$$L^2$$ boundedness of Hilbert transforms along variable flat curves

In this paper, the $$L^2$$ boundedness of Hilbert transform along variable flat curve $$(t,P(x_1)\gamma (t))$$ $$\begin{aligned} H_{P,\gamma }f(x_1,x_2):=\mathrm {p.\,v.}\int _{-\infty }^{\infty }f(x_1-t,x_2-P(x_1)\gamma (t))\,\frac{\text {d}t}{t},\quad \forall \, (x_1,x_2)\in {\mathbb {R}}^2, \end{aligned}$$ is studied, where P a real polynomial on $${\mathbb {R}}$$ . A new sufficient condition $$\gamma $$ introduced.