Ridged domains, embedding theorems and Poincare inequalities

We study the embeddings E : W(X(Omega),Y(Omega)) curved right arrow Z(Omega ), where X(Omega), Y(Omega) and Z(Omega) are rearrangement-invariant Banach function spaces (BFS) defined on a generalized ridged domain Omega, and W denotes a first-order Sobolev-type space. We obtain two-sided estimates for the measure of non-compactness of E when Z(Omega) = X(Omega) and, in turn, necessary and sufficient conditions for a Poincare-type inequality to be v alid and also for E to be compact. The results are used to analyse the exam ple of a trumpet-shaped domain Omega in Lorentz spaces. We consider the pro blem of determining the range of possible target spaces Z(Omega), in which case we prove that the problem is equivalent to an analogue on the generali zed ridge Gamma of Omega. The range of target spaces Z(Omega) is determined amongst a scale of (weighted) Lebesgue spaces for "rooms and passages" and trumpet-shaped domains.