Teodorescu transform for slice monogenic functions and applications

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.