In this paper we present the following result on regularity of solutions of the second order parabolic equation \(\partial _t u - {{\,\textrm{div}\,}}(A \nabla u)+B\cdot \nabla u=0\) on cylindrical domains of the form \(\Omega ={\mathcal {O}}\times {\mathbb {R}}\) where \({\mathcal {O}}\subset {\mathbb {R}}^n\) 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 \(\tilde{N}(\nabla u)\) belongs to \(L^p(\partial \Omega )\) for some \(p>1\) . Furthermore, assume that for \(u|_{\partial \Omega }=f\) we have that \(D^{1/2}_tf\in L^p(\partial \Omega )\) . Then both \(\tilde{N}(D^{1/2}_t u)\) and \(\tilde{N}(D^{1/2}_tH_t u)\) also belong to \(L^p(\partial \Omega )\) , where \(D^{1/2}_t\) 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.
