$L^p(\mathbb{R}^2)$-boundedness of Hilbert Transforms and Maximal Functions along Plane Curves with Two-variable Coefficients (2007.05356v1)
Abstract: In this paper, for general plane curves $\gamma$ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus $Lp(\mathbb{R}2)$-boundedness of the Hilbert transforms $H\infty_{U,\gamma}$ along the variable plane curves $(t,U(x_1, x_2)\gamma(t))$ and the $Lp(\mathbb{R}2)$-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 $M{\varepsilon_0}_{U,\gamma}$.