Function spaces in lipschitz domains and on lipschitz manifolds. Characteristic functions as pointwise multipliers

Abstract

Function spaces of type Bs pq and Fs pq cover as special cases classical and fractional Sobolev spaces, classical Besov spaces, Holder- zgmund spaces and inhomogeneous Hardy spaces. In the last or 3 decades they haven been studied preferably on Rn and in mooth bounded domains in Rn including numerous applications to pseudodi_erential operators, elliptic boundary value problems etc. To a lesser extent spaces of this type have been considered in Lipschitz domains. But in recent times there is a growing interest to study and to use spaces of this type in Lipschitz domains and on their boundaries. It is the aim of this paper to deal with function spaces of Bs pq and Fs pq type in Lipschitz domains and on Lipschitz manifolds in a systematic (although not comprehensive) way: We describe and comment on known results, seal some gaps, give new proofs, and add a few new results of relevant aspects.