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 ${\partial }_{t}u-\text{div}\left(A\nabla u\right)+B·\nabla u=0$ on cylindrical domains of the form $\Omega =O\times R$ where $O\subset R^n$ is a uniform domain (it satisfies both interior corkscrew and Harnack chain conditions) and has a boundary that is $n-1$ -Ahlfors regular. Let u be a solution of such PDE in $\Omega$ and the non-tangential maximal function of its gradient in spatial directions $\tilde{N}\left(\nabla u\right)$ belongs to $L^p\left(\partial \Omega \right)$ for some $p>1$ . Furthermore, assume that for ${u|}_{\partial \Omega }=f$ we have that $D_t^{1/2}f\in L^p\left(\partial \Omega \right)$ . Then both $\tilde{N}\left(D_t^{1/2}u\right)$ and $\tilde{N}\left(D_t^{1/2}{H}_{t}u\right)$ also belong to $L^p\left(\partial \Omega \right)$ , where $D_t^{1/2}$ and $H_t$ 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 $L^p$ 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.