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Embedding theorem for weighted Sobolev classes on a John domain with weights that are functions of the distance to some h-set

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Abstract

Let Ω be a John domain, let Γ ⊂ ∂Ω be an h-set, and let g and v be weights on Ω that are distance functions to the set Γ of special form. In the paper, sufficient conditions are obtained under which the Sobolev weighted class W r p,g (Ω) is continuously embedded in the space L q,v (Ω). Moreover, bounds for the approximation of functions in W r p,g (Ω) by polynomials of degree not exceeding r − 1 in L q,v (\(\tilde \Omega \)) are found, where \(\tilde \Omega \) is a subdomain generated by a subtree of the tree T defining the structure of Ω.

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  1. Department of Mechanics and Mathematics, Moscow State University, Moscow, 119991, Russia

    A. A. Vasil’eva

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Supported by RFBR grants nos. 13-01-00022 and 12-01-00554.

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Vasil’eva, A.A. Embedding theorem for weighted Sobolev classes on a John domain with weights that are functions of the distance to some h-set. Russ. J. Math. Phys. 20, 360–373 (2013). https://doi.org/10.1134/S1061920813030102

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