Bilinear multipliers of weighted Wiener amalgam spaces and variable exponent Wiener amalgam spaces

Abstract

Let omega(1), omega(2) be slowly increasing weight functions, and let omega(3) be any weight function on R-n. Assume that m(xi, eta) is a bounded, measurable function on R-n x R-n. We define B-m(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 C-c(infinity)(R-n). We say that m(xi, eta) is a bilinear multiplier on R-n of type (W(p(1), q(1), omega(1); p(2), q(2), omega(2); p(3), q(3), omega(3))) if B-m is a bounded operator from W(L-p1, L-omega 1(q1)) x W(L-p2, L-omega 2(q2)) to W(L-p3, L-omega 3(q3)), where 1 <= p(1) <= q(1) < infinity, 1 <= p(2) <= q(2) < infinity, 1 < p3, q(3) <= infinity. We denote by BM(W(p(1), q(1), omega(1); p(2), q(2), omega(2); p(3), q(3), omega(3))) the vector space of bilinear multipliers of type (W(p(1), q(1), omega(1); p(2), q(2), omega(2); p(3), q(3), omega(3))). In the first section of this work, we investigate some properties of this space and we give some examples of these bilinear multipliers. In the second section, by using variable exponent Wiener amalgam spaces, we define the bilinear multipliers of type (W(p(1)(x), q(1), omega(1); p(2)(x), q(2), omega(2); p(3)(x), q(3), omega(3))) from W(L-p1(x), L-omega 1(q1)) x W(L-p2(x), L-omega 2(q2)) to W(L-p3(x), L-omega 3(q3)), where p*(1), p*(2), p*(3) < infinity, p(1)(x) <= q(1), p(2)(x) <= q(2), 1 <= q(3) <= infinity for all p(1)(x), p(2)(x), p(3)(x) is an element of P(R-n). We denote by BM(W(p(1)(x), q(1), omega(1); p(2)(x), q(2), omega(2); p(3)(x), q(3), omega(3))) the vector space of bilinear multipliers of type (W(p(1)(x), q(1), omega(1); p(2)(x), q(2), omega(2); p(3)(x), q(3), omega(3))). Similarly, we discuss some properties of this space.