$L^2$ Boundedness of Hilbert Transforms along Variable Flat Curves

Abstract

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