Optimal embeddings and compact embeddings of Bessel-potential-type spaces
Let \(g_\sigma\), \(0 < \sigma <1\), be the Bessel kernels in \(\mathbb R^n\), \(\widehat{g_\sigma} (\xi) = c (1+ |\xi|^2)^{-\sigma/2}\). Let \(X(\mathbb R^n)\) be a rearrangement-invariant Banach function space, preferably a Lorentz-Zygmund space or, more generally, a Lorentz-Karamata space. Then the spaces \[ H^\sigma X(\mathbb R^n) = \{ u: u=f *g_\sigma, \;f\in X(\mathbb R^n) \} \] generalize the well-known classical Bessel-potential spaces \(H^\sigma_p (\mathbb R^n)\), \(1<p<\infty\), including the classical Sobolev spaces. It is the main aim of the paper to study embeddings of type \[ H^\sigma X(\mathbb R^n) \subset Y(\Omega), \] where \(Y(\Omega)\) is a suitable Banach space on the set \(\Omega\) in \(\mathbb R^n\). Here, \(Y(\Omega)\) are preferably generalized Hölder spaces. The authors provide necessary and sufficient conditions for continuous embeddings. They study their compactness.
scientific article; zbMATH DE number 5566982
scientific article