Holomorphic Functions on the Lie Ball and Their Monogenic Counterparts

The Cauchy integral formula in Clifford analysis allows us to associate a holomorphic function \({\tilde{f}}:L_n\rightarrow {\mathbb C}\) $\tilde{f}:L_n\to \mathbb{C}$ on the Lie ball \(L_n\) $L_n$ in \({\mathbb C}^n\) ${\mathbb{C}}^n$ with its monogenic counterpart \(f:B_1(0)\rightarrow {\mathbb C}^{n+1}\) $f:B_1(0)\to {\mathbb{C}}^{n+1}$ via the formula \({\tilde{f}}(z) = \int _{S^n}G_{\omega }(z){\varvec{n}}({\omega })f({\omega })\,d\mu ({\omega })\) $\tilde{f}(z)={\int }_{S^n}{G}_{\omega }(z)\mathit{n}(\omega )f(\omega )d\mu (\omega )$ , \(z\in L_n\) $z\in L_n$ . The inverse map \(\tilde{f}\mapsto f\) $\tilde{f}\mapsto f$ is constructed here using the Cauchy-Hua formula for the Lie ball following the work of Morimoto (Analytic Functionals on the Sphere. Translations of Mathematical Monographs, vol. 339. American Mathematical Society, Providence, 1998).