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

TL;DR

The paper establishes quasi-Banach $L^r$ bounds for the curved trilinear Hilbert transform $H_{3,\vec{\alpha},\vec{\beta}}$ and boundedness for the curved $n$-linear maximal operator $\mathcal{M}_{n,\vec{\alpha},\vec{\beta}}$ along curves $\vec{\beta} t^{\vec{\alpha}}$ under the non-resonant regime of pairwise distinct $\alpha_j$. The authors deploy the Rank II LGC method with a detailed time-frequency decomposition into diagonal/off-diagonal, high/low oscillatory, and stationary/non-stationary regimes, deriving local smoothing-type estimates and curved square-function controls, then interpolate restricted weak-type bounds to obtain quasi-Banach ranges. The trilinear operator is shown to map $L^{p_1}\times L^{p_2}\times L^{p_3}$ to $L^r$ when $1<p_j<\infty$ and $1/r = \sum 1/p_j$ with $r>1/2$, while the maximal operator is bounded for $1<p_j\le\infty$, $1/2<r\le\infty$ under the same sum constraint, with endpoint extensions for the maximal case. The work extends the curved operator theory into the quasi-Banach regime and lays groundwork for higher $n$ via generalizations, highlighting a flexible LGC framework that couples smoothing inequalities with structured decompositions. The results have implications for harmonic analysis on curved singular integrals and related ergodic/number-theoretic analogues, offering tools for handling non-resonant curvature scenarios in multilinear settings.

Abstract

Let $n\in\mathbb{N}$, $\vecα=(α_1,\ldots,α_n)\in (0,\infty)^n$, $\vecβ=(β_1,\ldots,β_n)\in (\mathbb{R}\setminus\{0\})^n$, $\vec{f}:=(f_1,\ldots, f_n)\in \mathcal{S}^n(\mathbb{R})$ and set $$H_{n,\vecα,\vecβ}(\vec{f})(x):=p.v. \int_{\mathbb{R}} f_1(x+β_1 t^{α_1})\ldots f_n(x+β_n t^{α_n}) \frac{dt}{t}, \quad x \in \mathbb{R}\,,$$ with $\mathcal{M}_{n,\vecα,\vecβ}$ 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 $\{α_j\}_{j=1}^n$ are pairwise distinct we show that $$\|H_{3,\vecα,\vecβ}(\vec{f})\|_{L^r}\lesssim_{\vecα,\vecβ} \prod_{j=1}^3\|f_j\|_{L^{p_j}}\qquad \textrm{and} \qquad \|\mathcal{M}_{n,\vecα,\vecβ}\|_{L^r}\lesssim_{n,\vecα,\vecβ} \prod_{j=1}^n\|f_j\|_{L^{p_j}} \qquad \forall\:n\geq 2\,.$$ (For the maximal operator the boundedness range can be extended to include the endpoint $\infty$ for both $p_j$ and $r$.)

Theorems & Definitions (18)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Corollary 1.5
• Remark 1.7
• Remark 1.8
• Conjecture 1.9
• Remark 1.10
• Remark 2.1