Teodorescu transform for slice monogenic functions and applications

Abstract: In the past few years, the theory of slice monogenic functions has been developed rapidly mainly motivated by the applications to an elegant functional calculus for non-commuting operators. In this article, we introduce the Teodorescu transform in the theory of slice monogenic functions, which turns out to be the right inverse of a slice Cauchy-Riemann operator. The boundednesses of the Teodorescu transform and its derivatives are investigated as well. A Hodge decomposition of the $\mathcal{L}^p$ space and a generalized Bergman projection are introduced at the end as applications.