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Rough Bilinear Hypersingular Integrals

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Abstract

We study the rough bilinear hypersingular integral operator

$$ T_{s}(f,g)(x)=\textrm{p.v.}{\int}_{\mathbb{R}^{2n}}\frac{\Omega((y,z)^{\prime})}{|(y,z)|^{2n+s}}f(x-y)g(x-z)dydz, $$

defined on all test functions f,g, where s ≥ 0, Ω is a function in Lq(S2n− 1) satisfying certain cancellation condition. For s ≥ 0, we obtain boundedness for Ts with Ω in \(L^{\infty }(\mathbf {S}^{2n-1})\). The result extends some known results on bilinear singular integrals and linear hypersingular integrals.

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Funding

The second author was supported by the NSF of China Grant 11501169; The third author was supported partly by NSF of Henan Grant 202300410184. Another two authors were not supported by any fundings.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan, 454003, People’s Republic of China

    Yige Cui, Honghai Liu, Zengyan Si & Hanbin Wang

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  1. Yige Cui
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  2. Honghai Liu
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  3. Zengyan Si
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  4. Hanbin Wang
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Correspondence to Honghai Liu.

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The second author was supported by the NSF of China Grant 11501169; The third author was supported partly by NSF of Henan Grant 202300410184.

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Cui, Y., Liu, H., Si, Z. et al. Rough Bilinear Hypersingular Integrals. Potential Anal 59, 1547–1569 (2023). https://doi.org/10.1007/s11118-022-10020-1

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  • DOI: https://doi.org/10.1007/s11118-022-10020-1

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