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Classical and Multilinear Harmonic Analysis

Book Description

This two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained and will be useful to graduate students and researchers in both pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. This first volume starts with classical one-dimensional topics: Fourier series; harmonic functions; Hilbert transform. Then the higher-dimensional Calderón–Zygmund and Littlewood–Paley theories are developed. Probabilistic methods and their applications are discussed, as are applications of harmonic analysis to partial differential equations. The volume concludes with an introduction to the Weyl calculus. The second volume goes beyond the classical to the highly contemporary and focuses on multilinear aspects of harmonic analysis: the bilinear Hilbert transform; Coifman–Meyer theory; Carleson's resolution of the Lusin conjecture; Calderón's commutators and the Cauchy integral on Lipschitz curves. The material in this volume has not previously appeared together in book form.

Table of Contents

  1. Cover
  2. Series
  3. Title
  4. Copyright
  5. Table of Contents
  6. Preface
  7. Acknowledgements
  8. 1: Fourier series: convergence and summability
    1. 1.1. The basics: partial sums and the Dirichlet kernel
      1. 1.1.1. Definitions
      2. 1.1.2. Dirichlet kernel
      3. 1.1.3. Convolution
    2. 1.2. Approximate identities, Fejér kernel
      1. 1.2.1. Cesáro means of partial sums
      2. 1.2.2. Convergence properties of approximate identities
    3. 1.3. The L[sup(p)] convergence of partial sums
      1. 1.3.1. Failure of uniform convergence
    4. 1.4. Regularity and Fourier series
      1. 1.4.1. Bernstein's inequality
      2. 1.4.2. Convex Fourier coefficients
      3. 1.4.3. Smoothness and Fourier coefficients
      4. 1.4.4. Smoothness and decay
      5. 1.4.5. Sobolev spaces and embeddings
      6. 1.4.6. Optimality of the embedding
    5. 1.5. Higher dimensions
    6. 1.6. Interpolation of operators
    7. Notes
    8. Problems
  9. 2: Harmonic functions; Poisson kernel
    1. 2.1. Harmonic functions
      1. 2.1.1. Mean-value property and maximum principle
    2. 2.2. The Poisson kernel
      1. 2.2.1. Derivation of the Poisson kernel
      2. 2.2.2. The Poisson kernel as an approximate identity
      3. 2.2.3. The boundary-value problem
    3. 2.3. The Hardy–Littlewood maximal function
      1. 2.3.1. The boundedness properties of the maximal function
    4. 2.4. Almost everywhere convergence
      1. 2.4.1. The case of measures
    5. 2.5. Weighted estimates for maximal functions
      1. 2.5.1. Changing measures
      2. 2.5.2. Weighted estimates for the maximal function
      3. 2.5.3. The Calderón–Zygmund decomposition
      4. 2.5.4. The weak bound for A[sub(p)] weights
      5. 2.5.5. The strong bound for A[sub(p)] weights
    6. Notes
    7. Problems
  10. 3: Conjugate harmonic functions; Hilbert transform
    1. 3.1. Hardy spaces of analytic functions
      1. 3.1.1. Subharmonic functions
      2. 3.1.2. Sub-mean-value property
      3. 3.1.3. Maximal function F[sup(*)]
    2. 3.2. Riesz theorems
      1. 3.2.1. The three versions of the first F. and M. Riesz theorem
      2. 3.2.2. Second F. and M. Riesz theorem
    3. 3.3. Definition and simple properties of the conjugate function
    4. 3.4. The weak-L[sup(1)] bound on the maximal function
    5. 3.5. The Hilbert transform
      1. 3.5.1. The weak-L[sup(1)] bound
      2. 3.5.2. The L[sup(p)] bound
      3. 3.5.3. Kernel representation of the Hilbert transform
      4. 3.5.4. The Hilbert transform on L[sup(∞)](T)
    6. 3.6. Convergence of Fourier series in L[sup(p)]
    7. Notes
