L2-boundedness of Hilbert transforms along variable curves

For the L2-boundedness of the Hilbert transforms along variable curves Hϕ,γ(f)(x1,x2)= p.v. ∫−∞+∞f(x1−t,x2−ϕ(x1)γ(t))dtt where γ∈C2(R1), odd or even, γ(0)=γ′(0)=0, convex on (0,∞), if ϕ≡1, A. Nagel, J. Vance, S. Wainger and D. Weinberg got a necessary and sufficient condition on γ; if ϕ is a polynomial, J.M. Bennett got a sufficient condition on γ. In this paper, we shall first give a counter-example to show that under the condition of Nagel-Vance-Wainger-Weinberg on γ, the L2-boundedness of Hϕ,γ may fail even if ϕ∈C∞(R1). On the other hand, we improve Bennett's result by relaxing the condition on γ and simplifying the proof.