Expansion of the heisenberg integral mean via iterated Kohn Laplacians: A Pizzetti-type formula

摘要

A generalization and some applications of the so-called Pizzetti's Formula (which expresses the integral mean of a smooth function over an Euclidean ball as a power series w.r.t. the radius of the ball, having the iterated of the ordinary Laplace operator as coefficients) is given for Delta(H), the Kohn Laplace operator on the Heisenberg group. A formula expressing the n-th power of Delta(H) is also proved. In the case of the ordinary Laplace operator, by Pizzetti's formula, we prove in a simple way that the only nonnegative polyharmonic functions are polynomials. [References: 13]