Bilinear multipliers of weighted Lebesgue spaces and variable exponent Lebesgue spaces

Abstract

Let 1 <= p(1), p(2) < infinity, 0 < p(3) <= infinity and omega(1), omega(2), omega(3) be weight functions on R-n. Assume that omega(1), omega(2) are slowly increasing functions. We say that a bounded function m(xi, eta) defined on R-n x R-n is a bilinear multiplier on R-n of type (p(1), omega(1); p(2), omega(2); p(3), omega(3)) (shortly (omega(1), omega(2), omega(3))) if 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 is a bounded bilinear operator from L-omega 1(p1) (R-n) x L-omega 2(p2) (R-n) to L-omega 3(p3) (R-n). We denote by BM(p(1), omega(1); p(2), omega(2); p(3), omega(3)) (shortly BM(omega(1), omega(2), omega(3))) the vector space of bilinear multipliers of type (omega(1), omega(2), omega(3)). In this paper first we discuss some properties of the space BM(omega(1), omega(2), omega(3)). Furthermore, we give some examples of bilinear multipliers. At the end of this paper, by using variable exponent Lebesgue spaces L-p1(x)(R-n), L-p2(x)(R-n) and L-p3((x))(R-n), we define the space of bilinear multipliers from L-p1((x))(R-n) x L-p2((x))(R-n) to L-p3((x))(R-n) and discuss some properties of this space.