$L^p$ Boundedness of Hilbert Transforms and Maximal Functions along Variable Curves

Junfeng Li, Haixia Yu

Published 2018-08-04Version 1

In this paper, the $L^p$ boundedness of the Hilbert transform along variable curve $(t, P(x_1)\gamma(t))$ $$H_{P,\gamma}f(x_1,x_2):=\mathrm{p.\,v.}\int_{-\infty}^{\infty}f(x_1-t,x_2-P(x_1)\gamma(t))\,\frac{\textrm{d}t}{t}$$ and the corresponding maximal function $$M_{P,\gamma}f(x_1,x_2):=\sup_{\varepsilon>0}\frac{1}{2\varepsilon}\int_{-\varepsilon}^{\varepsilon}|f(x_1-t,x_2-P(x_1)\gamma(t))|\,\textrm{d}t$$ are obtained, where $p\in (1,\infty)$, $P$ is a real polynomial on $\mathbb{R}$. It gives a positive answer to a concern proposed by Bennett in \cite{BJ}.