ON THE BILINEAR SQUARE FOURIER MULTIPLIER OPERATORS ASSOCIATED WITH gλ* FUNCTION

This paper will be devoted to study a class of bilinear squarefunction Fourier multiplier operator associated with a symbol m defined by L-lambda,L-m (f(1),f(2))(x) =(integral integral(R+n+1) )t/vertical bar x - z vertical bar + t)n(lambda) x vertical bar integral(2)((Rn)) e(2 pi ix.(xi 1+xi 2))m(t xi 1,t xi 2) (f) over cap (1) (xi(1))(f) over cap (2), (xi(2)) d xi(1) d xi(2)vertical bar(2) dz dt / t(n+1))(1/2). A basic fact about L-lambda,L-m is that it is closely associated with the multilinear Littlewood{Paley g(lambda)* function. In this paper we first investigate the boundedness of L-lambda,L-m on products of weighted Lebesgue spaces. Then, the weighted endpoint L log L type estimate and strong estimate for the commutators of L-lambda,L-m will be demonstrated.