The Boundedness of the Bilinear Fractional Integrals along Curves

Abstract

In this paper, for general curves $(t,\gamma (t))$ satisfying some suitable curvature conditions, we obtain some $L^p(\mathbb{R})\times L^q(\mathbb{R})\to L^r(\mathbb{R})$ estimates for the bilinear fractional integrals $H_{\alpha ,\gamma }$ along the curves $(t,\gamma (t))$, where $$H_{\alpha ,\gamma }(f,g)(x):={\int }_0^{\infty }f(x-t)g(x-\gamma (t))\frac{\text{d}t}{t^{1-\alpha }}$$ and $\alpha \in (0,1)$. At the same time, we also establish an almost sharp Hardy-Littlewood-Sobolev inequality, i.e., the $L^p(\mathbb{R})\to L^q(\mathbb{R})$ estimate, for the fractional integral operators $I_{\alpha ,\gamma }$ along the curves $(t,\gamma (t))$, where $$I_{\alpha ,\gamma }f(x):={\int }_0^{\infty }\left|f(x-\gamma (t))\right|\frac{\text{d}t}{t^{1-\alpha }}.$$

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• The Boundedness of the Bilinear Fractional Integrals along Curves

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• The Boundedness of the Bilinear Fractional Integrals along Curves