Optimal embeddings and compact embeddings of Bessel-potential-type spaces

Abstract

First, we establish necessary and sufficient conditions for embeddings of Bessel potential spaces H^ σ X(R^n) with order of smoothness less than one, modelled upon rearrangement invariant Banach function spaces X(R^n), into generalized Hölder spaces. To this end, we derive a sharp estimate of modulus of smoothness of the convolution of a function f in X(R^n) with the Bessel potential kernel gσ , 0 < s < 1. Such an estimate states that if gσ belongs to the associate space of X, then ω(f* gσ,t) precsim \int\limits_0^{t^n}s^{\frac{\σ}{n}-1}f^*(s)\,ds \quad {\rm for\,all} \quad t\in(0,1) \quad {\rm and\,every}\quad f in X(R^n). Second, we characterize compact subsets of generalized Hölder spaces and then we derive necessary and sufficient conditions for compact embeddings of Bessel potential spaces Hσ X(R^n) into generalized Hölder spaces. We apply our results to the case when X(R^n) is the Lorentz–Karamata space {L_{p,q;b}(R^n)}. In particular, we are able to characterize optimal embeddings of Bessel potential spaces {H^{σ}L_{p,q;b}(R^n)} into generalized Hölder spaces and also compact embeddings of spaces in question. Applications cover both superlimiting and limiting cases.