Pizzetti's formula for H-type groups
Let H be a H-type group, \(\Delta_H\) be the orthogonal sub-Laplacian on H. The paper under reviewing give a Pizzetti's formula on the H-type mean integral operator. An explicit formula for the \(k\)-th powers of \(\Delta_H\) is also given in proving the Pizzetti's formula. The corresponding results on the Heisenberg group, which is a special H-type group, can be found in the previous paper of the author [Potential Anal. 17, No. 2, 165--180 (2002; Zbl 1021.31003)]. And the Pizzetti type fromula in arbitrary Riemann manifolds, which don't contain the H-type groups, is established by \textit{A. Gray} and \textit{T. J. Willmore} [Proc. R. Soc. Edinb., Sect. A 92, 343--364 (1982; Zbl 0495.53040)]. The Pizzetti-type formula in the present paper may advantage the investigations and the computations of Numercial Analysis in the non-Euclidean setting of H-type and Heisenberg groups; and it may be applied in the study of asymptotic solutions of PDE and spherical harmonics in the non-Euclidean context.
scientific article; zbMATH DE number 5625027
scientific article