# On the boundedness of the bilinear Hilbert transform along “non-flat” smooth curves. The Banach triangle case $(L_{r},1≤r<∞)$

### Victor Lie

Purdue University, West Lafayette, USA and Institute of Mathematicsl of the Romanian Academy, Bucharest, Romania

## Abstract

We show that the bilinear Hilbert transform $H_{Γ}$ along curves $Γ=(t,−γ(t))$ with $γ∈NF_{C}$ is bounded from $L_{p}(R)×L_{q}(R)→L_{r}(R)$ where $p,q,r$ are Hölder indices, i.e., $1/p+1/q=1/r$, with $1<p<∞$, $1<q≤∞$ and $1≤r<∞$. Here $NF_{C}$ 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 $H_{Γ}$ to any triple of indices $(1/p,1/q,1/r_{′})$ within the Banach triangle. Our result is optimal up to end-points.

## Cite this article

Victor Lie, On the boundedness of the bilinear Hilbert transform along “non-flat” smooth curves. The Banach triangle case $(L_{r},1≤r<∞)$. Rev. Mat. Iberoam. 34 (2018), no. 1, pp. 331–353

DOI 10.4171/RMI/987