Hostname: page-component-7dd5485656-2pp2p Total loading time: 0 Render date: 2025-10-29T05:43:19.681Z Has data issue: false hasContentIssue false
  • English
  • Français

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 , Dalian, 116024, People’s Republic of China e-mail: junfengli@dlut.edu.cn
Haixia Yu*
Affiliation:
Department of Mathematics, Shantou University , Shantou, 515821, People’s Republic of China e-mail: 22mqzhao@stu.edu.cn
Minqun Zhao
Affiliation:
Department of Mathematics, Shantou University , Shantou, 515821, People’s Republic of China e-mail: 22mqzhao@stu.edu.cn
*
e-mail: hxyu@stu.edu.cn
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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).

References

Bennett, J. M., Hilbert transforms and maximal functions along variable flat curves . Trans. Amer. Math. Soc. 354(2002), no. 12, 4871–4892.CrossRefGoogle Scholar
Carbery, A., Christ, M., Vance, J., Wainger, S., and Watson, D., Operators associated to flat plane curves: ${L}^p$ estimates via dilation methods. Duke Math. J. 59(1989), no. 3, 675–700.10.1215/S0012-7094-89-05930-9CrossRefGoogle Scholar
Carbery, A., Vance, J., Wainger, S., and Watson, D., The Hilbert transform and maximal function along flat curves, dilations, and differential equations . Amer. J. Math. 116(1994), no. 5, 1203–1239.CrossRefGoogle Scholar
Carbery, A., Wainger, S., and Wright, J., Hilbert transforms and maximal functions along variable flat plane curves . J. Fourier Anal. Appl. Special Issue (1995), 119–139.Google Scholar
Carlsson, H., Christ, M., Córdoba, A., Duoandikoetxea, J., de Francia, J. L. Rubio, Vance, J., Wainger, S., and Weinberg, D., ${L}^p$ estimates for maximal functions and Hilbert transforms along flat convex curves in ${\mathbb{R}}^2$ . Bull. Amer. Math. Soc. (N.S.) 14(1986), no. 2, 263–267.CrossRefGoogle Scholar
Chen, X. and Guo, J., A polynomial Roth theorem for corners in ${\mathbb{R}}^2$ and a related bilinear singular integral operator. Math. Ann. 390(2024), no. 1, 255–301.10.1007/s00208-023-02763-yCrossRefGoogle Scholar
Christ, M., Durcik, P., and Roos, J., Trilinear smoothing inequalities and a variant of the triangular Hilbert transform . Adv. Math. 390(2021), Paper No. 107863. 60 pp.10.1016/j.aim.2021.107863CrossRefGoogle Scholar
Córdoba, A., Nagel, A., Vance, J., Wainger, S., and Weinberg, D., ${L}^p$ bounds for Hilbert transforms along convex curves . Invent. Math. 83(1986), no. 1, 59–71.CrossRefGoogle Scholar
Di Plinio, F., Guo, S., Thiele, C., and Zorin-Kranich, P., Square functions for bi-Lipschitz maps and directional operators . J. Funct. Anal. 275(2018), no. 8, 2015–2058.10.1016/j.jfa.2018.07.005CrossRefGoogle Scholar
Grafakos, L., Classical Fourier analysis. 3rd ed., Graduate Texts in Mathematics, 249, Springer, New York, 2014.10.1007/978-1-4939-1194-3CrossRefGoogle Scholar
Grafakos, L., Modern Fourier analysis. 3rd ed., Graduate Texts in Mathematics, 250, Springer, New York, 2014.CrossRefGoogle Scholar
Grafakos, L., On multilinear fractional integrals . Studia Math. 102(1992), no. 1, 49–56.10.4064/sm-102-1-49-56CrossRefGoogle Scholar
Grafakos, L., Strong type endpoint bounds for analytic families of fractional integrals . Proc. Amer. Math. Soc. 117(1993), no. 3, 653–663.10.1090/S0002-9939-1993-1100652-3CrossRefGoogle Scholar
Grafakos, L. and Kalton, N., Some remarks on multilinear maps and interpolation . Math. Ann. 319(2001), no. 1, 151–180.10.1007/PL00004426CrossRefGoogle Scholar
Grafakos, L. and Li, X., Uniform bounds for the bilinear Hilbert transforms, I . Ann. Math. 159(2004), no. 3, 889–933.CrossRefGoogle Scholar
Gressman, P., Convolution and fractional integration with measures on homogeneous curves in ${\mathbb{R}}^n$ . Math. Res. Lett. 11(2004), nos. 5–6, 869–881.10.4310/MRL.2004.v11.n6.a11CrossRefGoogle Scholar
Guo, J. and Xiao, L., Bilinear Hilbert transforms associated with plane curves . J. Geom. Anal. 26(2016), no. 2, 967–995.10.1007/s12220-015-9580-zCrossRefGoogle Scholar
Guo, S., Hickman, J., Lie, V., and Roos, J., Maximal operators and Hilbert transforms along variable non-flat homogeneous curves . Proc. London Math. Soc. 115(2017), no. 1, 177–219.10.1112/plms.12037CrossRefGoogle Scholar
Iosevich, A., Maximal operators associated to families of flat curves in the plane . Duke Math. J. 76(1994), no. 2, 633–644.10.1215/S0012-7094-94-07622-9CrossRefGoogle Scholar
