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

Abstract:

Let $v$ be a potential in the unit ball of ${{\mathbf {R}}^n}$, and ${\mathcal {M}_p}(v;r)$ be its $p$th order mean over the sphere of radius $r$ centred at the origin. It is shown that, as $r \to 1 -$, the function $f(r) = {(1 - r)^{(n - 1)(1 - 1/p)}}{\mathcal {M}_p}(v;r)$ has limit 0 when $1 \leq p{\text { < }}(n - 1)/(n - 2)$, and has lower limit 0 when $n \geq 3$ and $(n - 1)/(n - 2) \leq p{\text { < }}(n - 1)/(n - 3)$. This extends a result of Stoll, who showed that, when $n = 2$ and $p = + \infty ,\lim {\inf _{r \to 1 - }}f(r) = 0$. Examples are given to show that the theorems presented are best possible.