On parabolic measures and subparabolic functions

Let D be a domain in R x n × R t 1 R_x^n \, \times \, R_t^1 and ∂ p D {\partial _p}D be the parabolic boundary of D. Suppose ∂ p D {\partial _p}D is composed of two parts B and S: B is given locally by t = τ t = \tau and S is given locally by the graph of x n = f ( x 1 , x 2 , ⋯ , x n − 1 , t ) {x_n} = f({x_1},{x_2}, \cdots ,{x_{n - 1}},t) where f is Lip 1 with respect to the local space variables and Lip 1 2 \tfrac {1} {2} with respect to the universal time variable. Let σ \sigma be the n-dimensional Hausdorff measure in R n + 1 {R^{n + 1}} and σ ′ \sigma ’ be the ( n − 1 ) (n - 1) -dimensional Hausdorff measure in R n {\textbf {R}^n} . And let d m ( E ) = d σ ( E ∩ B ) + d σ ′ × d t ( E ∩ S ) dm(E) = d\sigma (E \cap B) + d{\sigma ’} \times dt(E \cap S) for E ⊆ ∂ p D E \subseteq {\partial _p}D . We study (i) the relation between the parabolic measure on ∂ p D {\partial _p}D and the measure dm on ∂ p D {\partial _p}D and (ii) the boundary behavior of subparabolic functions on D.