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Weighted Modular Inequalities for Hardy Type Operators
Author(s) -
Lai Qinsheng
Publication year - 1999
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/s0024611599012010
Subject(s) - mathematics , type (biology) , operator (biology) , hardy space , mathematics subject classification , maximal operator , norm (philosophy) , lorentz transformation , combinatorics , function (biology) , pure mathematics , discrete mathematics , mathematical analysis , ecology , biochemistry , chemistry , physics , repressor , classical mechanics , evolutionary biology , biology , political science , transcription factor , law , bounded function , gene
Given weight functions θ, w , ρ and v , the weighted modular inequalityQ − 1 ( ∫ 0 ∞ Q ( θ ( x ) T f ( x ) ) w ( x ) d x ) ⩽ P − 1 ( ∫ 0 ∞ P ( C ρ ( x ) f ( x ) ) v ( x ) d x ) is characterized. Here Q is a strictly increasing function with Q (0) = 0, Q (∞) = ∞ and 2 Q ( x ) ⩽ Q ( C x ), P is a Young's function, and T is the Hardy operator or a Hardy type operator. In particular, a characterizing condition for the Hardy type operator to map L p ( w ) to L q ( v ) when 0 < q < 1 ⩽ p < ∞ is deduced. In addition, a new proof for the Maz'ja‐Sinnamon theorem is given, and weighted Lorentz norm inequalities for Hardy type operators are established. 1991 Mathematics Subject Classification : primary 26D15, 42B25; secondary 26A33, 46E30.