Bochner–Riesz Means of Morrey Functions

This paper concerns both norm estimation and pointwise approximation for the Bochner–Riesz means of an arbitrary Morrey function on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{\mathbb {R}}}}^n}$$\end{document}—Theorems 1.1 and 1.2 for Lp,λ(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p,\lambda }({{{{\mathbb {R}}}}^n})$$\end{document}—thereby generalizing the corresponding results for Lp(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p({{{{\mathbb {R}}}}^n})$$\end{document} 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∈Lp,λ(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in L^{p,\lambda }({{{{\mathbb {R}}}}^n})$$\end{document} under 2-1(n+1)<λ≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 2^{-1}(n+1)<\lambda \le n$$\end{document} is compactly supported, then ‖f^‖L2(Sn-1)≲‖f‖Lp,λ(Rn)for4λn+1+2λ≤p<2λn+1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {\hat{f}}\Vert _{L^2({\mathbb {S}}^{n-1})}\lesssim \Vert f\Vert _{L^{p,\lambda }({{{{\mathbb {R}}}}^n})}\ \ \hbox {for}\ \ \frac{4\lambda }{n+1+2\lambda }\le p<\frac{2\lambda }{n+1}. \end{aligned}$$\end{document}