Parabolic mean values and maximal estimates for gradients of temperatures
type: Item
instance of: scholarly article
title: Parabolic mean values and maximal estimates for gradients of temperatures
zbMATH Open document ID: 1171.35020
author: Hugo Aimar
author: I. Gómez
author: Bibiana Iaffei
published in: Journal of Functional Analysis
publication date: 2008-12-22T00:00:00Z
full work available at URL: http://cimec.org.ar/ojs/index.php/cmm/article/view/2385
review text: The authors prove a pointwise inequality on a weighted space-time gradients of temperature functions on a cylinder domain in \(\mathbb{R}^{d+1}\). The gradient is shown to be dominated by an iterated maximal function of the temperature at each point in the domain. Specifically suppose that \( u(x,t)\) is a solution to \(\left( \frac{\partial }{\partial t}-\Delta \right) u(x,t)=0\) on \(\Omega =D\times \mathbb{R}^{+}\), with \(D\) being an open set in \(\mathbb{R}^{d}\). Let \(\left( \nabla ^{2,1}\right) ^{n}v(x,t)\) denote the vector of all derivatives of parabolic order \(2n\) of a smooth function \(v\). So\ \(\left( \nabla ^{2,1}v\right) =\left( \nabla ^{2}v,\frac{\partial v}{ \partial t}\right) \). Let \(\partial _{par}\Omega =\left( D\times \{0\}\right) \cup \left( \partial D\times \mathbb{R}^{+}\right) \) denote the parabolic boundary of \(\Omega \), and \(\delta (x,t)=\inf \{\max \left( \left| x-y\right| ,\sqrt{\left| t-s\right| }\right) :(y,s)\in \partial _{par}\Omega \}\), so \(\delta (x,t)\) denotes the parabolic distance from the point \((x,t)\) to the parabolic boundary of the domain. For \(f\in L_{loc}^{1}\left( \mathbb{R}\right) \), let \(M_{\mathbb{R} ^{+}}^{-}\left( f\right) \left( t\right) =\sup \{\frac{1}{h} \int_{t-h}^{t}\left| f(s)\right| ds:0<h<t\}\) denote the one-sided maximal operator restricted to \(\mathbb{R}^{+}\). If \(h\) is a smooth function defined on \(D\subset \mathbb{R}^{d}\), let \[ M_{D}^{\#,\lambda ,k}\left( h\right) \left( x\right) =\sup \{\frac{1}{\left| B(x;\delta )\right| ^{\frac{d+\lambda }{d}}}\int_{B(x;\delta )}\left| h(y)-P_{x}(y)\right| dy:0<\delta <\delta (x)\} \] be a local version of the Calderón-Scott maximal operator of order \(\lambda \). Here \(\delta (x)=\inf \{\left| x-z\right| :z\in \partial D\}\) and \(P_{x}(y)\) is the Taylor polynomial of degree \(k-1\) for \(h(y)\), expanded around the point \( x\). The authors' main result is that, for \(0<\lambda <2n<\lambda +d\), there exists a constant \(C>0\) such that …
reviewed by: Caroline Sweezy
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 5480722
zbMATH Keywords: maximal operators
zbMATH Keywords: mean value formula
zbMATH Keywords: caloric mean value kernel
MaRDI profile type: Publication
OpenAlex ID: W1990193396
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DOI: 10.1016/J.JFA.2008.06.006
description: scientific article; zbMATH DE number 5480722