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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 useful to graduates and researchers in pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. The 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. Cambridge Studies in Advanced Mathematics 138
  3. Title Page
  4. Copyright
  5. Table of Contents
  6. Preface
  7. Acknowledgements
  8. 1: Leibnitz rules and the generalized Korteweg–de Vries equation
    1. 1.1 Conserved quantities
    2. 1.2. Dispersive estimates for the linear equation
    3. 1.3. Dispersive estimates for the nonlinear equation
    4. 1.4. Wave packets and phase-space portraits
    5. 1.5. The phase-space portraits of e[sup(2πix[sup(2)])] and e[sup(2πix[sup(3)])]
    6. 1.6. Asymptotics for the Airy function
      1. Notes
      2. Problems
  9. 2: Classical paraproducts
    1. 2.1. Paraproducts
    2. 2.2. Discretized paraproducts
    3. 2.3. Discretized Littlewood–Paley square-function operator
    4. 2.4. Dualization of quasi-norms
    5. 2.5. Two particular cases of Theorem 2.3
    6. 2.6. The John–Nirenberg inequality
    7. 2.7. L[sup(1,∞)] sizes and L[sup(1,∞)] energies
    8. 2.8. Stopping-time decompositions
    9. 2.9. Generic estimate of the trilinear paraproduct form
    10. 2.10. Estimates for sizes and energies
    11. 2.11. L[sup(p)] bounds for the first discrete model
    12. 2.12. L[sup(p)] bounds for the second discrete model
    13. 2.13. The general Coifman–Meyer theorem
    14. 2.14. Bilinear pseudodifferential operators
      1. Notes
      2. Problems
  10. 3: Paraproducts on polydisks
    1. 3.1. Biparameter paraproducts
    2. 3.2. Hybrid square and maximal functions
    3. 3.3. Biparameter BMO
    4. 3.4. Carleson's counterexample
    5. 3.5. Proof of Theorem 3.1; part 1
    6. 3.6. Journé's lemma
    7. 3.7. Proof of Theorem 3.1; part 2
    8. 3.8. Multiparameter paraproducts
    9. 3.9. Proof of Theorem 3.1; a simplification
    10. 3.10. Proof of the generic decomposition
      1. Notes
      2. Problems
  11. 4: Calderón commutators and the Cauchy integral on Lipschitz curves
    1. 4.1. History
      1. 4.1.1. Calderón commutators
      2. 4.1.2. Cauchy integral on Lipschitz curves
      3. 4.1.3. Dirichlet problem on Lipschitz domains
    2. 4.2. The first Calderón commutator
      1. 4.2.1. Quadratic decay estimates for the symbol of C[sub(1)]
      2. 4.2.2. A discrete theorem
      3. 4.2.3. Proof of the discrete theorem
      4. 4.2.4. Logarithmic estimates for the shifted maximal operator
      5. 4.2.5. Logarithmic estimates for the shifted square function operator
    3. 4.3. Generalizations
    4. 4.4. The Cauchy integral on Lipschitz curves
      1. 4.4.1. Extension by duality
      2. 4.4.2. A few remarks on the symbols of C[sub(d)] for d ≥ 2
      3. 4.4.3. Some heuristical arguments
      4. 4.4.4. Discrete minimal models
      5. 4.4.5. Reduction to the discrete minimal model
      6. 4.4.6. Noncompact Littlewood–Paley projections
      7. 4.4.7. The generic decomposition of C[sub(d)]
      8. 4.4.8. Case I: i[sub(1)] = 0 and i[sub(2)] = 1
      9. 4.4.9. Case II: j[sub(1)] = 0 and j[sub(2)] = d+1
      10. 4.4.10. Case III: j[sub(1)] = 2 and j[sub(2)] = 3
    5. 4.5. Generalizations
      1. Notes
      2. Problems
  12. 5: Iterated Fourier series and physical reality
    1. 5.1. Iterated Fourier series
    2. 5.2 Physical reality
    3. 5.3. Generic L[sup(p)] AKNS systems for 1≤ p < 2
    4. 5.4 Generic L[sup(2)] AKNS systems
      1. Notes
      2. Problems
  13. 6: The bilinear Hilbert transform
    1. 6.1 Discretization
    2. 6.2 The particular scale-1 case of Theorem 6.5
    3. 6.3 Trees, L[sup(2)] sizes, and L[sup(2)] energies
    4. 6.4 Proof of Theorem 6.5
    5. 6.5. Bessel-type inequalities
    6. 6.6 Stopping-time decompositions
    7. 6.7 Generic estimate of the trilinear BHT form
    8. 6.8 The 1/2 < r < 2/3 counterexample
    9. 6.9 The bilinear Hilbert transform on polydisks
      1. Notes
      2. Problems
  14. 7: Almost everywhere convergence of Fourier series
    1. 7.1 Reduction to the continuous case
    2. 7.2. Discrete models
    3. 7.3. Proof of Theorem 7.2 in the scale-1 case
    4. 7.4. Estimating a single tree
    5. 7.5. Additional sizes and energies
    6. 7.6. Proof of Theorem 7.2
    7. 7.7. Estimates for Carleson energies
    8. 7.8. Stopping-time decompositions
    9. 7.9. Generic estimate of the bilinear Carleson form
    10. 7.10. Fefferman's counterexample
      1. Notes
      2. Problems
  15. 8: Flag paraproducts
    1. 8.1. Generic flag paraproducts
    2. 8.2 Mollifying a product of two paraproducts
    3. 8.3 Flag paraproducts and quadratic NLS
    4. 8.4 Flag paraproducts and U-statistics
    5. 8.5 Discrete operators and interpolation
    6. 8.6 Reduction to the model operators
    7. 8.7 Rewriting the 4-linear forms
    8. 8.8 The new size and energy estimates
    9. 8.9 Estimates for T[sub(1)] and T[sub(1,ℓ[sub(0)])] near A[sub(4)]
    10. 8.10 Estimates for T[sup(*[sup(3)][sub(1)])] and T[sup(*[sup(3)][sub(1,ℓ[sub(0)])])] near A[sub(31)] and A[sub(32)]
    11. 8.11 Upper bounds for flag sizes
    12. 8.12. Upper bounds for flag energies
      1. Notes
      2. Problems
  16. 9: Appendix: Multilinear Interpolation
    1. Notes
  17. References
  18. Index