Intrinsic Atomic Characterizations of Function Spaces on Domains

this paper to seal the gap by providing intrinsic atomic characterizations for all these spaces under very mild and natural restrictions on the (non-- smooth) domain\Omega\Gamma A second problem connected with the approach (i) is the question whether there exists a (linear bounded) extension operator from the spaces on\Omega into the corresponding spaces on

A Fourier analytical characterization of the Hausdorff dimension of a closed set and of related Lebesgue spaces

. Let \Gamma be a closed set in R n with the Lebesgue measure j\Gammaj = 0. The first aim of the paper is to give a Fourier analytical characterization of the Hausdorff dimension of \Gamma. Let 0 ! d ! n. If there exist a Borel measure ¯ with supp ¯ ae \Gamma and constants c 1 ? 0 and c 2 ? 0 such that c 1 r d ¯(B(x; r)) c 2 r d for all 0 ! r ! 1 and all x 2 \Gamma, where B(x; r) is a ball with centre x and radius r, then \Gamma is called d- set. The second aim of the paper is to provide a link between the related Lebesgue spaces L p (\Gamma) , 0 ! p 1, with respect to that measure ¯ on the one hand and the Fourier analytically defined Besov spaces B s p;q (R n ) (s 2 R, 0 ! p 1, 0 ! q 1) on the other hand. 1. Introduction Let 0 ! oe ! 1. Then C oe (R n ) stands for the usual Holder space on R n , that means for the collection of all complex- valued continuous functions f on R n such that kf j C oe (R n ) k = sup x2R n jf(x)j + sup x6=y jf(x) \Gamma..