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On improved fractional Sobolev-Poincaré inequalities. (English) Zbl 1362.35012

Summary: We study a certain improved fractional Sobolev-Poincaré inequality on domains, which can be considered as a fractional counterpart of the classical Sobolev-Poincaré inequality. We prove the equivalence of the corresponding weak and strong type inequalities; this leads to a simple proof of a strong type inequality on John domains. We also give necessary conditions for the validity of an improved fractional Sobolev-Poincaré inequality, in particular, we show that a domain of finite measure, satisfying this inequality and a ‘separation property’, is a John domain.

MSC:

35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
35R11 Fractional partial differential equations
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