The boundedness of the bilinear oscillatory integral along a parabola

Abstract In this paper, the $L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})$ boundedness of the bilinear oscillatory integral along parabola \begin{equation*} T_\beta(f, g)(x)=p.v.\int_{{\mathbb R}} f(x-t)g(x-t^{2})\,{\rm e}^{i |t|^{\beta}}\,\frac{{\rm d}t}{t} \end{equation*}is set up, where β > 1 or β < 0, $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ and $\frac{1}{2}\lt r\lt\infty$, p > 1 and q > 1. The result for the case β < 0 extends the $L^\infty\times L^2\to L^2$ boundedness obtained by Fan and Li (D. Fan and X. Li, A bilinear oscillatory integral along parabolas, Positivity 13(2) (2009), 339–366) by confirming an open question raised in it.