The Riemann Mapping Theorem states existence of a conformal homeomorphism φ of a simply connected plane domain Ω⊂C with non-empty boundary onto the unit disc D⊂C. In the first part of the paper we study embeddings of Sobolev spaces W1p∘(Ω) into weighted Lebesgue spaces Lq(Ω,h) with an {}"universal" weight that is Jacobian of φ i.e. h(z):=J(z,φ)=|φ′(z)|2. Weighted Lebesgue spaces with such weights depend only on a conformal structure of Ω. By this reason we call the weights h(z) conformal weights. In the second part of the paper we prove compactness of embeddings of Sobolev spaces W12∘(Ω) into Lq(Ω,h) for any 1≤q<∞. With the help of Brennan's conjecture we extend these results to Sobolev spaces W1p∘(Ω). In this case q is not arbitrary and depends on p and the summability exponent for Brennan's conjecture. Applications to elliptic boundary value problems are demonstrated in the last part of the paper.

Original language | English |
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State | Published - 2013 |
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Name | arXiv preprint arXiv:1302.4054 |
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