A ug 2 00 9 One - radius results for supermedian functions on R d , d ≤ 2

نویسندگان

  • Wolfhard Hansen
  • Nikolai Nikolov
چکیده

A classical result states that every lower bounded superharmonic function on R2 is constant. In this paper the following (stronger) one-circle version is proven. If f : R2 → (−∞,∞] is lower semicontinuous, lim inf |x|→∞ f(x)/ ln |x| ≥ 0, and, for every x ∈ R2, 1/(2π) ∫ 2π 0 f(x + r(x)e) dt ≤ f(x), where r : R2 → (0,∞) is continuous, supx∈R2(r(x) − |x|) < ∞, and infx∈R2(r(x) − |x|) = −∞, then f is constant. Moreover, it is shown that, with respect to the assumption r ≤ c| · |+M on Rd, there is a striking difference between the restricted volume mean property for the cases d = 1 and d = 2. 2000 Mathematics Subject Classification: 31A05

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تاریخ انتشار 2009