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Weighted Sobolev inequalities and harmonic measure associated with quasiregular functions. (English) Zbl 0719.31002
Let \(U\subset {\mathbb{R}}^ n\) be a bounded domain, \(\phi\) a quasi regular function on U and \(J_{\phi}\) its Jacobian determinant. Then a weighted Sobolev inequality of the form \[ \int_{U}| u(x)|^ 2J_{\phi}(x)dx\quad \leq \quad C\int_{U}| \nabla u(x)|^ 2J_{\phi}^{1-2/n}(x)dx \] for all \(u\in C_ 0^{\infty}(U)\) is derived. The estimate is used to prove existence of the harmonic measure of the diffusion \(X_ t\) associated to \(\phi\). As an application a new result about boundary values of quasi regular functions is proved.
Reviewer: R.Leis (Bonn)

MSC:
31C15 Potentials and capacities on other spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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