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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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


Author: S. J. Gardiner
Journal: Proc. Amer. Math. Soc. 103 (1988), 861-869
MSC: Primary 31B25
DOI: https://doi.org/10.1090/S0002-9939-1988-0947671-3
MathSciNet review: 947671
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0947671-3
Article copyright: © Copyright 1988 American Mathematical Society