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Reduction theorems for Sobolev embeddings into the spaces of Hölder, Morrey and Campanato type
Author(s) -
Holík Miloslav
Publication year - 2016
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/mana.201500043
Subject(s) - mathematics , sobolev space , embedding , interpolation space , pure mathematics , differentiable function , banach space , birnbaum–orlicz space , type (biology) , sobolev inequality , invariant (physics) , mathematical analysis , functional analysis , ecology , biochemistry , chemistry , artificial intelligence , biology , computer science , mathematical physics , gene
Let X be a rearrangement‐invariant Banach function space on Q where Q is a cube in R n and letV 1 X ( Q )be the Sobolev space of real‐valued weakly differentiable functions f satisfying | ∇ f | ∈ X ( Q ) . We establish a reduction theorem for an embedding of the Sobolev spaceV 1 X ( Q )into spaces of Campanato, Morrey and Hölder type. As a result we obtain a new characterization of such embeddings in terms of boundedness of a certain one‐dimensional integral operator on representation spaces.