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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the capacity of sets of divergence associated with the spherical partial integral operator
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by Emmanuel Montini PDF
Trans. Amer. Math. Soc. 355 (2003), 1415-1441 Request permission

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
  • Email: emmanuel@montini.ca
  • 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)
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 355 (2003), 1415-1441
  • MSC (2000): Primary 42B05, 31B15
  • DOI: https://doi.org/10.1090/S0002-9947-02-03144-6
  • MathSciNet review: 1946398