Bilinear Multipliers on Banach Function Spaces

Let X 1 , X 2 , X 3 be Banach spaces of measurable functions in L 0 ( R ) and let m ( ξ , η ) be a locally integrable function in R 2 . We say that m ∈ B M ( X 1 , X 2 , X 3 ) ( R ) if B m ( f , g ) ( x ) = ∫ R ∫ R f ^ ( ξ ) g ^ ( η ) m ( ξ , η ) e 2 π i < ξ + η , x > d ξ d η , defined for f a...

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Bibliographic Details
Published in:Journal of function spaces 2019-01, Vol.2019 (2019), p.1-11
Main Author: Blasco, Oscar
Format: Article
Language:English
Subjects:
Banach spaces
Fourier analysis
Fourier transforms
Function space
Mathematical functions
Mathematics
Multipliers
Operators (mathematics)
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Summary:Let X 1 , X 2 , X 3 be Banach spaces of measurable functions in L 0 ( R ) and let m ( ξ , η ) be a locally integrable function in R 2 . We say that m ∈ B M ( X 1 , X 2 , X 3 ) ( R ) if B m ( f , g ) ( x ) = ∫ R ∫ R f ^ ( ξ ) g ^ ( η ) m ( ξ , η ) e 2 π i < ξ + η , x > d ξ d η , defined for f and g with compactly supported Fourier transform, extends to a bounded bilinear operator from X 1 × X 2 to X 3 . In this paper we investigate some properties of the class B M ( X 1 , X 2 , X 3 ) ( R ) for general spaces which are invariant under translation, modulation, and dilation, analyzing also the particular case of r.i. Banach function spaces. We shall give some examples in this class and some procedures to generate new bilinear multipliers. We shall focus on the case m ( ξ , η ) = M ( ξ - η ) and find conditions for these classes to contain nonzero multipliers in terms of the Boyd indices for the spaces.
ISSN:2314-8896
2314-8888
DOI:10.1155/2019/7639380