A Characterization of Solvability of the Parabolic $L^p$ Dirichlet Problem on Lipschitz Graph Domains Via Carleson Measure Estimates of Bounded Solutions

In this paper, we show that if the bounded solutions to the parabolic Dirichlet problem on a Lipshitz-$[1,\frac{1}{2}]$ domain obey a Carleson measure estimate, then the corresponding parabolic measure on the boundary will belong to class $A^{\mathrm{\infty }}$, which is equivalent to $L^p$ solvability for some $p<\mathrm{\infty }$. This improves the existing literature which places additional assumptions on the parabolic uniform rectifiability or, equivalently, on the half-order time derivative of the function whose graph defines the boundary of the domain.