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

Victor Lie

Published 2015-12-31Version 1

We show that the bilinear Hilbert transform $H_{\Gamma}$ along curves $\Gamma=(t,-\gamma(t))$ with $\gamma\in\n\f$ is bounded from $L^{p}(\R)\times L^{q}(\R)\,\rightarrow\,L^{r}(\R)$ where $p,\,q,\,r$ are H\"older indices, i.e. $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, with $1<p<\infty$, $1<q\leq\infty$ and $1\leq r<\infty$. Here $\n\f$ 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 $(\frac{1}{p},\,\frac{1}{q},\,\frac{1}{r'})$ within the Banach triangle. Our result is optimal up to end-points.