Abstract. We show that a domain is an extension domain for a Haj lasz-Besov or for a Haj lasz-Triebel-Lizorkin space if and only if it satisfies a measure density condition. We use a modification of the Whitney extension where integral averages are replaced by median values, which allows us to handle also the case 0 < p < 1. The necessity of the measure density condition is derived from embedding theorems; in the case of Haj lasz-Besov spaces we apply an optimal Lorentz-type Sobolev embedding theorem which we prove using a new interpolation result. This interpolation theorem says that Haj lasz-Besov spaces are intermediate spaces between L p and Haj laszSobolev spaces. Our results are proved in the setting of a metric measure space, but most of them are new even in the Euclidean setting, for instance, we obtain a characterization of extension domains for classical Besov spaces B s p,q , 0 < s < 1, 0 < p < ∞, 0 < q ≤ ∞, defined via the L p -modulus of smoothness of a function.
We prove a certain improved fractional Sobolev-Poincaré inequality on John domains; the proof is based on the equivalence of the corresponding weak and strong type inequalities. We also give necessary conditions for the validity of an improved fractional Sobolev-Poincaré inequality, in particular, we show that a domain having a finite measure and satisfying this inequality, and a 'separation property', is a John domain.2010 Mathematics Subject Classification. 46E35 (26D10).
Abstract. Let X be a metric space equipped with a doubling measure. We consider weights w(x) = dist(x, E) −α , where E is a closed set in X and α ∈ R. We establish sharp conditions, based on the Assouad (co)dimension of E, for the inclusion of w in Muckenhoupt's A p classes of weights, 1 ≤ p < ∞. With the help of general A p -weighted embedding results, we then prove (global) Hardy-Sobolev inequalities and also fractional versions of such inequalities in the setting of metric spaces.
Abstract. We prove fractional order Hardy inequalities on open sets under a combined fatness and visibility condition on the boundary. We demonstrate by counterexamples that fatness conditions alone are not sufficient for such Hardy inequalities to hold. In addition, we give a short exposition of various fatness conditions related to our main result, and apply fractional Hardy inequalities in connection to the boundedness of extension operators for fractional Sobolev spaces.
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