A Characterization of Solvability of the Parabolic Lᵖ 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-[1,1/2] domain obey a Carleson measure estimate, then the corresponding parabolic measure on the boundary will belong to class A^∞, which is equivalent to Lᵖ solvability for some p<∞. 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.