Growth properties of $p$th means of potentials in the unit ball

S. J. Gardiner
1988 Proceedings of the American Mathematical Society  
Let v be a potential in the unit ball of R", and jrv(v;r) be its pth order mean over the sphere of radius r centred at the origin. It is shown that, as r -* 1-, the function f(r) = (1 -r)(n_1)(1_1/p).- 3 and (n -l)/(n -2) < p < (n -l)/(n -3). This extends a result of Stoll, who showed that, when n = 2 and p = +oo, liminfr-.1-f(r) = 0. Examples are given to show that the theorems presented are best possible.
doi:10.1090/s0002-9939-1988-0947671-3 fatcat:gakgyff3brealkr6drkykoyq2y