Bilinear Multipliers on Banach Function Spaces

摘要

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.