\(\ell^p\)-improving inequalities for discrete spherical averages
type: Item
instance of: scholarly article
title: \(\ell^p\)-improving inequalities for discrete spherical averages
zbMATH Open document ID: 1449.42033
author: Robert M. Kesler
author: Michael T. Lacey
published in: Analysis Mathematica
publication date: 2020-02-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1804.09845
review text: For \( \lambda ^2 \in \mathbb{N} \), let \(\mathbb{S}^d_\lambda := \{ n\in \mathbb{Z}^d \;:\; | n| = \lambda\}\). For a function \(f:\mathbb{Z} ^{d} \to \mathbb{R} \), define \[A _{\lambda } f (x) = | \mathbb{S}^d_\lambda | ^{-1} \sum_{n \in \mathbb{S}^d_\lambda } f(x-n).\] The following estimate is the main result of the paper under review: \[| A _{\lambda }| _{\ell ^{p} \to \ell ^{p'}} \leq C _{d,p, \omega (\lambda ^2 )} \lambda ^{d (1-\frac{2}p)}, \tfrac{d-1}{d+1} < p \leq \frac{d} {d-2}, d\geq 4.\] In dimension \(d=4\) this estimate proved for odd \(\lambda ^2\). Here \(\omega (\lambda ^2)\) is the number of distinct prime factors of \(\lambda^2\). \par This inequality is a discrete version of a classical inequality of \textit{W. Littman} [Partial diff. Equ., Berkeley 1971, Proc. Sympos. Pure Math. 23, 479--481 (1973; Zbl 0263.44006)] and \textit{R. S. Strichartz} [J. Funct. Anal. 5, 218--235 (1970; Zbl 0189.40701)] on the \(L^{p}\) improving property of spherical averages on \(\mathbb{R} ^{d}\).
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 7175020
zbMATH Keywords: discrete spherical average
zbMATH Keywords: maximal function
MaRDI profile type: Publication
OpenAlex ID: W3006314687
arXiv ID: 1804.09845
DOI: 10.1007/S10476-020-0019-9
description: scientific article