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Optimal embeddings and compact embeddings of Bessel-potential-type spaces
Authors:Amiran Gogatishvili  Júlio S Neves  Bohumír Opic
Institution:(1) Mathematical Institute, Academy of Sciences of the Czech Republic, Z̆itná 25, 11567 Prague 1, Czech Republic;(2) CMUC, Department of Mathematics, University of Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal;(3) Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 46117 Liberec, Czech Republic
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 Hölder 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 Hölder spaces and then we derive necessary and sufficient conditions for compact embeddings of Bessel potential spaces \({H^{\sigma}X(\mathbb R^n)}\) into generalized Hölder 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 Hölder spaces and also compact embeddings of spaces in question. Applications cover both superlimiting and limiting cases.
Keywords:
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