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 $$M_{\Gamma }(f,g)(x):=\underset{\epsilon >0}{\sup }\frac{1}{2\epsilon }{\int }_{-\epsilon }^{\epsilon }|f(x-t)g(x+\gamma (t))|dt$$ is bounded from $L^p(\mathbb{R})\times L^q(\mathbb{R})\to L^r(\mathbb{R})$ for all $p,q$ and $r$ Hölder indices, that is $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, with $1<p,q\le \infty$ and $1\le r\le \infty$. This is the maximal boundedness range for $M_{\Gamma }$, that is, our result is sharp.

23 pages, 1 figure. Some minor typos have been removed. This paper was accepted for publication in JFAA