Abstract
A classical result states that every lower bounded superharmonic function on \({\mathbb{R}^{2}}\) is constant. In this paper the following (stronger) one-circle version is proven. If \({f : \mathbb{R}^{2} \to (-\infty,\infty]}\) is lower semicontinuous, lim inf|x|→∞ f (x)/ ln |x| ≥ 0, and, for every \({x \in \mathbb{R}^{2}}\) , \({1/(2\pi) \int_0^{2\pi} f(x + r(x)e^{it}) \, dt \le f(x)}\) , where \({r : \mathbb{R}^{2} \to (0,\infty)}\) is continuous, \({{\rm sup}_{x \in \mathbb{R}^{2}} (r(x) - |x|) < \infty},\) , and \({{\rm inf}_{x \in \mathbb{R}^{2}} (r(x)-|x|)=-\infty}\) , then f is constant. Moreover, it is shown that, assuming r ≤ c| · | + M on \({\mathbb{R}^d}\) , d ≤ 2, and taking averages on \({\{y \in \mathbb{R}^{d} : |y-x| \le r(x)\}}\) , such a result of Liouville type holds for supermedian functions if and only if c ≤ c 0, where c 0 = 1, if d = 2, whereas 2.50 < c 0 < 2.51, if d = 1.
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The research of N. Nikolov was partially supported by a CNRS-grant at the Paul Sabatier University, Toulouse. He is indebted to P.J. Thomas for stimulating discussions on the subject.
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Hansen, W., Nikolov, N. One-radius results for supermedian functions on \({\mathbb R^d}\) , d ≤ 2. Math. Ann. 348, 565–575 (2010). https://doi.org/10.1007/s00208-010-0488-4
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DOI: https://doi.org/10.1007/s00208-010-0488-4