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

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