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

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