MÖbius Transformations and Multiplicative Representations for Spherical Potentials

F. G. Avkhadiev

Publications de l'Institut Mathématique, _N_S_75 (2004) no. 89, p. 253

Résumé

For the unit spheres $S^n\subset\mathbf R^{n+1}$ and $S^{2n-1}\subset\mathbf R^{2n}=\mathbf C^n$ we prove the following identities for two classical potentials $ \int_{S^n}\frac{f(y)}{|x-y|^{n+\alpha}}d\sigma_y =\frac{1}{|1-|x|^2|^\alpha} \int_{S^n}\frac{f(T_{n,x}(y))}{|x-y|^{n-\alpha}}d\sigma_y, $ $ \int_{S^{2n-1}}\frac{F(\zeta)d\sigma_\zeta}{|1-(z,\zeta)|^{n+\alpha}}= \frac{1}{(1-|z|^2)^\alpha}\int_{S^{2n-1}} \frac{F(\Phi_{n,z}(\zeta))d\sigma_\zeta}{|1-(z,\zeta)|^{n-\alpha}}, $ where $x\in\mathbf R^{n+1}$ ($|x|\ne0$ and $|x|\ne1$), $z\in\mathbf C^n$ ($|z|1$), $T_{n,x}$ and $\Phi_{n,z}$ are explicit involutions of $S^n$ and $S^{2n-1}$ respectively. Some applications of these formulas are also considered.

Détail

DOI : 10.2298/PIM0475253A

Classification : 31B25 30C65
Keywords: spherical potentials