Compact Embeddings of Bessel-Potential-Type Spaces into Generalized Hölder Spaces Involving $k$-Modulus of Smoothness

Abstract

We present conditions which are necessary and sufficient for compact embeddings of Bessel potential spaces $H^{\sigma }X({\mathbb{R}}^n)$, modelled upon a rearrangement-invariant Banach function spaces $X({\mathbb{R}}^n)$, into generalized Hölder spaces involving k-modulus of smoothness. To this end, we derive a characterization of compact subsets of generalized Hölder spaces. We apply our results to the case when $X({\mathbb{R}}^n)$ is a Lorentz–Karamata space $L_{p,q;b}({\mathbb{R}}^n)$. Applications cover both superlimiting and limiting cases.