On the Boundedness of The Bilinear Hilbert Transform along "non-flat" smooth curves. The Banach triangle case ($L^r,1\le r<\infty$)

Abstract

We show that the bilinear Hilbert transform $H_{\Gamma }$ along curves $\Gamma =(t,-\gamma (t))$ with $\gamma \in \mathcal{N}{\mathcal{F}}^C$ is bounded from $L^p(\mathbb{R})\times L^q(\mathbb{R})\to L^r(\mathbb{R})$ where $p,q,r$ are Hölder indices, i.e. $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, with $1<p<\infty$, $1<q\le \infty$ and $1\le r<\infty$. Here $\mathcal{N}{\mathcal{F}}^C$ stands for a wide class of smooth "non-flat" curves near zero and infinity whose precise definition is given in Section 2. This continues author's earlier work on this topic, extending the boundedness range of $H_{\Gamma }$ to any triple of indices $(\frac{1}{p},\frac{1}{q},\frac{1}{r^{\prime }})$ within the Banach triangle. Our result is optimal up to end-points.

arXiv comment25 pages, one figure. To appear in Revista Matematica Iberoamericana