Parabolic mean values and maximal estimates for gradients of temperatures
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 for every temperature function \(u(x,t)\) defined on \(\Omega \), and for every \(\left( x,t\right) \in \Omega \), then \[ \delta ^{2n-\lambda }\left( x,t\right) \left| \left( \nabla ^{2,1}\right) ^{n}u(x,t)\right| \leq CM_{\mathbb{R}^{+}}^{-}\left[ M_{D}^{\#,\lambda ,2n}\left( u\right) \right] (x,t). \] The authors start their proof with a basic mean value property for a temperature function, go on to derive estimates for the spatial derivatives of the caloric mean value kernel, and then establish the maximal operator estimates. If \(D\) is a bounded Lipschitz domain, they obtain mixed norm inequalities for \(\delta ^{2n-\lambda }\left( x,t\right) \left| \left( \nabla ^{2,1}\right) ^{n}u(x,t)\right| \) as well. They note that their paper is part of a larger project to extend results of \textit{S. Dahlke} and \textit{R. A. DeVore} [Comm. Partial Differential Equations 22, No.~1--2, 1--16 (1997; Zbl 0883.35018)] to the parabolic setting. Other related work is contained in \textit{N. Suzuki} and \textit{N A. Watson} [Colloq. Math. 1998, No.~1, 87--96 (2003; Zbl 1047.35049)], \textit{A. P. Calderón} and \textit{R. Scott} [Studia Math. 62, No.~1, 75--92 (1978; Zbl 0399.46031)] and \textit{D. Jerison} and \textit{C. E. Kenig} [J. Funct. Anal. 130, No.~1, 161--219 (1995; Zbl 0832.35034)].
scientific article; zbMATH DE number 5480722