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On the capacity of sets of divergence associated with the spherical partial integral operator

Author: Emmanuel Montini
Journal: Trans. Amer. Math. Soc. 355 (2003), 1415-1441
MSC (2000): Primary 42B05, 31B15
Published electronically: November 14, 2002
MathSciNet review: 1946398
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Abstract: In this article, we study the pointwise convergence of the spherical partial integral operator $S_Rf(x)=\int_{B(0,R)} \hat{f} (y) e^{2\pi ix\cdot y}dy$ when it is applied to functions with a certain amount of smoothness. In particular, for $f\in \mathcal{L}_{\alpha}^p(\mathbb{R} ^n)$, $\tfrac{n-1}{2} <\alpha\leq\tfrac{n}{p}$, $2\leq p<\tfrac{2n}{n-1}$, we prove that $S_Rf(x)\to G_{\alpha} *g(x)$ $C_{\alpha,p}$-quasieverywhere on $\mathbb{R} ^n$, where $g\in L^p({\mathbb{R} }^n )$ is such that $f=G_{\alpha}*g$ almost everywhere. A weaker version of this result in the range $0<\alpha\leq\tfrac{n-1}{2}$ as well as some related localisation principles are also obtained. For $1\leq p<2-\tfrac{1}{n}$ and $0\leq\alpha <\tfrac{(2-p)n-1}{2p}$, we construct a function $f\in\mathcal{L}_\alpha^p(\mathbb{R} ^n)$ such that $S_Rf(x)$ diverges everywhere.

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Additional Information

Emmanuel Montini
Affiliation: Department of Mathematics and Statistics, University of Edinburgh, J.C.M.B., King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom
Address at time of publication: NEKS Technologies Inc., 230 rue Bernard-Belleau, Bureau 221, Laval (Québec) H7V 4A9, Canada

Received by editor(s): August 31, 2000
Published electronically: November 14, 2002
Additional Notes: The author was supported in part by a Commonwealth Academic Staff Fellowship (CA0355)
Article copyright: © Copyright 2002 American Mathematical Society