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

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 $D = \{ (x,t):x > f(t)\}$ in ${\mathcal {R}^2}$, where the boundary function $f$ is in ${I_{1/2}}({\text {BMO}})$. 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 $1 < p < \infty$, the boundary single-layer potential operator for $D$ maps ${L^p}$ into the homogeneous Sobolev space ${I_{1/2}}({L^p})$. This regularity result is obtained by studying the regularity properties of a related family of commutators. Along the way, we prove ${L^p}$ estimates for a class of singular integral operators to which the ${\text {T1}}$ Theorem of David and Journé does not apply. The necessary estimates are obtained by a variety of real-variable methods.