Solvability of the ${\mathrm{L}}^2$ Dirichlet problem for parabolic operators with spatially periodic coefficients

We prove global reverse Hölder estimates in ${\mathrm{L}}^2$ for the Poisson kernel of scalar divergence-form parabolic operators in the upper half-space. The coefficient matrix is real, symmetric, and periodic in every spatial variable $X=(\lambda ,x)$, while no periodicity in time is assumed. If its boundary trace $A^0=A^0(x,t)$ has a quantitative uniform modulus of continuity in time and its convergence to $A^0$ in the transverse variable is controlled by a square-Dini modulus, then the Poisson kernel satisfies a uniform ${\mathrm{R}\mathrm{H}}_2$ estimate at scales below the period. Full spatial periodicity is then used to propagate this estimate to every scale. The main new ingredient in the large-scale argument is a positive adjoint height coordinate ${\mathrm{\Phi }}^{\ast }=\lambda +O(1)$. Boundary comparison with ${\mathrm{\Phi }}^{\ast }$ yields the required large-scale decay of the Green function and hence the global ${\mathrm{R}\mathrm{H}}_2$ estimate. Consequently, the ${\mathrm{L}}^2$ Dirichlet problem is uniquely solvable among weak solutions $u$ satisfying $N_{\ast }u\in {\mathrm{L}}^2$. We also establish, for fully spatially periodic coefficients, a nonsymmetric $A_{\mathrm{\infty }}$ theorem under square-Dini convergence to a transverse-independent boundary trace, with no temporal regularity beyond measurability. We explain how the results transfer to time-independent Lipschitz graph domains under the stated coefficient and periodicity-compatibility hypotheses, and we derive scale-uniform boundary estimates for the periodic homogenization families considered below. The condition $D_t^{1\mathrm{/}2}{A}^0\in {\mathrm{L}}_x^{\mathrm{\infty }}({\mathrm{B}\mathrm{M}\mathrm{O}}_t)$ provides a natural sufficient criterion for the temporal continuity required in the symmetric theorem.