One-radius results for supermedian functions on mathbb Rd{mathbb R^d} , d ≤ 2

摘要

A classical result states that every lower bounded superharmonic function on is constant. In this paper the following (stronger) one-circle version is proven. If is lower semicontinuous, lim inf_|x|→∞ f (x)/ ln |x| ≥ 0, and, for every , , where is continuous, src="/content/RW4175UQ77556523/208_2010_488_Article_IEq7.gif" alt="$${{rm sup}_{x in mathbb{R}^{2}} (r(x) - |x|) , and , then f is constant. Moreover, it is shown that, assuming r ≤ c| · | + M on , d ≤ 2, and taking averages on , 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.