Abstract
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 \(\Delta _{\mathbb{H}^N } \), the Kohn Laplace operator on the Heisenberg group. A formula expressing the n-th power of \(\Delta _{\mathbb{H}^N } \) 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.
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Bonfiglioli, A. Expansion of the Heisenberg Integral Mean via Iterated Kohn Laplacians: A Pizzetti-type Formula. Potential Analysis 17, 165–180 (2002). https://doi.org/10.1023/A:1015763601757
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DOI: https://doi.org/10.1023/A:1015763601757