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

Abstract

Let $n\in \mathbb{N}$, $\vec{\alpha }=({\alpha }_1,\dots ,{\alpha }_n)\in (0,\infty {)}^n$, $\vec{\beta }=({\beta }_1,\dots ,{\beta }_n)\in (\mathbb{R}\setminus \{0\}{)}^n$, $\vec{f}:=(f_1,\dots ,f_n)\in {\mathcal{S}}^n(\mathbb{R})$ and set $$H_{n,\vec{\alpha },\vec{\beta }}(\vec{f})(x):=p.v.{\int }_{\mathbb{R}}{f}_1(x+{\beta }_1{t}^{{\alpha }_1})\dots f_n(x+{\beta }_n{t}^{{\alpha }_n})\frac{dt}{t},x\in \mathbb{R},$$ 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 $$\Vert H_{3,\vec{\alpha },\vec{\beta }}(\vec{f}){\Vert }_{L^r}{\lesssim }_{\vec{\alpha },\vec{\beta }}\prod _{j=1}^3\Vert f_j{\Vert }_{L^{p_j}}\text{and}\Vert {\mathcal{M}}_{n,\vec{\alpha },\vec{\beta }}{\Vert }_{L^r}{\lesssim }_{n,\vec{\alpha },\vec{\beta }}\prod _{j=1}^n\Vert f_j{\Vert }_{L^{p_j}}\forall n\ge 2.$$ (For the maximal operator the boundedness range can be extended to include the endpoint $\infty$ for both $p_j$ and $r$.)

30 pages, no figures