For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula:

$$ f(\zeta) = \frac1{2\pi i}\int_{\partial D}\frac{f(z)}{z-\zeta}dz \ + \ \frac1{2\pi i}\int_D \frac{\partial f}{\partial \overline{z}}(z) \frac1{z-\zeta}dz\wedge d\overline{z} $$

If $f$ is holomorphic then it has certain rigid properties imposed by the Cauchy-Riemann equations, like being infinitely differentiable and having the maximum-modulus principle hold. The result of this rigidity is that $f$ can be determined from its values on the boundary $\partial D$, and so the equation reduces to the first term.

I was recently interested to find that there exists an analogous result in Clifford analysis, where functions from $\mathbb{R}^n$ to $C\ell_n$ are considered. Like holomorphic functions, elements of the kernel of the left and right Dirac operators $\sum_{j=1}^n e_j \frac{\partial}{\partial x_j}$, $\sum_{j=1}^n \frac{\partial}{\partial x_j} e_j$ can be recovered from expressions involving their boundary behavior, notably using the Clifford-Cauchy integral formulas

$$ f(y) = \frac1{\omega_n}\int_S G(x-y) \ n(x) \ f(x) \ d\mu(x) $$

$$ g(y) = \frac1{\omega_n}\int_S g(x) \ n(x) \ G(x-y) \ d\mu(x) $$

where $f,g$ are $C^1$ functions from a subset of $\mathbb{R}^n$ to $C\ell_n$, $S$ is the nice boundary of some domain in $\mathbb{R}^n$ where these things are defined, $n(x)$ is the outward normal to $S$, $\omega_n$ is the area of the unit sphere in $\mathbb{R}^n$, and $G(x) = \frac{x}{\|x\|^n}$. In this example $f$ satisfies $\sum_{j=1}^n e_j \frac{\partial f}{\partial x_j}\equiv 0$ (called left monogenic or left regular), and $g$ satisfies $\sum_{j=1}^n \frac{\partial g}{\partial x_j}e_j \equiv 0$ (right monogenic).

Like complex holomorphic functions, left and right monogenic functions inherit a variety of properties from this expression, like a Taylor series and a Cauchy theorem. It also becomes possible to develop Hardy spaces for Clifford algebras using these ideas.

The impetus for these properties is the fact that such functions (including complex holomorphic functions) can be thought of as lying in the kernel of one of these Dirac operators, and so my question is what is the natural setting to study such operators, Cauchy-type integral formulas, and generalized Hardy spaces? Are Clifford algebras the appropriate way to approach such a topic, or are they simply one kind of structure where such operators are easy to express?

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