A note on Gaussian maximal functions
type: Item
instance of: scholarly article
title: A note on Gaussian maximal functions
zbMATH Open document ID: 1325.42023
author: Jonas Teuwen
published in: Indagationes Mathematicae. New Series
publication date: 2014-11-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1311.5032
review text: Let \(\gamma\) be the Gaussian measure \(d\gamma(x)=\pi^{-\frac{d}2}e^{-|x|^2}\,d\lambda(x)\), where \(\lambda\) is the Lebesgue measure, and let \(L:=\frac12\Delta-\langle x,\nabla\rangle\) be the Ornstein-Uhlenbeck operator on the Gaussian measure space \((\mathbb{R}^d,d\gamma)\). In this paper, it is proved that the non-tangential maximal function of the Ornstein-Uhlenbeck semigroup is bounded pointwise by the Gaussian Hardy-Littlewood maximal function, namely, \[ \sup_{(y,t)\in \Gamma^{(A,a)}_x}|e^{t^2L}u(y)|\lesssim \sup_{r>0}\frac1{\gamma(B(x,r))}\int_{B(x,r)}|u|\,d\gamma, \] where \(\Gamma^{(A,a)}_x\) denotes the set \[ \{(y,t)\in \mathbb{R}^{d+1}_+:\;\;|x-y|<AT,\;\;t\leq am(x)\} \] with \(A,a>0\) and \(m(x)=\min\{1, 1/|x|\}\). This generalizes a former result by \textit{E. Pineda} and \textit{W. Urbina R.} [Divulg. Mat. 16, No. 1, 107--124 (2008; Zbl 1217.60028)].
reviewed by: Wen Yuan
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 28C20
zbMATH DE Number: 6374450
zbMATH Keywords: Gaussian maximal function
zbMATH Keywords: non-tangential maximal function
zbMATH Keywords: Ornstein-Uhlenbeck semigroup
zbMATH Keywords: Mehler kernel
zbMATH Keywords: admissible cones
MaRDI profile type: Publication
OpenAlex ID: W2964067717
arXiv ID: 1311.5032
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DOI: 10.1016/J.INDAG.2014.07.017
description: scientific article