$$L^p$$ bounds for Stein’s spherical maximal operators

IF 1.3 2区数学 Q1 MATHEMATICS

Mathematische Annalen Pub Date : 2024-05-04 DOI:10.1007/s00208-024-02884-y

Naijia Liu, Minxing Shen, Liang Song, Lixin Yan

引用次数: 0

Abstract

Let ${\mathfrak {M}}^\alpha $ be the spherical maximal operators of complex order $\alpha $ on ${{\mathbb {R}}^n}$. In this article we show that when $n\ge 2$, suppose

$$\begin{aligned} \Vert {\mathfrak {M}}^{\alpha } f \Vert _{L^p({{\mathbb {R}}^n})} \le C\Vert f \Vert _{L^p({{\mathbb {R}}^n})} \end{aligned}$$

holds for some $\alpha $ and $p\ge 2$, then we must have that $\textrm{Re}\,\alpha \ge \max \{1/p-(n-1)/2,\ -(n-1)/p \}.$ In particular, when $n=2$, we prove that $ \Vert {\mathfrak {M}}^{\alpha } f \Vert _{L^p({{\mathbb {R}}^2})} \le C\Vert f \Vert _{L^p({{\mathbb {R}}^2})}$ if $\textrm{Re}\ \! \alpha >\max \{1/p-1/2,\ -1/p\}$, and consequently the range of $\alpha $ is sharp in the sense that the estimate fails for $\textrm{Re}\ \alpha <\max \{1/p-1/2, -1/ p\}.$

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斯坦因球面最大算子的 $L^p$$ 边界

让（{\mathfrak {M}}^\alpha \）成为（{\mathbb {R}}^n}\) 上复阶（\alpha \）的球面最大算子。在本文中，我们将证明当（n\ge 2\) 时，假设 $$\begin{aligned}{\Vert {\mathfrak {M}}^{\alpha } f \Vert _{L^p({{\mathbb {R}}^n})}\le C\Vert f \Vert _{L^p({{\mathbb {R}}^n})}\end{aligned}$$holds for some $α $ and $p\ge 2$, then we must have that $textrm{Re\},α \ge \max \{1/p-(n-1)/2,\ -(n-1)/p \}.$特别地，当（n=2）时，我们证明（（（ Vert {\mathfrak {M}}^{\alpha } f \Vert _{L^p({\{mathbb {R}}^2})}\le C\Vert f \Vert _{L^p({{\mathbb {R}}^2})}\) if （textrm{Re}\！\max （1/p-1/2,-1/p\}），因此 $\alpha $的范围是尖锐的，即 $\textrm{Re}\alpha <\max （1/p-1/2,-1/p\}.$ 的估计失败。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematische Annalen 数学-数学

CiteScore

2.90

自引率

7.10%

发文量

181

审稿时长

4-8 weeks

期刊介绍： Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin. The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin. Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.