On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains

The restrictions B-pq(s)(Omega) and F-pq(s)(Omega) of the Besov and Triebel-Lizorkin spaces of tempered distributions B-pq(s)(R-n) and F-pq(s)(R-n) to Lipschitz domains Omega subset of R-n are studied. For general values of parameters (s is an element of R, p > 0, q > 0) a 'universal' linear bounded extension operator from B-pq(s)(Omega) and F-pq(s)(Omega) into the corresponding spaces on R-n is constructed. The construction is based on a new variant of the Calderon reproducing formula with kernels supported in a fixed cone. Explicit characterizations of the elements of B-pq(s)(Omega) and F-pq(s)(Omega) in terms of their values in Omega are also obtained.