×

Parabolic mean values and maximal estimates for gradients of temperatures. (English) Zbl 1171.35020

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 S. Dahlke and 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 N. Suzuki and N A. Watson [Colloq. Math. 1998, No. 1, 87–96 (2003; Zbl 1047.35049)], A. P. Calderón and R. Scott [Studia Math. 62, No. 1, 75–92 (1978; Zbl 0399.46031)] and D. Jerison and C. E. Kenig [J. Funct. Anal. 130, No. 1, 161–219 (1995; Zbl 0832.35034)].

MSC:

35B45 A priori estimates in context of PDEs
35K05 Heat equation
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Calderón, A. P.; Scott, R., Sobolev type inequalities for \(p > 0\), Studia Math., 62, 1, 75-92 (1978) · Zbl 0399.46031
[2] Dahlke, S.; DeVore, R. A., Besov regularity for elliptic boundary value problems, Comm. Partial Differential Equations, 22, 1-2, 1-16 (1997) · Zbl 0883.35018
[3] de Guzmán, M., Real Variable Methods in Fourier Analysis, North-Holland Math. Stud., vol. 46 (1981), North-Holland: North-Holland Amsterdam, Notas de Matemática [Mathematical Notes], 75 · Zbl 0449.42001
[4] DeVore, R. A.; Sharpley, R. C., Maximal functions measuring smoothness, Mem. Amer. Math. Soc., 47, 293 (1984), viii+115 · Zbl 0529.42005
[5] Evans, L. C., Partial Differential Equations, Grad. Stud. Math., vol. 19 (1998), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI
[6] Fulks, W., A mean value theorem for the heat equation, Proc. Amer. Math. Soc., 17, 6-11 (1966) · Zbl 0152.10503
[7] Jerison, D.; Kenig, C. E., The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130, 1, 161-219 (1995) · Zbl 0832.35034
[8] Martín-Reyes, F. J., New proofs of weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Proc. Amer. Math. Soc., 117, 3, 691-698 (1993) · Zbl 0771.42011
[9] Peetre, J., New Thoughts on Besov Spaces, Duke Univ. Math. Ser., vol. 1 (1976), Duke Univ. Press: Duke Univ. Press Durham, NC · Zbl 0356.46038
[10] Sawyer, E., Weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Trans. Amer. Math. Soc., 297, 1, 53-61 (1986) · Zbl 0627.42009
[11] Suzuki, N.; Watson, N. A., Mean value densities for temperatures, Colloq. Math., 98, 1, 87-96 (2003) · Zbl 1047.35049
[12] Watson, N. A., A theory of subtemperatures in several variables, Proc. London Math. Soc. (3), 26, 385-417 (1973) · Zbl 0253.35045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.