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The boundedness of the bilinear fractional integrals along curves
- Part of
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 22 August 2025, pp. 1-31
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- Article
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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 and
$$ \begin{align*}H_{\alpha,\gamma}(f,g)(x):=\int_{0}^{\infty}f(x-t)g(x-\gamma(t))\,\frac{\textrm{d}t}{t^{1-\alpha}}\end{align*} $$
$\alpha \in (0,1)$. 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 
$$ \begin{align*}I_{\alpha,\gamma}f(x):=\int_{0}^{\infty}\left|f(x-\gamma(t))\right|\,\frac{\textrm{d}t}{t^{1-\alpha}}.\end{align*} $$
