On the interpolation, embedding, and extension of spaces of functions of variable smoothness. (English. Russian original) Zbl 1272.46016

Besov, O. V.

Dokl. Math. 71, No. 2, 163-167 (2005); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 401, No. 1, 7-11 (2005).

Summary: We study the spaces \(B^s_{p,q}(G)\) and \(L^s_{p,q}(G)=F^s_{p,q}(G)\) of functions defined on a domain \(G\) of the Euclidean \(n\)-space \(\mathbb{R}^n\); here, either \(G=\mathbb{R}^n\) or \(G\subset\mathbb{R}^n\) has the form \[ G=\{x=(x',x_n)\in\mathbb{R}^n: x'\in\mathbb{R}^{n-1},\;x_n>\varphi(x')\},\tag{1} \] where \(\varphi\) is a Lipschitz function on \(\mathbb{R}^{n-1}\), i.e., \[ \exists\Lambda>0:|\varphi (x')-\varphi(y')\leq\Lambda|x-y|\quad \forall x',y'\in\mathbb{R}^{n-1}.\tag{2} \]

MSC:

46E15	Banach spaces of continuous, differentiable or analytic functions
46B70	Interpolation between normed linear spaces
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems