Commutative Harmonic Analysis IV: Harmonic Analysis in Rn
V.P. Khavin, N.K. Nikol'skii
Springer Science & Business Media, Nov 15, 1991 - Mathematics - 228 pages
In this volume of the series "Commutative Harmonie Analysis", three points mentioned in the preface to the first volume are realized: 1) Multiple Fourier series and Fourier integrals; 2) The machinery of singular integrals; 3) Exeeptional sets in harmonie analysis. The first theme is the subjeet matter of the eontribution by Sh. A. Ali mov, R. R. Ashurov, A. K. Pulatov, whieh in an obvious way eonstitutes the "multidimensional parallel" to S. V. Kislyakov's article in Volume I, devoted to the "inner" questions of Fourier analysis of funetions of one variable. The passage to the analysis of functions defined on ]Rn, n > 1, teIls us something essential about the nature of the problem under study. The eontribution by E. M. Dyn'kin, the beginning of which was already published in Volume I of this subseries, is devoted to singular integrals. Be sides classical material (Calder6n-Zygmund and Littlewood-Paley theory), this article eontains an exposition of reeent results, whieh in an essential way have widened the seope of the whole area and have made it possible to solve many old problems, thereby sometimes transeending the very frames of harmonie analysis in its eanonieal interpretation.
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a. e. convergence absolute convergence Alimov analogue analytic functions arbitrary assume bounded in L2 Calderon Calderon-Zygmund operator Carleson measure Cauchy integral Chapter characteristic function Coifman commutators condition const constant construction converges a. e. Corollary corresponding David defined definition Dirichlet kernel divergence elliptic operators English translation estimate example exceptional sets exists Fefferman finite follows formula Fourier coefficients Fourier integrals Fourier transform function f halfplane harmonic analysis Hilbert transform Hormander Il'in inequality Laplace operator Lemma linear Lipschitz curves Lipschitz domain Littlewood-Paley theory localization principle Lp(TN Luzin function martingale Math maximal function Meyer multiple Fourier series multiple trigonometric Nikishin norm obtain partial sums Pisier problem proved rectangular partial sums remark result Riesz means satisfies Sect sequence Sidon sets singular integral operator space spectral expansions spherical partial sums Stein subset sufficiently summable summation Tl-Theorem trigonometric series uniform convergence uniformly variables zero Zygmund