The boundedness of the bilinear fractional integrals along curves

Abstract In this article, for general curves $(t,\gamma (t))$ satisfying some suitable curvature conditions, we obtain some $L^p(\mathbb {R})\times L^q(\mathbb {R}) \rightarrow L^r(\mathbb {R})$ estimates for the bilinear fractional integrals $H_{\alpha ,\gamma }$ along the curves $(t,\gamma (t))$ , where $$ \begin{align*}H_{\alpha,\gamma}(f,g)(x):=\int_{0}^{\infty}f(x-t)g(x-\gamma(t))\,\frac{\textrm{d}t}{t^{1-\alpha}}\end{align*} $$ and $\alpha \in (0,1)$ . At the same time, we also establish an almost sharp Hardy–Littlewood–Sobolev inequality, i.e., the $L^p(\mathbb {R})\rightarrow L^q(\mathbb {R})$ estimate, for the fractional integral operators $I_{\alpha ,\gamma }$ along the curves $(t,\gamma (t))$ , where $$ \begin{align*}I_{\alpha,\gamma}f(x):=\int_{0}^{\infty}\left|f(x-\gamma(t))\right|\,\frac{\textrm{d}t}{t^{1-\alpha}}.\end{align*} $$