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We study the relationship between the Regularity and Dirichlet boundary valueproblems for parabolic equations of the form $Lu=\text{div}(A \nabla u)-u_t=0$in Lip$(1,1/2)$ time-varying cylinders, where the coefficient matrix $A =\left[ a_{ij}(X,t)\right] $ is uniformly elliptic and bounded.  We show that if the Regularity problem $(R)_p$ for the equation $Lu=0$ issolvable for some $1<p<\infty$ then the Dirichlet problem $(D^*)_{p'}$ for theadjoint equation $L^*v=0$ is also solvable, where $p'=p/(p-1)$.  This result is an analogue of the result established in the elliptic case byKenig and Pipher.  In the parabolic settings in the special case of the heat equation inslightly smoother domains this has been established by Hofmann and Lewis andNystr\"om for scalar parabolic systems. In comparison, our result is abstractwith no assumption on the coefficients beyond the ellipticity condition and isvalid in more general class of domains.

Parabolic Regularity and Dirichlet boundary value problems