerations 3 Overview of the Methods of the Paper 3.1 Continuous Perspective 3.2 Discrete Perspective 4 Continuous Perspective: Proof of Theorem 1 4.1 Fourier Transform Estimates: Proof of Theorem 6 4.2 Proof of Inequality (58) 4.2.1 Proof of Inequality (62) for p=2 4.2.2 Proof of Inequality (62) for p(1, 2) 4.3 Proof of Inequality (59) 4.3.1 Proof of Inequality (69) for p=2 4.3.2 Proof of Inequality (69) for p(3/2, 2) 5 Discrete Perspective: Proof of Theorem 2 References Potential Spaces on Lie Groups 1 Introduction 2 Basic Facts and Definitions 3 Triebel–Lizorkin and Besov Spaces on G 4 Finite Differences Characterizations 4.1 Characterization of Triebel–Lizorkin Norm by Differences 4.2 Characterization of Besov Norm by Differences 5 A Density Result 6 Isomorphisms of Triebel–Lizorkin and Besov Spaces 7 Final Remarks and Open Problems References On Fourier Restriction for Finite-Type Perturbations of the Hyperbolic Paraboloid 1 Introduction 2 Reduction to Perturbations of Cubic Type 3 Transversality Conditions and Admissible Pairs of Sets 3.1 Admissible Pairs of Sets U1, U2 on which Transversalities Are of a Fixed Size: An Informal Discussion 3.2 Precise Definition of Admissible Pairs within QQ 4 The Bilinear Estimates 4.1 A Prototypical Admissible Pair in the Curved Box Case and the Crucial Scaling Transformation 4.2 Reduction to the Prototypical Case 5 The Whitney-Decomposition and Passage to Linear Restriction Estimates: Proof of Theorem 1 References On Young's Convolution Inequality for Heisenberg Groups 1 Introduction 1.1 Young's Inequality for Euclidean Groups 1.2 Young's Inequality for Heisenberg Groups 2 Definitions and Main Theorem 2.1 The Symplectic Group 2.2 Symmetries 2.3 Special Ordered Triples of Gaussians on Hd 2.4 Main Theorem 3 Approximate Solutions of Functional Equations 4 Analogue for Twisted Convolution 5 Nonexistence of Maximizers, and Value of the Optimal Constant 6 Sufficiency 7 Two Ingredients 8 Proof of Theorem 7 for Nonnegative Functions 9 The Complex-Valued Case 10 Some Matrix Algebra 11 Integration of Difference Relations 12 A Final Lemma 13 On Twisted Convolution References Young's Inequality Sharpened 1 Statements of Theorems 2 Negative Result 3 Reduction to the Perturbative Case 4 Perturbative Expansion 5 Analysis of Quadratic Forms 5.1 Diagonalization of Scalar Operators 5.2 Diagonalizing Qp+ 5.3 Eigenvalue Analysis for Qp+ 5.4 Analysis for Qp- 6 Balancing 7 Conclusion of Proof 8 Hermite Functions and Singular Values 9 The Case pi=2 10 Bilinear Variant 11 The Periodic Case References Strongly Singular Integrals on Stratified Groups 1 Introduction 1.1 Notation 2 The Euclidean Setting Rn 2.1 Our Multiplier Conditions 2.2 Our Multiplier Classes 2.3 The Basic Decomposition 2.4 Mn Versus (3) 2.5 An Interlude 2.6 The Results 3 The Stratified Lie Group Setting 3.1 The Multiplier Classes 3.2 The Main Result 3.3 The Interlude: Revisited 4 Preliminaries 4.1 Some Basics 4.2 A Weighted L2 Bound 4.3 Fefferman–Stein Inequality 4.4 Our Basic Decomposition 5 The Proof of Theorem 4: The Weak-Type (1,1) Bound 6 The Proof of Theorem 4: The Hardy Space Bound References Singular Brascamp–Lieb: A Survey 1 Brascamp–Lieb Forms and Inequalities 2 Singular Brascamp–Lieb Inequalities 3 Inequalities with One Singular Kernel and Hölder Scaling 4 Method of Rotations and More General Kernels 5 Inequalities with Two Singular Kernels and Hölder Scaling References On the Restriction of Laplace–Beltrami Eigenfunctions and Cantor-Type Sets 1 Introduction 2 Literature Review 3 Main Results 4 Remarks 5 Overview of the Proof References Basis Properties of the Haar System in Limiting Besov Spaces 1 Introduction 1.1 Dyadic Averaging Operators 1.2 Guide Through this Paper and Discussion of Further Quantitative Results 2 Preparatory Results 2.1 Besov Quasi-norms 2.2 Local Estimates 3 Upper Bounds for p≤1 3.1 Proofs 4 Upper Bounds for 1≤p≤∞ 4.1 Proofs 5 Necessary Conditions for Boundedness when s=1/p 5.1 Tensorized Test Functions 5.2 Proof of Proposition 25 6 Necessary Conditions for Boundedness when s=1 6.1 Proof of the Upper Bounds in Theorem 27 6.2 Proof of the Lower Bounds in Theorem 27 7 Necessary Conditions for Boundedness when s≤0 7.1 The Case 11 7.2 The Case p=∞, s=-1, q≤1 8 Density and Approximation 8.1 The Case s=1 8.2 An Approximation Result for b1/pp,∞ when 1