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Classical and Multilinear Harmonic Analysis.
This contemporary graduate-level text in harmonic analysis introduces the reader to a wide array of analytical results and techniques.
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
2013.
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| Colección: | Cambridge studies in advanced mathematics.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Preface; Acknowledgements; 1 Fourier series: convergence and summability; 1.1 The basics: partial sums and the Dirichlet kernel; 1.2 Approximate identities, Fej ́er kernel; 1.3 The Lp convergence of partial sums; 1.4 Regularity and Fourier series; 1.5 Higher dimensions; 1.6 Interpolation of operators; Notes; Problems; Problems; Problems; Problems; Problems; Problems; Problems; Problems; Problems; Problems; Problems; Problems; 2 Harmonic functions; Poisson kernel; 2.1 Harmonic functions; 2.2 The Poisson kernel; 2.3 The Hardy-Littlewood maximal function.
- 2.4 Almost everywhere convergence2.5 Weighted estimates for maximal functions; Notes; 3 Conjugate harmonic functions; Hilbert transform; 3.1 Hardy spaces of analytic functions; 3.2 Riesz theorems; 3.3 Definition and simple properties of the conjugate function; 3.4 The weak-L1 bound on the maximal function; 3.5 The Hilbert transform; 3.6 Convergence of Fourier series in Lp; Notes; 4 The Fourier transform on Rd and on LCA groups; 4.1 The Euclidean Fourier transform; 4.2 Method of stationary or nonstationary phases; 4.3 The Fourier transform on locally compact Abelian groups; Notes.
- 5 Introduction to probability theory5.1 Probability spaces; independence; 5.2 Sums of independent variables; 5.3 Conditional expectations; martingales; Notes; 6 Fourier series and randomness; 6.1 Fourier series on L1(T): pointwise questions; 6.2 Random Fourier series: the basics; 6.3 Sidon sets; Notes; 7 Calder ́on-Zygmund theory of singular integrals; 7.1 Calder ́on-Zygmund kernels; 7.2 The Laplacian: Riesz transforms and fractional integration; 7.3 Almost everywhere convergence; homogeneous kernels; 7.4 Bounded mean oscillation space; 7.5 Singular integrals and Ap weights.
- 7.6 A glimpse of H1-BMO duality and further remarksNotes; 8 Littlewood-Paley theory; 8.1 The Mikhlin multiplier theorem; 8.2 Littlewood-Paley square-function estimate; 8.3 Calderon-Zygmund H ̈older spaces, and Schauder estimates; 8.4 The Haar functions; dyadic harmonic analysis; 8.5 Oscillatory multipliers; Notes; 9 Almost orthogonality; 9.1 Cotlar's lemma; 9.2 Calderon-Vaillancourt theorem; 9.3 Hardy's inequality; 9.4 The T(1) theorem via Haar functions; 9.5 Carleson measures, BMO, and T(1); Notes; 10 The uncertainty principle; 10.1 Bernstein's bound and Heisenberg's uncertainty principle.
- 10.2 The Amrein-Berthier theorem10.3 The Logvinenko-Sereda theorem; 10.4 Solvability of constant-coefficient linear PDEs; Notes; 11 Fourier restriction and applications; 11.1 The Tomas-Stein theorem; 11.2 The endpoint; 11.3 Restriction and PDE; Strichartz estimates; 11.4 Optimal two-dimensional restriction; Notes; 12 Introduction to the Weyl calculus; 12.1 Motivation, definitions, basic properties; 12.2 Adjoints and compositions; 12.3 The L2 theory; 12.4 A phase-space transform; Notes; References; Index.