Besov-type spaces of variable smoothness on rough domains

The paper puts forward new Besov spaces of variable smoothness Bp,qφ0(G,{tk}) and B˜p,q,rl(Ω,{tk}) on rough domains. A domain  G is either a bounded Lipschitz domain in  Rn 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 Bp,qφ0(Rn,{tk}) and B˜p,q,rl(Rn,{tk}) on domains G and  Ω, respectively. The extension operator Ext1:Bp,qφ0(G,{tk})→Bp,qφ0(Rn,{tk}) is linear, the operator Ext2:B˜p,q,rl(Ω,{tk})→B˜p,q,rl(Rn,{tk}) 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.