GEORGIAN MATHEMATICAL JOURNAL, vol.23, no.3, pp.351-362, 2016 (SCI-Expanded)
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