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., $1/p+1/q=1/r$, 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 below. This continues author's earlier works, extending the boundedness range of $H_{\Gamma }$ to any triple of indices $(1/p,1/q,1/r^{\prime })$ within the Banach triangle. Our result is optimal up to end-points.