Interpolation, Embedding, and Extension of Spaces of Functions of Variable Smoothness

Interpolation, embedding, and extension theorems are proved for Banach spaces $B_{p,q}^s(G)$ and $L_{p,q}^s(G)=F_{p,q}^s(G)$, $1< p,q<\infty$, of functions that have a variable smoothness $s=s(x)$ and are defined on a domain $G\subset \mathbb R ^n$ with a Lipschitz boundary.