The Spaces of Bilinear Multipliers of Weighted Lorentz Type Modulation Spaces

Abstract

Fix a nonzero window g € S(Rn), a weight function w on R2n and 1 p,q ∞. The weighted Lorentz type modulation space M(p, q, w)(Rn) consists of all tempered distributions f € S'(Rn) such that the short time Fourier transform V<inf>g</inf>f is in the weighted Lorentz space L(p,q, wdμ)(R2n). The norm on M(p, q, w)(Rn) is

f

M(p,q,w) =

V<inf>g</inf>f

pq,w. This space was frstly defned and some of its properties were investigated for the unweighted case by Gürkanli in [9] and generalized to the weighted case by Sandikçi and Gürkanli in [16]. Let 1 p1,p2 ∞, 1 ≤ q3, q3 ≤ ∞, 1 ∞, ω1, ω2 be polynomial weights and ω3 be a weight function on R2n. In the present paper, we defne the bilinear multiplier operator from M(p1, q1, ω1)(Rn) x M(p2, q2, ω2)(Rn) to M(p3, q3, ω3)(Rn) in the following way. Assume that m(ξ, η) is a bounded function on R2n, and defne Bm(f,g)(x)= Rn∫Rn∫ f(ξ)g(η)m(ξ, η)e2πi ξ+η,x'ì dξdη forallf,g € S(Rn). The function m is said to be a bilinear multiplier on Rn of type (p1, q1, ω1,p1, q1, ω2;p3, q3,ω3) if B<inf>m</inf> is the bounded bilinear operator from M(p1, q1, ω1)(Rn) xM(p2, q1, ω2)(Rn) toM(p3, q1, ω3)(Rn). We denote by BM (p1, q1, ω1;p2, q 2, ω2)(Rn) the space of all bilinear multipliers of type (p1, q1,ω1;p2,q2,ωi;p3,q3, ω3), and defne

m

(p1,q1,ω1;p2,q2,ω2;p3,q3,ω3) =

B<inf>m</inf>

. We discuss the necessary and sufcient conditions for B<inf>m</inf> to be bounded. We investigate the properties of this space and we give some examples. © 2016 by De Gruyter.