Bilinear multipliers of weighted Lebesgue spaces and variable exponent Lebesgue spaces

摘要

Let 1 ≤ p 1 , p 2 < ∞ , 0 < p 3 ≤ ∞ and ω 1 , ω 2 , ω 3 be weight functions on R n . Assume that ω 1 , ω 2 are slowly increasing functions. We say that a bounded function m ( ξ , η ) defined on R n × R n is a bilinear multiplier on R n of type ( p 1 , ω 1 ; p 2 , ω 2 ; p 3 , ω 3 ) (shortly ( ω 1 , ω 2 , ω 3 ) ) if B m ( f , g ) ( x ) = ∫ R n ∫ R n f ? ( ξ ) g ? ( η ) m ( ξ , η ) e 2 π i 〈 ξ + η , x 〉 d ξ d η is a bounded bilinear operator from L ω 1 p 1 ( R n ) × L ω 2 p 2 ( R n ) to L ω 3 p 3 ( R n ) . We denote by BM ( p 1 , ω 1 ; p 2 , ω 2 ; p 3 , ω 3 ) (shortly BM ( ω 1 , ω 2 , ω 3 ) ) the vector space of bilinear multipliers of type ( ω 1 , ω 2 , ω 3 ) . In this paper first we discuss some properties of the space BM ( ω 1 , ω 2 , ω 3 ) . Furthermore, we give some examples of bilinear multipliers. At the end of this paper, by using variable exponent Lebesgue spaces L p 1 ( x ) ( R n ) , L p 2 ( x ) ( R n ) and L p 3 ( x ) ( R n ) , we define the space of bilinear multipliers from L p 1 ( x ) ( R n ) × L p 2 ( x ) ( R n ) to L p 3 ( x ) ( R n ) and discuss some properties of this space. MSC:42A45, 42B15, 42B35.