Bochner–Riesz Means of Morrey Functions

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

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