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

Abstract

We establish necessary and sufficient conditions for embeddings of Bessel potential spaces H σ X(IR n ) with order of smoothness σ ∈ (0, n), modelled upon rearrangement invariant Banach function spaces X(IR n ), into generalized Holder spaces (involving k-modulus of smoothness). We apply our results to the case when X(IR n ) is the Lorentz-Karamata space \(L_{p,q;b}({{\rm I\kern-.17em R}}^n)\). In particular, we are able to characterize optimal embeddings of Bessel potential spaces \(H^{\sigma}L_{p,q;b}({{\rm I\kern-.17em R}}^n)\) into generalized Holder spaces. Applications cover both superlimiting and limiting cases. We also show that our results yield new and sharp embeddings of Sobolev-Orlicz spaces W k + 1 L n/k(logL) α (IR n ) and W k L n/k(logL) α (IR n ) into generalized Holder spaces.