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

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 $1<pleq 2$, 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 $M^{varepsilon_0}_{U,gamma}$.