    8. Problems
  11. 4: The Fourier transform on R[sup(d)] and on LCA groups
    1. 4.1. The Euclidean Fourier transform
      1. 4.1.1. Basic definitions: Fourier transform, Schwartz space
      2. 4.1.2. The inversion theorem
      3. 4.1.3. Poisson summation formula
      4. 4.1.4. Plancherel theorem
      5. 4.1.5. Sobolev spaces
      6. 4.1.6. Trace lemma
    2. 4.2. Method of stationary or nonstationary phases
      1. 4.2.1. Fourier transform of surface-carried measures
      2. 4.2.2. Nonstationary phase
      3. 4.2.3. Stationary phase
    3. 4.3. The Fourier transform on locally compact Abelian groups
      1. 4.3.1. Generalities
      2. 4.3.2. Commutative Banach algebras
      3. 4.3.3. Inversion and Plancherel
    4. Notes
    5. Problems
  12. 5: Introduction to probability theory
    1. 5.1. Probability spaces; independence
    2. 5.2. Sums of independent variables
      1. 5.2.1. Strong law of large numbers
      2. 5.2.2. Sub-Gaussian bounds
      3. 5.2.3. Central limit theorem
      4. 5.2.4. Another look at the law of large numbers
      5. 5.2.5. The law of the iterated logarithm
      6. 5.2.6. Stopping times
    3. 5.3. Conditional expectations; martingales
      1. 5.3.1. Conditional expectation
      2. 5.3.2. Martingales
      3. 5.3.3. Doob convergence theorem
      4. 5.3.4. Applications
    4. Notes
    5. Problems
  13. 6: Fourier series and randomness
    1. 6.1. Fourier series on L[sup(1)](T): pointwise questions
      1. 6.1.1. Maximal functions and weak-L[sup(1)] bounds
      2. 6.1.2. Resonant measures for the Dirichlet kernel
    2. 6.2. Random Fourier series: the basics
      1. 6.2.1. The L[sup(2)] theory of random Fourier series
      2. 6.2.2. The L[sup(∞)] case
    3. 6.3. Sidon sets
      1. 6.3.1. Interpolating measures
      2. 6.3.2. Riesz measures
      3. 6.3.3. The Λ(p) property
      4. 6.3.4. Rider's characterization of Sidonicity
    4. Notes
    5. Problems
  14. 7: Calderón-Zygmund theory of singular integrals
    1. 7.1. Calderón-Zygmund kernels
      1. 7.1.1. The basic definition
      2. 7.1.2. The L[sup(2)]-boundedness of T
      3. 7.1.3. Calderón-Zygmund decomposition, weak-L[sup(1)] bound
      4. 7.1.4. The L[sup(p)] -boundedness of T
      5. 7.1.5. Boundedness on Hölder spaces
    2. 7.2. The Laplacian: Riesz transforms and fractional integration
    3. 7.3. Almost everywhere convergence; homogeneous kernels
      1. 7.3.1. A maximal function bound
      2. 7.3.2. Almost everywhere convergence
      3. 7.3.3. A characterization of homogeneous kernels
    4. 7.4. Bounded mean oscillation space
      1. 7.4.1. The most basic example: the (discrete) logarithm
      2. 7.4.2. Singular integrals on L[sup(∞)]
      3. 7.4.3. An interpolation result
      4. 7.4.4. The Hardy–Littlewood and sharp maximal functions
      5. 7.4.5. Exponential bounds: the John–Nirenberg inequality
      6. 7.4.6. Commutators of singular integral operators with BMO functions
    5. 7.5. Singular integrals and A[sub(p)] weights
      1. 7.5.1. The reverse Hölder inequality
      2. 7.5.2. The A[sub(∞)] condition
      3. 7.5.3. The singular integral bound
    6. 7.6. A glimpse of H[sup(1)]–BMO duality and further remarks
    7. Notes
    8. Problems
  15. 8: Littlewood–Paley theory
    1. 8.1. The Mikhlin multiplier theorem
      1. 8.1.1. A partition of unity over a geometric scale
      2. 8.1.2. The multiplier theorem
    2. 8.2. Littlewood–Paley square-function estimate
    3. 8.3. Calderón–Zygmund Hölder spaces, and Schauder estimates
      1. 8.3.1. A Besov characterization of Hölder spaces
      2. 8.3.2. Singular integrals on C[sup(α)]
      3. 8.3.3. An atomic decomposition in S
      4. 8.3.4. The Schauder estimate
