The generalized localization principle for Bochner-Riesz means (Q1207184)

Let \(f\in L^ p(\mathbb{R}^ n)\), \(1\leq p\leq 2\). The Bochner-Riesz means with index \(\alpha\) of \(f\) are defined via Fourier transform by \[ (B^ \alpha_ R f)^ \land(\xi)=\left(1-{| \xi|^ 2\over R^ 2}\right)^ \alpha_ +\widehat f(\xi),\quad 0<R<\infty,\quad\text{Re }\alpha>-1. \] In the paper it is proved the following generalized localization principle: Let \(\text{Re }\alpha=(n-1)(1/p-1/2)\). Then \(\lim_{R\to\infty} B^ \alpha_ R f(x)=0\), a.e. \(x\in\mathbb{R}^ n\backslash\text{supp }f\). The case \(p=2\) is due to \textit{A. J. Bastis} [Sov. Math., Dokl. 39, No. 1, 91-94 (1989); translation from Dokl. Akad. Nauk SSSR 304, No. 3, 526-529 (1989; Zbl 0685.42007)].