Compact Embeddings of Bessel-Potential-Type Spaces into Generalized Hölder Spaces Involving <i>k</i>-Modulus of Smoothness

Abstract

We present conditions which are necessary and sufficient for compact embeddings of Bessel potential spaces H_σ_X(ℝ_n_), modelled upon a rearrangement-invariant Banach function spaces X(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(ℝ_n_) is a Lorentz–Karamata space Lp,q;b(ℝ_n_). Applications cover both superlimiting and limiting cases.