The Boundedness of the (Sub)Bilinear Maximal Function along "non-flat" smooth curves

Let $\mathcal{N}\mathcal{F}$ be the class of smooth non-flat curves near the origin and near infinity previously introduced by the second author and let $\gamma \in \mathcal{N}\mathcal{F}$. We show - via a unifying approach relative to the correspondent bilinear Hilbert transform $H_{\mathrm{\Gamma }}$ - that the (sub)bilinear maximal function along curves $\mathrm{\Gamma }=(t,-\gamma (t))$ defined as $$M_{\mathrm{\Gamma }}(f,g)(x):=\underset{ϵ>0}{\sup }\frac{1}{2ϵ}{\int }_{-ϵ}^{ϵ}\mathrm{\mid }f(x-t)g(x+\gamma (t))\mathrm{\mid }dt$$ is bounded from $L^p(\mathbb{R})\times L^q(\mathbb{R})\to L^r(\mathbb{R})$ for all $p,q$ and $r$ H"older indices, that is $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, with $1<p,q\le \mathrm{\infty }$ and $1\le r\le \mathrm{\infty }$. This is the maximal boundedness range for $M_{\mathrm{\Gamma }}$, that is, our result is sharp.