Generalized Π-operator in the theory of slice monogenic functions and applications
Ziyi Sun, Chao Ding
TL;DR
This work extends the classical $\Pi$-operator to the theory of slice monogenic functions by defining a generalized operator $\Pi_{\Omega_D}$ as the composition of the slice Cauchy–Riemann operator with the Teodorescu transform. It develops integral representations, adjoints, and left/right inverses, and analyzes their mapping properties on weighted $L^p$ spaces, establishing how $\Pi_{\Omega_D}$ interacts with kernel spaces and providing explicit norm estimates. A key contribution is the application to a slice Beltrami equation, where solvability is tied to the norm of $\Pi_{\Omega_D}$ via a Banach fixed-point framework, yielding a concrete solvability condition in terms of $L^p(d\mu)$ norms and Calderón–Zygmund-type constants. The results unify high-dimensional hypercomplex analysis with Beltrami-type problems and recover the classical two-dimensional Beltrami equation as a special case, offering a robust analytic tool for slice-monogenic problems.
Abstract
The Π-operator plays an important role in complex analysis, especially in the theory of generalized analytic functions in the sense of Vekua. In this paper, we introduce a generalized Π-operator in the theory of slice monogenic functions, and some mapping properties of the generalized Π-operator are also introduced. Further, a left and right inverse and the adjoint operator of the generalized Π-operator are given. As an application, we introduce a slice Beltrami equation, which reduces to the classical complex Beltrami equation when the dimension is 2. We show details that the norm estimate of the generalized Π-operator can determine the existence of solutions of the slice Beltrami equation.