# Revista Matemática Iberoamericana

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**Volume 34, Issue 1, 2018, pp. 331–353**

**DOI: 10.4171/RMI/987**

Published online: 2018-02-06

On the boundedness of the bilinear Hilbert transform along “non-flat” smooth curves. The Banach triangle case $(L^r, 1 ≤ r < \infty)$

Victor Lie^{[1]}(1) Purdue University, West Lafayette, USA and Institute of Mathematicsl of the Romanian Academy, Bucharest, Romania

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

*Keywords: *Bilinear Hilbert transform, shifted square function

Lie Victor: On the boundedness of the bilinear Hilbert transform along “non-flat” smooth curves. The Banach triangle case $(L^r, 1 ≤ r < \infty)$. *Rev. Mat. Iberoamericana* 34 (2018), 331-353. doi: 10.4171/RMI/987