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

Renhui Wan

Mathematische Nachrichten, 2023, vol. 296, issue 4, 1669-1686

Abstract: We prove the Lp$L^p$ bound for the Hilbert transform along variable non‐flat curves (t,u(x)[t]α+v(x)[t]β)$(t,u(x)[t]^\alpha +v(x)[t]^\beta )$, where α and β satisfy α≠β,α≠1,β≠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 M[n]$\mathbf {M}^{[n]}$ to establish some pointwise estimates.

Date: 2023
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