On restrictions and extensions of the besov and triebel-lizorkin spaces with respect to lipschitz domains

摘要

The restrictions B_(bq)~s(#OMEGA#) and F_(bq)~s(#OMEGA#) of the Besov and Triebel-Lizorkin spaces of tempered distributions B_(bq)~s(R~n) and F_(bq)~s(R~n) to Lipschitz domains #OMEGA# is contained in R~n are studied. For general values of parameters (s, implied R, p > 0, q > 0)a universal linear bounded extension operator from B_(bq)~s(#OMEGA#) and F_(bq)~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_(bq)~s(#OMEGA#) and F_(bq)~s(#OMEGA#) in terms of their values in #OMEGA# are also obtained.