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Weighted Hardy‐Type Inequalities for Differences and the Extension Problem for Spaces with Generalized Smoothness
Author(s) -
Burenkov V. I.,
Evans W. D.
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798005626
Subject(s) - bounded function , mathematics , extension (predicate logic) , smoothness , domain (mathematical analysis) , sobolev space , boundary (topology) , embedding , pure mathematics , type (biology) , space (punctuation) , bounded operator , operator (biology) , discrete mathematics , combinatorics , mathematical analysis , computer science , ecology , biochemistry , chemistry , repressor , artificial intelligence , transcription factor , gene , biology , operating system , programming language
It is well known that there are bounded domains Ω⊂ℝ n whose boundaries ∂Ω are not smooth enough for there to exist a bounded linear extension for the Sobolev spaceW p 1 ( Ω ) intoW p 1 ( R n ) , but the embeddingW p 1 ( Ω ) ⊂ L p ( Ω ) is nevertheless compact. For the Lipγ boundaries (0<γ<1) studied in [ 3 , 4 ], there does not exist in general an extension operator ofW p 1 ( Ω ) intoW p 1 ( R n ) but there is a bounded linear extension ofW p 1 ( Ω ) intoW p γ ( R n ) and the smoothness retained by this extension is enough to ensure that the embeddingW p 1 ( Ω ) ⊂ L p ( Ω ) is compact. It is natural to ask if this is typical for bounded domains which are such thatW p 1 ( Ω ) ⊂ L p ( Ω ) is compact, that is, that there exists a bounded extension into a space of functions in ℝ n which enjoy adequate smoothness. This is the question which originally motivated this paper. Specifically we study the ‘extension by zero’ operator on a space of functions with given ‘generalized’ smoothness defined on a domain with an irregular boundary, and determine the target space with respect to which it is bounded.