Improved ℓp -Boundedness for Integral k -Spherical Maximal Functions

IF 1 3区数学 Q1 MATHEMATICS

Discrete Analysis Pub Date : 2017-07-26 DOI:10.19086/da.3675

T. Anderson, Brian Cook, Kevin A. Hughes, A. Kumchev

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引用次数: 22

Abstract

Improved $\ell^p$-Boundedness for Integral $k$-Spherical Maximal Functions, Discrete Analysis 2018:10, 18pp. An important role in harmonic analysis is played by the notion of a _maximal function_ (which is actually a non-linear operator on a space of functions). The best-known example is the _Hardy-Littlewood maximal function_, which takes a function $f:\mathbb R^d\to\mathbb C$ and replaces it by the function $Mf:\mathbb R^d\to\mathbb R$, which is defined by the formula $$Mf(x)=\sup_{r>0}\frac 1{|B_r(x)|}\int_{B_r(x)}|f(x)|dx,$$ where $B_r(x)$ is the ball of radius $r$ about $x$. In other words, $Mf$ is the largest average of $|f|$ over any ball centred at $x$. Particularly useful are inequalities bounding norms of $Mf$ in terms of norms of $f$: for example, it is known that if $1

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积分k -球面极大函数的改进的p -有界性

改进$\ell^p$ -积分有界性$k$ -球面极大函数，离散分析，2018:10,18页。极大函数的概念在调和分析中起着重要的作用(它实际上是函数空间上的非线性算子)。最著名的例子是＿Hardy-Littlewood极大函数＿，它将函数$f:\mathbb R^d\to\mathbb C$替换为函数$Mf:\mathbb R^d\to\mathbb R$，该函数由公式$$Mf(x)=\sup_{r>0}\frac 1{|B_r(x)|}\int_{B_r(x)}|f(x)|dx,$$定义，其中$B_r(x)$是关于$x$的半径为$r$的球。换句话说，$Mf$是$|f|$比任何以$x$为中心的球的最大平均值。特别有用的是用$f$的范数表示$Mf$的不等式边界范数:例如，已知如果$1

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来源期刊

Discrete Analysis Mathematics-Algebra and Number Theory

CiteScore

1.60

自引率

0.00%

发文量

审稿时长

17 weeks

期刊介绍： Discrete Analysis is a mathematical journal that aims to publish articles that are analytical in flavour but that also have an impact on the study of discrete structures. The areas covered include (all or parts of) harmonic analysis, ergodic theory, topological dynamics, growth in groups, analytic number theory, additive combinatorics, combinatorial number theory, extremal and probabilistic combinatorics, combinatorial geometry, convexity, metric geometry, and theoretical computer science. As a rough guideline, we are looking for papers that are likely to be of genuine interest to the editors of the journal.

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