On the boundedness of the curved trilinear Hilbert transform and the curved $n-$linear maximal operator in the quasi-Banach regime

TL;DR

This paper proves boundedness results for a class of curved multilinear operators, including the trilinear Hilbert transform, in the quasi-Banach regime under non-resonant conditions, extending previous understanding of their behavior.

Contribution

It establishes new boundedness results for curved n-linear maximal operators and trilinear Hilbert transforms in the quasi-Banach setting, assuming non-resonance among parameters.

Findings

Boundedness of the 3-linear Hilbert transform in L^r spaces.

Boundedness of the n-linear maximal operator in L^r spaces.

Extension of boundedness to include endpoint cases for the maximal operator.

Abstract

Let $n\in\mathbb{N}$, $\vec{\alpha}=(\alpha_1,\ldots,\alpha_n)\in (0,\infty)^n$, $\vec{\beta}=(\beta_1,\ldots,\beta_n)\in (\mathbb{R}\setminus\{0\})^n$, $\vec{f}:=(f_1,\ldots, f_n)\in \mathcal{S}^n(\mathbb{R})$ and set $$H_{n,\vec{\alpha},\vec{\beta}}(\vec{f})(x):=p.v. \int_{\mathbb{R}} f_1(x+\beta_1 t^{\alpha_1})\ldots f_n(x+\beta_n t^{\alpha_n}) \frac{dt}{t}, \quad x \in \mathbb{R}\,,$$ with $\mathcal{M}_{n,\vec{\alpha},\vec{\beta}}$ being its maximal operator counterpart. Assume that $1<p_j<\infty$ and $\frac{1}{2}<r<\infty$ satisfy $\sum_{j=1}^n \frac{1}{p_j}=\frac{1}{r}$. Under the \emph{non-resonant} assumption that $\{\alpha_j\}_{j=1}^n$ are pairwise distinct we show that $$\|H_{3,\vec{\alpha},\vec{\beta}}(\vec{f})\|_{L^r}\lesssim_{\vec{\alpha},\vec{\beta}} \prod_{j=1}^3\|f_j\|_{L^{p_j}}\qquad \textrm{and} \qquad \|\mathcal{M}_{n,\vec{\alpha},\vec{\beta}}\|_{L^r}\lesssim_{n,\vec{\alpha},\vec{\beta}} \prod_{j=1}^n\|f_j\|_{L^{p_j}} \qquad \forall\:n\geq 2\,.$$ (For the maximal operator the boundedness range can be extended to include the endpoint $\infty$ for both $p_j$ and $r$.)