On the Boundedness of The Bilinear Hilbert Transform along "non-flat" smooth curves

TL;DR

This paper establishes $L^2 imes L^2 o L^1$ bounds for the bilinear Hilbert transform along certain smooth, non-flat curves, advancing understanding of its boundedness properties in harmonic analysis.

Contribution

It proves boundedness of the bilinear Hilbert transform along non-flat smooth curves near zero and infinity, a significant extension in the study of multilinear singular integrals.

Findings

Proves $L^2 imes L^2 o L^1$ boundedness for specific curves.

Extends previous results to a broader class of non-flat curves.

Provides new techniques for analyzing multilinear operators along curves.

Abstract

We are proving $L^2(\R)\times L^2(\R)\,\rightarrow\,L^1(\R)$ bounds for the bilinear Hilbert transform $H_{\Gamma}$ along curves $\Gamma=(t,-\gamma(t))$ with $\gamma$ being a smooth "non-flat" curve near zero and infinity.