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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
##### Keywords:
weighted Sobolev inequality; harmonic measure
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##### References:
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