Estimates for the commutator of bilinear Fourier multiplier

Let b(1), b(2) is an element of BMO(R-n ) and T (sigma) be a bilinear Fourier multiplier operator with associated multiplier sigma satisfying the Sobolev regularity that sup(k is an element of Z) parallel to sigma(k)parallel to(Ws1), (s2)(R-2n) < infinity for some s(1), s(2) is an element of (n/2, n]. In this paper, the behavior on L-p1(R-n) x L-p2(R-n) (p(1), p(2) is an element of (1, infinity)), on H-1(R-n ) x L-p2(R-n) (p(2) is an element of [2, infinity)), and on H-1(R (n)) x H-1(R-n ), is considered for the commutator T-sigma,T-<(b)over right arrow> defined by T-sigma,T-(b) over right arrow(f(1), f(2))(x) = b(1)(x)T-sigma(f(1), f(2))(x) - T-sigma(b(1) f(1), f(2))(x) +b(2)(x)T-sigma(f(1), f(2))(x) - T-sigma(f(1), b(2)f(2))(x). By kernel estimates of the bilinear Fourier multiplier operators and employing some techniques in the theory of bilinear singular integral operators, it is proved that these mapping properties are very similar to those of the bilinear Fourier multiplier operator which were established by Miyachi and Tomita.