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

We establish necessary and sufficient conditions for embeddings of Bessel potential spaces HσX(IRn) with order of smoothness σ ∈ (0, n), modelled upon rearrangement invariant Banach function spaces X(IRn), into generalized Hölder spaces (involving k-modulus of smoothness). We apply our results to the case when X(IRn) is the Lorentz-Karamata space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{p,q;b}({{\rm I\kern-.17em R}}^n)$\end{document}. In particular, we are able to characterize optimal embeddings of Bessel potential spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{\sigma}L_{p,q;b}({{\rm I\kern-.17em R}}^n)$\end{document} into generalized Hölder spaces. Applications cover both superlimiting and limiting cases. We also show that our results yield new and sharp embeddings of Sobolev-Orlicz spaces Wk + 1Ln/k(logL)α(IRn) and WkLn/k(logL)α(IRn) into generalized Hölder spaces.