Convergence almost everywhere of certain singular integrals and multiple Fourier series
TLDR
In this paper, the authors generalize the L p es t imate of the opera tor M* in [7] to a Calderdn -Zygmund kernel defined in R', s ~_ 2, which has continuous derivatives of order s 41 outside the origin.Abstract:
T s Ts = o ( l o g l o g I~]), t~]-+ 0% for almost eve ry x in T,. I n Sections 1 to 3 in this pape r we prove among o ther things the following theorem, which generalizes the L p es t imate of the opera tor M* in [7]. TEEORE~. Assume that k is a Calderdn -Zygmund kernel defined in R', s ~_ 2, which has continuous derivatives of order ~ s 41 outside the origin. Let the operator M be defined byread more
Citations
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Journal ArticleDOI
Calderón-Zygmund theory for operator-valued kernels
Book
Classical and Multilinear Harmonic Analysis
Camil Muscalu,Wilhelm Schlag +1 more
TL;DR: In this article, a two-volume text in harmonic analysis introduces a wealth of analytical results and techniques, including Fourier series, harmonic functions, Hilbert transform, and Weyl calculus.
Journal ArticleDOI
Hilbert integrals, singular integrals, and Radon transforms I
Duong H. Phong,Elias M. Stein +1 more
TL;DR: The authors introduce deux nouvelles classes d'operateurs, etudie leur relation and montre comment on peut les appliquer a l'etude des problemes aux valeurs limites.
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On the convergence of multiple Fourier series
TL;DR: In this paper, the authors consider the special case of a triangle with a vertex at the origin, and assume that the characteristic function of any triangle is a linear combination of characteristic functions of triangles with vertices at zero.
Journal ArticleDOI
On the divergence of multiple Fourier series
References
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Journal ArticleDOI
On the existence of certain singular integrals
A. P. Calderón,A. Zygmund +1 more
TL;DR: In this article, it was shown that the integral can exist almost everywhere even if K is not integrable, and the most interesting special case is that of K (x) = 1/x.
Book ChapterDOI
On Singular Integrals
TL;DR: In this paper, the results obtained jointly with A. P. Calderon during the last few years are presented, but not all the results are accompanied by proofs, and for additional details the reader is referred to original papers.
Journal ArticleDOI
On the convergence of multiple Fourier series
TL;DR: In this paper, the authors consider the special case of a triangle with a vertex at the origin, and assume that the characteristic function of any triangle is a linear combination of characteristic functions of triangles with vertices at zero.