## On a covering property of rarefied sets at infinity in a cone.(English)Zbl 1119.31003

Aikawa, Hiroaki (ed.) et al., Potential theory in Matsue. Selected papers of the international workshop on potential theory, Matsue, Japan, August 23–28, 2004. Tokyo: Mathematical Society of Japan (ISBN 4-931469-33-7/hbk). Advanced Studies in Pure Mathematics 44, 233-244 (2006).
This paper is concerned with domains of the form $$C(\Omega )=\{rz:r>0,z\in \Omega \}$$, where $$\Omega$$ is a $$C^{2,\alpha }$$-domain in the unit sphere of $$\mathbb{R}^{n}$$. There is a Martin function for $$C(\Omega )$$, associated with the point at infinity, which has the form $$K(rz)=r^{\alpha }f(z)$$ for $$r>0$$ and $$z\in \Omega$$. If $$v$$ is a positive superharmonic function on $$C(\Omega )$$ such that $$\inf_{C(\Omega )}v/K=0$$, then the set $$E_{v}$$, defined as $$\{rz\in C(\Omega ):v(rz)\geq r^{\alpha }\}$$, cannot be large near infinity. The authors show that such a set $$E_{v}$$ can always be covered by a sequence of balls $$B(x_{k},r_{k})$$ such that $$\sum (r_{k}/\left| x_{k}\right| )^{n-1}<\infty$$. The case of a halfspace had previously been treated by V. S. Azarin [Mat. Sb. (N.S.) 66 (108), 248–264 (1965; Zbl 0135.32203)] and M. Essén, H. L. Jackson and P. J. Rippon [Hiroshima Math. J. 15, 393–410 (1985; Zbl 0594.31014)].
For the entire collection see [Zbl 1102.31001].

### MSC:

 31B25 Boundary behavior of harmonic functions in higher dimensions 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions

### Keywords:

superharmonic function; boundary behaviour

### Citations:

Zbl 0135.32203; Zbl 0594.31014