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

Abstract

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 $H_{\Gamma}$ - that the (sub)bilinear maximal function along curves $\Gamma=(t,-\gamma(t))$ defined as $$M_{\Gamma}(f,g)(x):=\sup\limits_{\epsilon>0} \frac{1}{2\epsilon} \int_{-\epsilon}^{\epsilon} |f(x-t)\,g(x+\gamma(t))|\,dt$$ 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 $1<p,\,q\leq\infty$ and $1\le r\leq\infty$. This is the maximal boundedness range for $M_{\Gamma}$, that is, our result is sharp.