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Regularity properties of commutators and layer potentials associated to the heat equation
Journal article   Open access   Peer reviewed

Regularity properties of commutators and layer potentials associated to the heat equation

John L. Lewis and Margaret A. M. Murray
Transactions of the American Mathematical Society, Vol.328(2), pp.815-842
01/01/1991
DOI: 10.1090/S0002-9947-1991-1020043-6
url
https://doi.org/10.1090/S0002-9947-1991-1020043-6View
Published (Version of record) Open Access

Abstract

In recent years there has been renewed interest in the solution of parabolic boundary value problems by the method of layer potentials. In this paper we consider graph domains D = { ( x , t ) : x > f ( t ) } D = \{ (x,t):x > f(t)\} in R 2 {\mathcal {R}^2} , where the boundary function f f is in I 1 / 2 ( BMO ) {I_{1/2}}({\text {BMO}}) . This class of domains would appear to be the minimal smoothness class for the solvability of the Dirichlet problem for the heat equation by the method of layer potentials. We show that, for 1 > p > ∞ 1 > p > \infty , the boundary single-layer potential operator for D D maps L p {L^p} into the homogeneous Sobolev space I 1 / 2 ( L p ) {I_{1/2}}({L^p}) . This regularity result is obtained by studying the regularity properties of a related family of commutators. Along the way, we prove L p {L^p} estimates for a class of singular integral operators to which the T1 {\text {T1}} Theorem of David and Journé does not apply. The necessary estimates are obtained by a variety of real-variable methods.

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