Tailored Besov spaces and h-sets

In this paper we define Besov-type spaces with generalised smoothness on a rather vast class of (isotropic) irregular sets of fractal type in the Euclidean space (h-sets). As a special case we shall obtain the definition of Besov spaces with the usual scalar-index of regularity. To deal with this problem we rely on the one hand on the measure-geometric theory we have developed for h-sets and, on the other, we rely on some known results for Besov spaces with generalised smoothness in R-n and on some advanced techniques concerning these function spaces (local means and atoms), which have been recently developed in full generality by W. Farkas and H.-G. Leopold. (C) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.