Higher Dimensional Versions of the Douglas-Ahlfors Identities

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.