Bochner–Riesz Means of Morrey Functions
Tóm tắt
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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