Cauchy–Pompeiu Formula for Multi-meta-weighted-monogenic Functions of first class

Abstract

In this paper we give a Cauchy–Pompeiu type integral formula for a class of functions called multi-meta-weighted-monogenic using a distance calculated via the quadratic form associated with an elliptic operator. This is used for the construction of the kernel over the domain $\mathbb{R}^{m+1}$, constructed by fixing the real part for all products of

$$\mathbb{R}^{m+1}=\mathbb{R}^{m_1}\times \mathbb{R}^{m_2}\times \cdots \times \mathbb{R}^{m_n}$$

Also, we present a section where we discuss the inhomogeneous meta-n-weighted- monogenic equation and a distributional solution for this equation is obtained. In some special cases, the distributional solution becomes a classical solution.