Embeddings of Sobolev-type spaces into generalized Hölder spaces involving $$k$$ k -modulus of smoothness

Abstract

We use 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 \(L^{n/k, 1}({\mathbb {R}}^n),\,k \in {\mathbb {N}},\,k\le n\), and we prove its reverse form to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces. These spaces are modelled upon rearrangement-invariant Banach function spaces \(X({\mathbb {R}}^n)\). Target spaces of our embeddings are generalized Holder spaces defined by means of the \(k\)-modulus of smoothness \((k\in {\mathbb {N}})\). General results are illustrated with examples.