A Pizzetti-type formula for the heat operator

We provide an asymptotic expansion of the integral mean of a smooth function over the Heat ball. Namely we generalize to the Heat operator the so-called Pizzetti’s Formula, which expresses the integral mean of a smooth function over an Euclidean ball in terms of a power series with respect to the radius of the ball having the iterated of the ordinary Laplace operator as coefficients. Similarly here, we express the heat integral mean as a power series with respect to the radius of the heat ball, whose coefficients are powers of a distorted heat operator. We also discuss sufficient conditions to have a finite sum. Received: 27 May 2005