Characterization of embeddings of Sobolev-type spaces into generalized Hölder spaces defined by $L_p$-modulus of smoothness

Abstract

We prove a sharp estimate for the $k$-modulus of smoothness, modelled upon a~\mbox{$p$-Lebesgue} space, of a function $f$ in $ W^kL^{\frac{pn}{n+kp},p}(\Omega)+W^kL^p(\Omega)$, where $\Omega$ is a domain with minimally smooth boundary and finite Lebesgue measure. This sharp estimate is used to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces into generalized H\"older spaces defined by means of the $k$-modulus of smoothness.

General results are illustrated with examples. In particular, we obtain a generalization of the classical Jawerth embeddings.

Keywords

• Characterization
• embeddings
• Sobolev-type
• into
• generalized
• Hölder
• spaces
• defined
• $L_p$-modulus
• smoothness