On the Regularity Problem for Parabolic Operators and the Role of Half-Time Derivative

In this paper we present the following result on regularity of solutions of the second order parabolic equation $$ on cylindrical domains of the form $$ where $$ is a uniform domain (it satisfies both interior corkscrew and Harnack chain conditions) and has a boundary that is $$ -Ahlfors regular. Let u be a solution of such PDE in $$ and the non-tangential maximal function of its gradient in spatial directions $$ belongs to $$ for some $$ . Furthermore, assume that for $$ we have that $$ . Then both $$ and $$ also belong to $$ , where $$ and $$ are the half-derivative and the Hilbert transform in the time variable, respectively. We expect this result will spur new developments in the study of solvability of the $$ parabolic Regularity problem as thanks to it it is now possible to formulate the parabolic Regularity problem on a large class of time-varying domains.