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| For $\phi \in C^1(R^1)$, $\gamma \in C^2(R^1)$, odd or even, $\gamma (0)=\gamma ^{\prime }(0)=0$, convex on $(0,\infty )$, define a Hilbert transform along variable curves by $$H_{\phi ,\gamma }(f)(x_1,x_2)=p.v.\int_{-\infty }^{+\infty }f(x_1-t,x_2-\phi (x_1)\gamma (t))\frac{dt}t. $$In this paper, we shall first give a counter-example to show that under the condition of Nagel-Vance-Wainger-Weinberg on $\gamma $, the $L^2-$boundedness of $H_{\phi ,\gamma }$ may fail even if $\phi \in C^\infty (R^1)$. Then, we relax Bennett |
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| Keywords:Hilbert transform, curves, $L^2$ boundedness |
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