Parabolic mean values and maximal estimates for gradients of temperatures

Hugo A. Aimar, Ivana Gómez, Bibiana Iaffei


The main result of this paper is a pointwise estimate for
the space time gradients of a temperature on a cylindrical
domain in terms of an iteration of two maximal operators. The
result is an extension to the parabolic setting of the elliptic
inequalities proved by D.Jerison and C.Kenig and also by S.Dahlke and
R.DeVore. After an improvement of the parabolic mean value formula and the
analysis of the kernel and the operator that provides the space
of temperatures, we obtain a pointwise estimate for space
gradients weighted by powers of the distance to the parabolic
boundary in terms of an iteration of two maximal operators which
are well known in harmonic analysis: the one-sided maximal
Hardy-Littlewood operator in the time variable and the
Calderón maximal operator in the space variable.

Published: Journal of Functional Analysis. Volume 255, Issue 8, 2008, Pages 1939-1956

Link: doi:10.1016/j.jfa.2008.06.006

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