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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.

On the boundedness of the bilinear Hilbert transform along “non-flat” smooth curves