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.

我们在Clifford分析框架中给出了Lebesgue积分范数中Cauchy变换的估计，这是Cauchy变换在复平面上的推广，主要建立了\((L^{p}, L^{q})\)有界使用 Clifford 代数和 Hardy–Littlewood 极大函数在欧几里得空间\({\mathbb {R}^{n+1}}\)中进行 Clifford Cauchy 变换。此外，我们证明了与 Clifford Cauchy 变换相关的 Hedberg 估计和 Kolmogorov 不等式。作为应用，直接得到了复平面上的一些相应结果。基于Clifford Cauchy变换的性质和一致有界原理，我们求解了四元数分析中带有Cauchy核的积分方程解的存在性。