$L^p(\mathbb{R}^2)$-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 $H^\infty_{U,\gamma}$ 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 $M^\infty_{U,\gamma}$, where $p>2$ and $U$ is a measurable function. The range on $p$ is sharp. Furthermore, for $1<p\leq 2$, under the additional conditions that $U$ is Lipschitz and making a $\varepsilon_0$-truncation with $\gamma(2 \varepsilon_0)\leq 1/4\|U\|_{\textrm{Lip}}$, we also obtain similar boundedness for these two operators $H^{\varepsilon_0}_{U,\gamma}$ and $M^{\varepsilon_0}_{U,\gamma}$.