We give one sufficient and two necessary conditions for boundedness between Lebesgue or Lorentz spaces of several classes of bilinear multiplier operators closely connected with the bilinear Hilbert transform.

MSC 2010: 42B10, 44A15In this paper we prove Bochner-Hecke theorems for the Weinstein transform and we give an application to homogeneous distributions.

The aim of this paper is to study singular integrals T generated by holomorphic kernels defined on a natural neighbourhood of the set $z{\zeta }^{-1}:z,\zeta \in ,z\ne \zeta$, where is a star-shaped Lipschitz curve, $=exp\left(iz\right):z=x+iA\left(x\right),A^{\text{'}}\in L^{\infty }[-\pi ,\pi ],A(-\pi )=A\left(\pi \right)$. Under suitable conditions on F and z, the operators are given by (1) $TF\left(z\right)=p.v.{\int }_{(}z{\eta }^{-1})F\left(\eta \right)(d\eta /\eta ).$ We identify a class of kernels of the stated type that give rise to bounded operators on $L^2(,|d\left|\right)$. We establish also transference results relating the boundedness of kernels on closed Lipschitz curves to corresponding results on periodic, unbounded curves.