Integral formulas and Teodorescu transform for generalized partial-slice monogenic functions
Manjie Hu, Chao Ding, Yifei Shen, Jiani Wang
TL;DR
This work develops a unified framework for generalized partial-slice monogenic functions, combining Clifford analysis with slice monogenic theory. It establishes core integral tools, including a Cauchy–Pompeiu formula, a Cauchy integral formula for exterior domains, and a Plemelj–Sokhotski form, all adapted to the generalized partial-slice setting via stem functions and the generalized CR equations. A Teodorescu transform is introduced with a corresponding kernel, and norm estimates demonstrate boundedness and mapping properties on $L^t$ spaces, including preservation of the generalized partial-slice structure. The results pave the way for further analytic developments such as Hodge-type decompositions, generalized projection operators, and Vekua-type systems within this broader Clifford-slice framework. The formalism provides tools for solving partial differential equations (e.g., Beltrami-type equations) and Dirichlet problems in settings that interpolate between classical Clifford analysis and slice-monogenic theory.
Abstract
The theory of generalized partial-slice monogenic functions is considered as a syhthesis of the classical Clifford analysis and the theory of slice monogenic functions. In this paper, we investigate the Cauchy integral formula and the Plemelj formula for generalized partial-slice monogenic functions. Further, we study some properties of the Teodorescu transform in this context. A norm estimation for the Teodorescu transform is discussed as well.