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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On parabolic measures and subparabolic functions

Author: Jang Mei G. Wu
Journal: Trans. Amer. Math. Soc. 251 (1979), 171-185
MSC: Primary 31C99; Secondary 31D05, 35K99
Erratum: Trans. Amer. Math. Soc. 259 (1980), 636-636.
MathSciNet review: 531974
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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.

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Keywords: Lipschitz domain, heat equation, parabolic measure, parabolic function, subparabolic function, Green's function, Schauder estimates, Harnack inequality, maximum principle, Brownian trajectories
Article copyright: © Copyright 1979 American Mathematical Society