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

Abstract

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