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Record Nr. |
UNINA9910786183003321 |
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Autore |
Muscalu Camil |
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Titolo |
Classical and multilinear harmonic analysis . Volume 1 / / Camil Muscalu, Wilhelm Schlag [[electronic resource]] |
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Pubbl/distr/stampa |
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Cambridge : , : Cambridge University Press, , 2013 |
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ISBN |
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1-107-23372-0 |
1-139-60986-6 |
1-139-61172-0 |
1-139-62102-5 |
1-283-94313-1 |
1-139-62474-1 |
1-139-61544-0 |
1-139-60838-X |
1-139-04708-6 |
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Descrizione fisica |
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1 online resource (xviii, 370 pages) : digital, PDF file(s) |
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Collana |
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Cambridge studies in advanced mathematics ; ; 137 |
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Disciplina |
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Soggetti |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Note generali |
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Title from publisher's bibliographic system (viewed on 05 Oct 2015). |
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Nota di bibliografia |
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Includes bibliographical references and index. |
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Nota di contenuto |
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Contents; 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 |
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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 |
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Sommario/riassunto |
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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. |
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