On parabolic measures and subparabolic functions

Abstract:

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