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Notes on bilinear multipliers on Orlicz spaces
Author(s) -
Blasco Oscar,
Osançlıol Alen
Publication year - 2019
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.201800551
Subject(s) - mathematics , bilinear interpolation , multiplier (economics) , lp space , pure mathematics , type (biology) , class (philosophy) , bilinear map , function space , symmetric bilinear form , bounded function , space (punctuation) , bilinear form , algebra over a field , mathematical analysis , banach space , computer science , ecology , statistics , biology , operating system , economics , macroeconomics , artificial intelligence
LetΦ 1 , Φ 2and Φ 3 be Young functions and letL Φ 1( R ) ,L Φ 2( R )andL Φ 3( R )be the corresponding Orlicz spaces. We say that a function m ( ξ , η ) defined on R × R is a bilinear multiplier of type ( Φ 1 , Φ 2 , Φ 3 ) ifB m ( f , g ) ( x ) = ∫ R ∫ R f ̂ ( ξ ) g ̂ ( η ) m ( ξ , η ) e 2 π i ( ξ + η ) x d ξ d η defines a bounded bilinear operator fromL Φ 1( R ) × L Φ 2( R )toL Φ 3( R ) . We denote byBM ( Φ 1 , Φ 2 , Φ 3 )( R )the space of all bilinear multipliers of type ( Φ 1 , Φ 2 , Φ 3 ) and investigate some properties of such a class. Under some conditions on the triple ( Φ 1 , Φ 2 , Φ 3 ) we give some examples of bilinear multipliers of type ( Φ 1 , Φ 2 , Φ 3 ) . We will focus on the case m ( ξ , η ) = M ( ξ − η ) and get necessary conditions on ( Φ 1 , Φ 2 , Φ 3 ) to get non‐trivial multipliers in this class. In particular we recover some of the known results for Lebesgue spaces.