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

Abstract

We prove the Lp$L^p$ bound for the Hilbert transform along variable non‐flat curves false(t,u(x)false[tfalse]α+v(x)false[tfalse]βfalse)$(t,u(x)[t]^\alpha +v(x)[t]^\beta )$, where α and β satisfy α≠β,0.33emα≠1,0.33emβ≠1$\alpha \ne \beta ,\ \alpha \ne 1,\ \beta \ne 1$. Compared with the associated theorem in the work (Guo et al. Proc. Lond. Math. Soc. 2017) investigating the case α=β≠1$\alpha =\beta \ne 1$, 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 boldM[n]$\mathbf {M}^{[n]}$ to establish some pointwise estimates.