Jones, B. F., A class of singular integrals . Amer. J. Math. 86(1964), 441–462.10.2307/2373175CrossRefGoogle Scholar
Kenig, C. E. and Stein, E. M., Multilinear estimates and fractional integration . Math. Res. Lett. 6(1999), no. 1, 1–15. Erratum to: Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999), no. 3–4, 467.CrossRefGoogle Scholar
Lacey, M. T. and Thiele, C., ${L}^p$ estimates on the bilinear Hilbert transform for $2<p<\infty$ . Ann. Math. 146(1997), no. 3, 693–724.CrossRefGoogle Scholar
Lacey, M. T. and Thiele, C., On Calderón’s conjecture . Ann. Math. 149(1999), no. 2, 475–496.10.2307/120971CrossRefGoogle Scholar
Li, G. and Li, J., The boundedness of a bilinear oscillatory integral . In: M. Milman, J. Xiao and B. Zegarlinski (eds.), Advanced analytic geometry, vol. 6, De Gruyter, Berlin, 2022, pp. 123–133.Google Scholar
Li, G. and Li, J., The boundedness of the bilinear oscillatory integral along a parabola . Proc. Edinb. Math. Soc. 66(2023), no. 1, 54–88.10.1017/S0013091523000032CrossRefGoogle Scholar
Li, J. and Liu, P., Bilinear fractional integral along homogeneous curves . J. Fourier Anal. Appl. 23(2017), no. 6, 1465–1479.10.1007/s00041-016-9511-9CrossRefGoogle Scholar
Li, J. and Yu, H., Bilinear Hilbert transforms and (sub) bilinear maximal functions along convex curves . Pacific J. Math. 310(2021), no. 2, 375–446.10.2140/pjm.2021.310.375CrossRefGoogle Scholar
Li, J. and Yu, H., ${L}^p$ boundedness of Carleson and Hilbert transforms along plane curves with certain curvature constraints . J. Fourier Anal. Appl. 28(2022), no. 1, Paper No. 11. 33 pp.10.1007/s00041-021-09902-6CrossRefGoogle Scholar
Li, J. and Yu, H., ${L}^2$ boundedness of Hilbert transforms along variable flat curves . Math. Z. 298(2021), nos. 3–4, 1573–1591.CrossRefGoogle Scholar
Li, X., Bilinear Hilbert transforms along curves I: The monomial case . Anal. PDE 6(2013), no. 1, 197–220.CrossRefGoogle Scholar
Li, X., Uniform bounds for the bilinear Hilbert transforms. II . Rev. Mat. Iberoam. 22(2006), no. 3, 1069–1126.CrossRefGoogle Scholar
Li, X. and Xiao, L., Uniform estimates for bilinear Hilbert transforms and bilinear maximal functions associated to polynomials . Amer. J. Math. 138(2016), no. 4, 907–962.CrossRefGoogle Scholar
Lie, V., On the boundedness of the bilinear Hilbert transform along “non-flat” smooth curves . Amer. J. Math. 137(2015), no. 2, 313–363.CrossRefGoogle Scholar
Lie, V., On the boundedness of the bilinear Hilbert transform along “non-flat” smooth curves. The Banach triangle case ( ${L}^r$ , $1\le r<\infty$ ) . Rev. Mat. Iberoam. 34(2018), no. 1, 331–353.CrossRefGoogle Scholar
Liu, N., Song, L., and Yu, H., ${{L}}^{\mathrm{p}}$ bounds of maximal operators along variable planar curves in the Lipschitz regularity. J. Funct. Anal. 280(2021), no. 5, Paper No. 108888, 40 pp.CrossRefGoogle Scholar
Liu, N. and Yu, H., Hilbert transforms along variable planar curves: Lipschitz regularity . J. Funct. Anal. 282(2022), no. 4, Paper No. 109340, 36 pp.10.1016/j.jfa.2021.109340CrossRefGoogle Scholar
Nagel, A., Vance, J., Wainger, S., and Weinberg, D., Hilbert transforms for convex curves . Duke Math. J. 50(1983), no. 3, 735–744.CrossRefGoogle Scholar
Pierce, L. B., A note on discrete fractional integral operators on the Heisenberg group . Int. Math. Res. Not. IMRN (2012), no. 1, 17–33.10.1093/imrn/rnr008CrossRefGoogle Scholar
Ricci, F. and Stein, E. M., Harmonic analysis on nilpotent groups and singular integrals. III. Fractional integration along manifolds . J. Funct. Anal. 86(1989), no. 2, 360–389.CrossRefGoogle Scholar
Secco, S., Fractional integration along homogeneous curves in ${\mathbb{R}}^3$ . Math. Scand. 85(1999), no. 2, 259–270.CrossRefGoogle Scholar
Seeger, A. and Wainger, S., Bounds for singular fractional integrals and related Fourier integral operators . J. Funct. Anal. 199(2003), no. 1, 48–91.10.1016/S0022-1236(02)00114-3CrossRefGoogle Scholar
Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Monographs in Harmonic Analysis III, Princeton University Press, Princeton, NJ, 1993.Google Scholar
Stein, E. M. and Wainger, S., Discrete analogues in harmonic analysis. II. Fractional integration . J. Anal. Math. 80(2000), 335–355.CrossRefGoogle Scholar
Stein, E. M. and Wainger, S., Problems in harmonic analysis related to curvature . Bull. Amer. Math. Soc. 84(1978), no. 6, 1239–1295.10.1090/S0002-9904-1978-14554-6CrossRefGoogle Scholar