The convergence of the Pizzetti series in potential theory (Q1206984)

If \(f\) is real-analytic on the unit ball \(B(1)\) of \(\mathbb{R}^ N\) \((N\geq 2)\), then the Pizzetti series of \(f\) is defined by \[ {\mathcal P}(f,r)=\sum^ \infty_{j=0}\{2^ jj!N(N+2)\cdots(N+2j-2)\}^{- 1}\Delta^ jf(O)r^{2j}, \] where \(\Delta^ j\) denotes the iterated Laplacian. This series has a positive radius of convergence. Further, if \(0<r<1\) and \({\mathcal P}(f,r)\) converges, then \({\mathcal P}(f,r)={\mathcal M}(f,r)\), where \({\mathcal M}(f,r)\) denotes the mean of \(f\) over the sphere of centre \(O\) and radius \(r\). This paper gives more precise information on the radius of convergence when \(f\) is a power of a positive harmonic function. It is shown that, if \(h\) is positive and harmonic on \(B(1)\) and \(p\) is a real number, then \({\mathcal P}(h^ p,r)={\mathcal M}(h^ p,r)\) when \(0<r<\tan(\pi/(2N))\). The sharpness of this result is illustrated by several related theorems.