The spaces of bilinear multipliers of weighted Lorentz type modulation spaces

Abstract

Fix a nonzero window g is an element of S(IRn), a weight function w on R-2n and 1 <= p, q <= infinity. The weighted Lorentz type modulation space M(p, q, w)(R-n) consists of all tempered distributions f is an element of S'(R-n) such that the short time Fourier transform V(g)f is in the weighted Lorentz space L(p, q, wd mu)(R-2n). The norm on M(p, q, w)(R-n) is vertical bar vertical bar f vertical bar vertical bar(M(p, q, w)) = vertical bar vertical bar V(g)f vertical bar vertical bar pq, w. This space was firstly defined and some of its properties were investigated for the unweighted case by Gurkanli in [ 9] and generalized to the weighted case by Sandikci and Gurkanli in [16]. Let 1 < p(1), p(2) < infinity, 1 <= q(1), q(2) < infinity, 1 <= p(3), q(3) <= infinity, omega(1), omega(2) be polynomial weights and omega(3) be a weight function on R-2n. In the present paper, we define the bilinear multiplier operator from M(p(1), q(1), omega(1))(R-n) x M(p(2), q(2), omega(2))(R-n) to M(p(3), q(3), omega(3))(R-n) in the following way. Assume that m(xi, eta) is a bounded function on R-2n, and define Bm(f, g)(x) = integral(Rn) integral(Rn) <(f)over cap>(xi)(g) over cap(eta)m(xi, eta)e(2 pi i(xi+eta,x))d xi d eta for all f,g is an element of S(R-n). The function m is said to be a bilinear multiplier on R-n of type (p(1), q(1), omega(1); p(2), q(2), omega(2); p(3), q(3), omega(3)) if B-m is the bounded bilinear operator from M(p(1), q(1), omega(1))(R-n) x M(p(2), q(2), omega(2))(R-n) to M(p(3), q(3), omega(3))(R n). We denote by BM(p(1), q(1), omega(1); p(2), q(2), omega(2))(R-n) the space of all bilinear multipliers of type (p(1), q(1), omega(1); p(2), q(2), omega(2); p(3), q(3), omega(3)), and define vertical bar vertical bar m vertical bar vertical bar((p1, q1, omega 1; p2, q2, omega 2; p3, q3,) omega 3) = vertical bar vertical bar B-m vertical bar vertical bar. We discuss the necessary and sufficient conditions for B-m to be bounded. We investigate the properties of this space and we give some examples.