Besov-type spaces of variable smoothness on rough domains. (November 2016) - Tyulenev, A.I.
Abstract: The paper puts forward new Besov spaces of variable smoothness B p, q φ 0 ( G, { t k } ) and B ˜ p, q, r l ( Ω, { t k } ) on rough domains. A domain  G is either a bounded Lipschitz domain in  R n or the epigraph of a Lipschitz function, a domain  Ω is an ( ε, δ ) -domain. These spaces are shown to be the traces of the spaces B p, q φ 0 ( R n, { t k } ) and B ˜ p, q, r l ( R n, { t k } ) on domains G and  Ω, respectively. The extension operator Ext 1 : B p, q φ 0 ( G, { t k } ) → B p, q φ 0 ( R n, { t k } ) is linear, the operator Ext 2 : B ˜ p, q, r l ( Ω, { t k } ) → B ˜ p, q, r l ( R n, { t k } ) is nonlinear. As a corollary, an exact description of the traces of 2-microlocal Besov-type spaces and weighted Besov-type spaces on rough domains is obtained.