$L^p$ bound for the Hilbert transform along variable non-flat curves

Abstract

We prove the $L^p$ bound for the Hilbert transform along variable non-flat curves $(t,u(x)[t]^\alpha+v(x)[t]^\beta)$, where $\alpha$ and $\beta$ satisfy $\alpha\neq \beta,\ \alpha\neq 1,\ \beta\neq 1.$ Comparing with the associated theorem in \cite{GHLJ} investigating the case $\alpha=\beta\neq 1$, our result is more general while the proof is more involved. To achieve our goal, we divide the frequency of the objective function into three cases and take different strategies to control these cases. Furthermore, we need to introduce a "short" shift maximal function $\mathfrak{M}^{[n]}$ to establish some pointwise estimate.