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Ridged Domains, Embedding Theorems and Poincaré Inequalities
Author(s) -
Evans W.D.,
Harris D.J.,
Pick L.
Publication year - 2001
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200101)221:1<41::aid-mana41>3.0.co;2-m
Subject(s) - mathematics , lp space , embedding , compact space , pure mathematics , interpolation space , function space , birnbaum–orlicz space , lorentz transformation , sobolev space , lorentz space , invariant (physics) , order (exchange) , banach space , mathematical analysis , functional analysis , biochemistry , chemistry , physics , finance , classical mechanics , artificial intelligence , computer science , economics , mathematical physics , gene
We study the embeddings E : W ( X (Ω), Y (Ω)) ↪ Z (Ω), where X (Ω), Y (Ω) and Z (Ω) are rearrangement–invariant Banach function spaces (BFS) defined on a generalized ridged domain Ω, and W denotes a first–order Sobolev–type space. We obtain two–sided estimates for the measure of non–compactness of E when Z (Ω) = X (Ω) and, in turn, necessary and sufficient conditions for a Poincaré–type inequality to be valid and also for E to be compact. The results are used to analyse the example of a trumpet–shaped domain Ω in Lorentz spaces. We consider the problem of determining the range of possible target spaces Z (Ω), in which case we prove that the problem is equivalent to an analogue on the generalized ridge Γ of Ω. The range of target spaces Z (Ω) is determined amongst a scale of (weighted) Lebesgue spaces for “rooms and passages” and trumpet–shaped domains.