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

TL;DR

This paper establishes $L^p$ bounds for the Hilbert transform along a broad class of variable non-flat curves, extending previous results and employing a detailed frequency analysis and a novel maximal function.

Contribution

It generalizes the $L^p$ boundedness of the Hilbert transform to more complex variable curves with distinct exponents, using advanced frequency decomposition techniques.

Findings

Proved $L^p$ bounds for the Hilbert transform along variable non-flat curves.

Introduced a 'short' shift maximal function for pointwise estimates.

Handled multiple frequency cases with tailored strategies.

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.