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Zbl 0651.31002
Kaufman, Robert; Wu, Jang-Mei
Parabolic measure on domains of class Lip $\frac12$. (English)
[J] Compos. Math. 65, No. 2, 201-207 (1988). ISSN 0010-437X; ISSN 1570-5846/e

A function, $x\sb n=f(x',t)$, Lip 1 with respect to $x'\in R\sp{n-1}$ and Lip $\frac12$ with respect to $t\in {\bbfR}$ can be equipped with a norm, $$ \Vert f\Vert \triangleq \sup \vert f(x',t)-f(x\sb 0',t\sb 0)\vert < M(\sum\sp{n-1}\sb{i=1}(x\sb i'-x'\sb{0\sb i})\sp 2+\vert t-t\sb 0\vert)\sp{1/2}. $$ This paper develops a domain, $\Omega \subset {\bbfR}\sp 2$, whose boundary is given by the graph of a Lip $\frac12$ function, $x=F(t)$, so that on $\partial \Omega$ the parabolic measure, $\omega$, and the adjoint parabolic measure, $\omega\sp*$ are concentrated on two disjoint sets, whose projection onto the t-axis has Hausdorff dimension strictly less than 1. The paper carefully examines a function of class Lip $$ and connects the naturality of class Lip $$ to the solutions of the heat equation. Carefully computed estimates permeate this paper. The two disjoint sets are carefully constructed by implementing an increasing sequence, $A\sb 0\subset A\sb 1\subset...A\sb k\subset...$, of algebras of subsets of the half open interval, $[0,1)\subset {\bbfR}\sp 1$, defined as $A\sb k\triangleq [2pN\sp{-2k},(2p+2)N\sp{-2k})$ where p is an integer subject to the constraint, $0\le 2p\le N\sp{2k}-2$. The construction is clever and carefully presented to the reader.
[J.Schmeelk]

MSC 2000:: *31B25 Boundary behavior of harmonic functions (higher-dim.)
31C05 Generalizations of harmonic (etc.) functions
35K05 Heat equation

Keywords: projection; Lip $\frac12$ function; parabolic measure; adjoint parabolic measure; Hausdorff dimension; heat equation

Cited in: Zbl 0826.35041 Zbl 0780.35049

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