$L^p({\mathbb{R}}^2)$-boundedness of Hilbert Transforms and Maximal Functions along Plane Curves with Two-variable Coefficients

Abstract

In this paper, for general plane curves $\gamma$ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus $L^p({\mathbb{R}}^2)$-boundedness of the Hilbert transforms $H_{U,\gamma }^{\infty }$ along the variable plane curves $(t,U(x_1,x_2)\gamma (t))$ and the $L^p({\mathbb{R}}^2)$-boundedness of the corresponding maximal functions $M_{U,\gamma }^{\infty }$, where $p>2$ and $U$ is a measurable function. The range on $p$ is sharp. Furthermore, for $1<p\le 2$, under the additional conditions that $U$ is Lipschitz and making a ${\epsilon }_0$-truncation with $\gamma (2{\epsilon }_0)\le 1/4\Vert U{\Vert }_{\text{Lip}}$, we also obtain similar boundedness for these two operators $H_{U,\gamma }^{{\epsilon }_0}$ and $M_{U,\gamma }^{{\epsilon }_0}$.

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