Fractional Sobolev-Poincaré Inequalities inIrregular Domains

Citation:

Chang-Yu GUO.Fractional Sobolev-Poincaré Inequalities inIrregular Domains[J].Chinese Annals of Mathematics B,2017,38(3):839~856
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Authors:

Chang-Yu GUO;

Foundation:

This work was supported by the Magnus Ehrnrooth Foundation.
Abstract: This paper is devoted to the study of fractional $(q,p)$-Sobolev-Poincar\'e inequalities in irregular domains. In particular, the author establishes (essentially) sharp fractional $(q,p)$-Sobolev-Poincar\'e inequalities in $s$-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tend to the results for the usual derivatives. Furthermore, the author verifies that those domains which support the fractional $(q,p)$-Sobolev-Poincar\'e inequalities together with a separation property are $s$-diam John domains for certain $s$, depending only on the associated data. An inaccurate statement in [Buckley, S. and Koskela, P., Sobolev-Poincar\'e implies John, \textit{Math. Res. Lett.}, \textbf{2}(5), 1995, 577--593]is also pointed out.

Keywords:

Fractional Sobolev-Poincaré inequality, $s$-John domain, Quasihyperbolic boundary condition

Classification:

46E35, 26D10
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