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

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