Spherical maximal functions and fractal dimensions of dilation sets

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2020-04-02 DOI:10.1353/ajm.2023.a902955

J. Roos, A. Seeger

引用次数: 24

Abstract

abstract:For the spherical mean operators $\scr{A}_t$ in $\Bbb{R}^d$, $d\ge 2$, we consider the maximal functions $M_Ef=\sup_{t\in E}|\scr{A}_t f|$, with dilation sets $E\subset [1,2]$. In this paper we give a surprising characterization of the closed convex sets which can occur as closure of the sharp $L^p$ improving region of $M_E$ for some $E$. This region depends on the Minkowski dimension of $E$, but also other properties of the fractal geometry such as the Assouad spectrum of $E$ and subsets of $E$. A key ingredient is an essentially sharp result on $M_E$ for a class of sets called (quasi-)Assouad regular which is new in two dimensions.

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膨胀集的球面极大函数与分形维数

摘要:对于球面平均算子$\scr{A}_t$在$\Bbb{R}^d$， $d\ ge2 $中，考虑极大函数$M_Ef=\sup_{t\在E}|\scr{A}_t f|$中，膨胀集$E\子集[1,2]$。本文给出了闭凸集的一个令人惊讶的特征，它可以作为$M_E$的锐利的$L^p$改进区域的闭包出现，对于某些$E$。这个区域依赖于E$的Minkowski维数，但也依赖于分形几何的其他性质，如E$的assad谱和E$的子集。一个关键的因素是对于一类叫做(拟-)正则集的在二维空间中新出现的关于$M_E$的一个本质上尖锐的结果。

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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.