    4. 8.4. The Haar functions; dyadic harmonic analysis
      1. 8.4.1. Basic definitions and properties
      2. 8.4.2. Multiplier operators for the Haar expansion
      3. 8.4.3. Burkholder's inequality
      4. 8.4.4. The square function
      5. 8.4.5. Dyadic H[sup(1)] and BMO spaces
      6. 8.4.6. The first duality estimate
      7. 8.4.7. The second duality estimate
      8. 8.4.8. The atomic decomposition in dyadic H[sup(1)]
    5. 8.5. Oscillatory multipliers
    6. Notes
    7. Problems
  16. 9: Almost orthogonality
    1. 9.1. Cotlar's lemma
      1. 9.1.1. Motivation of almost orthogonality
      2. 9.1.2. The precise formulation
      3. 9.1.3. Schur's lemma
      4. 9.1.4. Singular integrals on L[sup(2)]
    2. 9.2. Calderón-Vaillancourt theorem
    3. 9.3. Hardy's inequality
    4. 9.4. The T(1) theorem via Haar functions
      1. 9.4.1. Carleson's lemma
      2. 9.4.2. Paraproducts
      3. 9.4.3. Proof of Theorem 9.8
      4. 9.4.4. T(1) on the line
    5. 9.5. Carleson measures, BMO, and T(1)
      1. 9.5.1. Carleson measures
      2. 9.5.2. Bounded mean oscillation space and Carleson measures
      3. 9.5.3. Paraproducts and T(1)
    6. Notes
    7. Problems
  17. 10: The uncertainty principle
    1. 10.1. Bernstein’s bound and Heisenberg’s uncertainty principle
      1. 10.1.1. Motivation
      2. 10.1.2. Bernstein’s bound
      3. 10.1.3. Heisenberg’s inequality
    2. 10.2. The Amrein-Berthier theorem
    3. 10.3. The Logvinenko-Sereda theorem
      1. 10.3.1. A simple version
      2. 10.3.2. A refined version
      3. 10.3.3. Some facts from the theory of several complex variables
      4. 10.3.4. The proof of the Logvinenko-Sereda theorem
      5. 10.3.5. Negligible sets, and an equivalent version
    4. 10.4. Solvability of constant-coefficient linear PDEs
      1. 10.4.1. The Malgrange-Ehrenpreis theorem
      2. 10.4.2. A bound on polynomials
    5. Notes
    6. Problems
  18. 11: Fourier restriction and applications
    1. 11.1. The Tomas-Stein theorem
      1. 11.1.1. The restriction question
      2. 11.1.2. Duality and equivalent formulation
      3. 11.1.3. Optimality, Knapp example
      4. 11.1.4. Decay of σ[sub(S[sup(d-1)])]
      5. 11.1.5. Proof of the Tomas-Stein theorem, non-endpoint version
    2. 11.2. The endpoint
      1. 11.2.1. Complex interpolation
      2. 11.2.2. The fractional integration method
    3. 11.3. Restriction and PDE; Strichartz estimates
      1. 11.3.1. Schrödinger evolution and Fourier restriction
      2. 11.3.2. General Strichartz estimates
      3. 11.3.3. A nonlinear application
      4. 11.3.4. The wave equation
    4. 11.4. Optimal two-dimensional restriction
    5. Notes
    6. Problems
  19. 12: Introduction to the Weyl calculus
    1. 12.1. Motivation, definitions, basic properties
      1. 12.1.1. Quantization
      2. 12.1.2. Symbol classes
      3. 12.1.3. Commutators
      4. 12.1.4. Convergence of symbols
    2. 12.2. Adjoints and compositions
      1. 12.2.1. Adjoints
      2. 12.2.2. Moyal product of Schwarz symbols
      3. 12.2.3. The semiclassical calculus
      4. 12.2.4. Stationary phase expansions
      5. 12.2.5. The composition law and expansions in general symbol classes
      6. 12.2.6. Expansions relative to the order
      7. 12.2.7. Preservation of regularity
      8. 12.2.8. Inverses of symbols
    3. 12.3. The L[sup(2)] theory
      1. 12.3.1. The Calderón–Vaillancourt theorem
      2. 12.3.2. Mapping properties of elliptic operators
      3. 12.3.3. The sharp Gårding inequality
    4. 12.4. A phase-space transform
      1. 12.4.1. The Bargman transform
      2. 12.4.2. The S[sup(0)] calculus and the Bargman transform
    5. Notes
    6. Problems
  20. References
  21. Index