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The boundedness of the bilinear fractional integrals along curves

Part of: Harmonic analysis in one variable Harmonic analysis in several variables

Published online by Cambridge University Press:  22 August 2025

Junfeng Li ,
Haixia Yu  and
Minqun Zhao
Junfeng Li
Affiliation:
School of Mathematical Sciences, Dalian University of Technology https://ror.org/023hj5876 , Dalian, 116024, People’s Republic of China e-mail: junfengli@dlut.edu.cn
Haixia Yu*
Affiliation:
Department of Mathematics, Shantou University https://ror.org/01a099706 , Shantou, 515821, People’s Republic of China e-mail: 22mqzhao@stu.edu.cn
Minqun Zhao
Affiliation:
Department of Mathematics, Shantou University https://ror.org/01a099706 , Shantou, 515821, People’s Republic of China e-mail: 22mqzhao@stu.edu.cn
*
e-mail: hxyu@stu.edu.cn
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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
$$ \begin{align*}I_{\alpha,\gamma}f(x):=\int_{0}^{\infty}\left|f(x-\gamma(t))\right|\,\frac{\textrm{d}t}{t^{1-\alpha}}.\end{align*} $$

Keywords

Bilinear fractional integralconvolution type operatorbilinear interpolationrestricted weak type estimate

MSC classification

Primary: 42B15: Multipliers
Secondary: 42A85: Convolution, factorization

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 31
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

Junfeng Li is supported by the National Natural Science Foundation of China (Grant No. 12471090). Haixia Yu is supported by the National Natural Science Foundation of China (Grant Nos. 12201378 and 12471093), the Guangdong Basic and Applied Basic Research Foundation (Grant Nos. 2023A1515010635 and 2024A1515010468).

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