A relation between the Dirichlet and the Regularity problem for Parabolic equations

We study the relationship between the Dirichlet and Regularity problem for parabolic operators of the form $ L = \mbox{div}(A\nabla\cdot) - \partial_t $ on cylindrical domains $ \Omega = \mathcal O \times \mathbb R $, where the base $ \mathcal O \subset \mathbb R^{n} $ is a $1$-sided chord arc domain (and for one result Lipschitz) in the spatial variables. In the paper we answer the question when the solvability of the $L^p$ Regularity problem for $L$ (denoted by $ (R_L)_{p} $) can be deduced from the solvability of the $ L^{p'} $ Dirichlet problem for the adjoint operator $L^*$ (denoted $ (D_L^*)_{p'} $). We show that this holds if for at least of $q\in(1,\infty)$ the problem $ (R_L)_{q} $ is solvable. This result is a parabolic equivalent of two elliptic results of Kenig-Pipher (1993) and Shen (2006), the combination of which establishes the elliptic version of our result. For the converse, see Dindo\v{s}-Dyer (2019) where it is shown that $ (R_L)_{p}$ implies $ (D_L^*)_{p'}$ on parabolic $\mbox{Lip}(1,1/2)$ domains.