L2-boundedness of Hilbert transforms along variable curves

For the L 2 -boundedness of the Hilbert transforms along variable curves H ϕ , γ ( f ) ( x 1 , x 2 ) =  p.v.  ∫ − ∞ + ∞ f ( x 1 − t , x 2 − ϕ ( x 1 ) γ ( t ) ) d t t where γ ∈ C 2 ( R 1 ) , 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 L 2 -boundedness of H ϕ , γ may fail even if ϕ ∈ C ∞ ( R 1 ) . On the other hand, we improve Bennett’s result by relaxing the condition on γ and simplifying the proof.