Characterization of embeddings of Sobolev-type spaces into generalized Hölder spaces defined by \(L^{p}\)-modulus of smoothness
In the last years, a lot of attention has been devoted to embeddings of Sobolev-type spaces \(W^kX(\Omega)\), where \(X(\Omega)\) is a rearrangement invariant Banach function space, \(\Omega\) is a domain in \(\mathbb{R}^n\), and \(k \in \mathbb{N}\). Methods based on rearrangement technics, which reduce Sobolev-type embeddings into rearrangement invariant Banach function spaces to one dimensional Hardy-type inequalities, have been extensively studied. Embeddings of Sobolev-type spaces \(W^kX(\Omega)\) with Hölder-type spaces as targets have been also intensively investigated. In [Ann. Mat. Pura Appl. (4) 194, No. 2, 425--450 (2015; Zbl 1318.26045)], the authors proved the reverse form of an estimate of the \(k\)-modulus of smoothness of a function \(f\) such that the norm of its distributional gradient \(| \nabla^kf|\) belongs locally to the Lorentz space \(L^{n/k,1}(\mathbb{R}^n)\), \(k \in \mathbb{N}\), \(k \leq n\). This helped to establish necessary and sufficient conditions for continuous embeddings of Sobolev type spaces, modelled upon rearrangement invariant Banach function spaces \(X(\mathbb{R}^n)\), into Hölder spaces defined by means of the \(k\)-modulus of continuity (\(L^\infty\)-modulus). A sharp estimate involving the \(L^p\)-modulus of smoothness of order \(k\), with \(p < +\infty\), was not available. The aim of this paper is to derive such a sharp estimate and use it to establish necessary and sufficient conditions for continuous embeddings of Sobolev spaces \(W^kX(\Omega)\), modelled upon rearrangement invariant Banach function spaces \(X(\Omega)\), into generalized Hölder spaces defined by means of the \(L^p\)-modulus of smoothness of order \(k\;(k \in \mathbb{N})\). The general results are illustrated with examples. In particular, a consequence is a generalization of the classical Jawerth embeddings.
rearrangement-invariant Banach function spaces
scientific article; zbMATH DE number 6988451
scientific article