Teodorescu transform for slice monogenic functions and applications
Chao Ding, Zhenghua Xu
TL;DR
This work develops a Teodorescu-type transform for slice monogenic functions by showing that the Teodorescu transform $T_{\\Omega_D}$ is the right inverse of the slice Cauchy–Riemann operator $G$ on appropriate spaces. It proves $L^p$ boundedness of $T_{\\Omega_D}$ for $p>\\max\{m,2\\}$ and demonstrates that $G_{\\boldsymbol{q}}T_{\\Omega_I}f$ recovers $f$ on slice functions, establishing a robust operator calculus in this setting. The authors then construct a Hodge decomposition of the Banach space $\\mathcal{L}^p(\\Omega_D)$ into a slice Bergman component $\\mathcal{A}^p(\\Omega_D)$ and a complementary range $|\\underline{\\boldsymbol{x}}|^{1-m}G\\mathcal{L}^p_0(\\Omega_D)$, with projections $\\boldsymbol{P}$ and $\\boldsymbol{Q}$ and a generalized Bergman projection. Boundary-value analysis is developed via a Plemelj formula, connecting interior Teodorescu data to boundary traces and enabling slice monogenic continuation and trace-extension results. Collectively, these results extend Clifford-analysis techniques to the slice-monogenic setting and underpin a functional-calculus framework for noncommuting operators.
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.