Abstract
We give estimates of the Cauchy transform in Lebesgue integral norms in Clifford analysis framework which are the generalizations of Cauchy transform in complex plane, and mainly establish the \((L^{p}, L^{q})\)-boundedness of the Clifford Cauchy transform in Euclidean space \({\mathbb {R}^{n+1}}\) using the Clifford algebra and the Hardy–Littlewood maximal function. Furthermore, we prove Hedberg estimate and Kolmogorov’s inequality related to Clifford Cauchy transform. As applications, some respective results in complex plane are directly obtained. Based on the properties of the Clifford Cauchy transform and the principle of uniform boundedness, we solve existence of solutions to integral equations with Cauchy kernel in quaternionic analysis.
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Acknowledgements
The authors cordially thank two anonymous referees for their valuable comments that led to the improvement of this paper. The research was supported by NSF of Shandong Province (No. ZR2021MA079) and NSF of China (Grant No. 12171221) and AMEP of Linyi University.
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Communicated by Sören Krausshar.
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This work was partially supported by NSF of China (Grant No. 11401287) and AMEP of Linyi University
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Gu, L. Some Estimates for the Cauchy Transform in Higher Dimensions. Adv. Appl. Clifford Algebras 33, 50 (2023). https://doi.org/10.1007/s00006-023-01294-8
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DOI: https://doi.org/10.1007/s00006-023-01294-8