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

We show that the bilinear Hilbert transform $H_{\mathrm{\Gamma }}$ along curves $\mathrm{\Gamma }=(t,-\gamma (t))$ with $\gamma \in \mathcal{N}{\mathcal{F}}^C$ is bounded from $L^{p}($\mathbb{R}$)\times L^{q}($\mathbb{R}$),\rightarrow,L^{r}($\mathbb{R}$)$where$p,,q,,r$ are H"older indices, i.e. $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, with $1<p<\mathrm{\infty }$, $1<q\le \mathrm{\infty }$ and $1\le r<\mathrm{\infty }$. 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 $H_{\mathrm{\Gamma }}$ to any triple of indices $(\frac{1}{p},\frac{1}{q},\frac{1}{r^{\mathrm{\prime }}})$ within the Banach triangle. Our result is optimal up to end-points.