Higher Dimensional Versions of the Douglas-Ahlfors Identities

TL;DR

This paper extends the Douglas-Ahlfors identities from the unit disc in the complex plane to higher dimensions, exploring harmonic, quaternionic, and Clifford monogenic functions and their associated integral identities.

Contribution

It generalizes classical identities to higher-dimensional spaces and different function theories, revealing new relations especially in Clifford algebra settings.

Findings

Identifies higher-dimensional analogues of the Douglas-Ahlfors identities.

Shows equivalence relations hold for harmonic and quaternionic functions in higher dimensions.

Reveals that Clifford algebra cases require different relations for dimensions greater than two.

Abstract

Denote by ${\mathcal D}$ the open unit disc in the complex plane and $\partial {\mathcal D}$ its boundary. Douglas showed through an identical quantity represented by the Fourier coefficients of the concerned function $u$ that \begin{eqnarray}\label{abs} A(u)=\int_{\mathcal D}|\bigtriangledown U|^2dxdy&=&\frac{1}{2\pi}\int\int_{\partial {\mathcal D}\times \partial {\mathcal D}} \left|\frac{u(z_1)-u(z_2)}{z_1-z_2}\right|^2|dz_1||dz_2|,\end{eqnarray} \end{abstract} where $u\in L^2(\partial {\mathcal D}), U$ is the harmonic extension of $u$ into ${\mathcal D}$. Ahlfors gave a fourth equivalence form of $A(u)$ in (\ref{more}) via a different proof. The present article studies relations between the counterpart quantities in higher dimensional spheres with several different but commonly adopted settings, namely, harmonic functions in the Euclidean ${\mathbb R}^n, n\ge 2,$ regular functions in the quaternionic algebra, and Clifford monogenic functions with the real-Clifford algebra ${\mathcal{CL}}_{0, n-1},$ the latter being generated by the multiplication anti-commutative basic imaginary units ${\e}_1, {\e}_2, \cdots , {\e}_{n-1}$ with ${\e}_j^2=-1, j=1, 2, \cdots, n-1.$ It is noted that, while exactly the same equivalence relations hold for harmonic functions in ${\mathbb R}^n$ and regular functions in the quaternionic algebra, for the Clifford algebra setting $n>2,$ the relation (\ref{more}) has to be replaced by essentially a different rule.