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Symmetrization Inequalities for Composition Operators of Carathéodory Type
Author(s) -
Hajaiej H.,
Stuart C. A.
Publication year - 2003
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611503014473
Subject(s) - symmetrization , mathematics , context (archaeology) , type (biology) , identity (music) , mathematics subject classification , composition (language) , function (biology) , pure mathematics , measure (data warehouse) , combinatorics , inequality , mathematical analysis , physics , paleontology , database , evolutionary biology , computer science , acoustics , ecology , linguistics , philosophy , biology
Let F :(0, ∞) × [0, ∞) → R be a function of Carathéodory type. We establish the inequality∫R NF ( | x | , u ( x ) ) d x ⩽ ∫R NF ( | x | , u ∗ ( x ) ) d x . where u * denotes the Schwarz symmetrization of u , under hypotheses on F that seem quite natural when this inequality is used to obtain existence results in the context of elliptic partial differential equations. We also treat the case where R N is replaced by a set of finite measure. The identity∫R NG ( u ( x ) ) d x = ∫R NG ( u ∗ ( x ) ) d x is also discussed under the assumption that G : [0,∞) → R is a Borel function. 2000 Mathematics Subject Classification 26D20, 42C20, 46E30.