Rough Hypoellipticity for the Heat Equation in Dirichlet Spaces

TL;DR

This paper establishes local boundedness and continuity of heat equation solutions within Dirichlet spaces under weak assumptions, advancing understanding of heat kernel existence and solution regularity in this mathematical framework.

Contribution

It proves local boundedness and continuity of solutions in Dirichlet spaces with mild conditions, extending previous results and providing applications to heat kernel existence and solution structure.

Findings

Solutions are locally bounded and continuous under weak assumptions.

Existence of locally bounded heat kernels is demonstrated.

Results include $L^ abla$-structure for ancient solutions.

Abstract

This paper aims at proving the local boundedness and continuity of solutions of the heat equation in the context of Dirichlet spaces under some rather weak additional assumptions. We consider symmetric local regular Dirichlet forms which satisfy mild assumptions concerning (a) the existence of cut-off functions, (b) a local ultracontractivity hypothesis, and (c) a weak off-diagonal upper bound. In this setting, local weak solutions of the heat equation, and their time derivatives, are shown to be locally bounded; they are further locally continuous, if the semigroup admits a locally continuous density function. Applications of the results are provided including discussion on the existence of locally bounded heat kernel; $L^\infty$ structure results for ancient solutions of the heat equation. The last section presents a special case where the $L^\infty$ off-diagonal upper bound follows from the ultracontractivity property of the semigroup. This paper is a continuation of [7].