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Growth properties of \(p\)th means of potentials in the unit ball. (English) Zbl 0672.31005

Let \(v\) be a potential on the unit ball of \(R^ n\), and let \(M_ p(v,r)\) be its \(p\)th order mean over the concentric sphere of radius \(r\). Recently, M. Stoll [Proc. Am. Math. Soc. 93, 567-568 (1985; Zbl 0561.31003)] proved that \(\lim_{r\to 1-} \inf (1-r)M_{\infty}(v,r)=0\) when \(n=2\), which generalized an earlier result of M. Heins [Duke Math. J. 14, 179-215 (1947; Zbl 0030.05001)] concerning the logarithm of the modulus of a Blaschke product. Since a potential on the unit ball of \(R^ n\), \(n\geq 3\), can be identically infinite on a radius, there is no obvious analogue for higher dimensions. The present author proves that \(\lim_{r\to 1-} \inf (1-r)M_{(n-1)/(n-2)}(v,r)=0\) when \(n\geq 3\). More generally, he proves that
(i) if \(n\geq 2\) and \(1<p<(n-1)/(n-2),\) then \[ \lim_{r\to 1-}(1- r)^{(n-1)(1-1/p)} M_ p(v,r)=0, \] and (ii) if \(n\geq 3\) and \((n-1)/(n- 2)\leq p<(n-1)/(n-3)\) then \[ \lim_{r\to 1-} \inf (1-r)^{(n-1)(1-1/p)} M_ p(v,r)=0. \] He also gives examples to show that these results are best possible.
Reviewer: N.A.Watson

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
30D50 Blaschke products, etc. (MSC2000)
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