Some Estimates for the Cauchy Transform in Higher Dimensions

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.