Atomic decomposition on Lipschitz domains and compact composition operators on Dirichlet spaces

Abstract

The boundedness from below, the criterion for continuity and compactness of composition operators acting on α- Bloch spaces are shown. The essential norms of composition operators on and between μ-Bloch spaces are also shown. We study the atomic representations in function spaces and applications to pointwise multipliers , Lipschitz diffeomorphisms.The characteristic functions, non – smooth atomic decompositions, traces on Lipschitz domains, pointwise multipliers in function spaces and spaces on Lipschitz manifolds are given. We study the level sets and determined the approximation numbers of composition operators on weighted Dirichlet spaces.The Hausdorff measures, capacities of sets of contact points and compact composition operators on the Dirichlet space are discussed.