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Bochner–Riesz Means of Morrey Functions

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Abstract

This paper concerns both norm estimation and pointwise approximation for the Bochner–Riesz means of an arbitrary Morrey function on \({{{{\mathbb {R}}}}^n}\)—Theorems 1.1 and 1.2 for \(L^{p,\lambda }({{{{\mathbb {R}}}}^n})\)—thereby generalizing the corresponding results for \(L^p({{{{\mathbb {R}}}}^n})\) in Stein (Acta Math 100:93–147, 1958) and Carbery et al. (J Lond Math Soc 38:513–524, 1988). As a side note, this paper also establishes Lemma 4.1 of Tomas–Stein type—if \(f\in L^{p,\lambda }({{{{\mathbb {R}}}}^n})\) under \( 2^{-1}(n+1)<\lambda \le n\) is compactly supported, then

$$\begin{aligned} \Vert {\hat{f}}\Vert _{L^2({\mathbb {S}}^{n-1})}\lesssim \Vert f\Vert _{L^{p,\lambda }({{{{\mathbb {R}}}}^n})}\ \ \hbox {for}\ \ \frac{4\lambda }{n+1+2\lambda }\le p<\frac{2\lambda }{n+1}. \end{aligned}$$

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Correspondence to Jie Xiao.

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Jie Xiao was supported by NSERC of Canada (#20171864).

Communicated by Mieczyslaw Mastylo.

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Adams, D.R., Xiao, J. Bochner–Riesz Means of Morrey Functions. J Fourier Anal Appl 26, 7 (2020) doi:10.1007/s00041-019-09712-x

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Keywords

  • Bochner–Riesz means
  • Fourier transforms
  • Morrey functions

Mathematics Subject Classification

  • 42B15
  • 42B35