One-radius results for supermedian functions on R-d, d <= 2

Hansen W, Nikolov N (2010)
Mathematische Annalen 348(3): 565-575.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Hansen, WolfhardUniBi; Nikolov, Nikolai
Abstract / Bemerkung
A classical result states that every lower bounded superharmonic function on R-2 is constant. In this paper the following (stronger) one-circle version is proven. If f : R-2 --> (-infinity, infinity] is lower semicontinuous, lim inf(|x| --> infinity) f (x)/ln |x| >= 0, and, for every x --> R-2, 1/(2 pi) integral(2 pi)(0) f (x + r (x)e(it)) dt <= f (x), where r : R-2 --> (0,infinity) is continuous, sup(x)is an element of(R2) (r (x) - |x|) < infinity, and inf(x is an element of R2) (r (x) - |x|) = -infinity, then f is constant. Moreover, it is shown that, assuming r <= c|.| + M on R-d, d <= 2, and taking averages on {y is an element of R-d : | y - x| <= 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.
Erscheinungsjahr
2010
Zeitschriftentitel
Mathematische Annalen
Band
348
Ausgabe
3
Seite(n)
565-575
ISSN
0025-5831
Page URI
https://pub.uni-bielefeld.de/record/1794100

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Hansen W, Nikolov N. One-radius results for supermedian functions on R-d, d <= 2. Mathematische Annalen. 2010;348(3):565-575.
Hansen, W., & Nikolov, N. (2010). One-radius results for supermedian functions on R-d, d <= 2. Mathematische Annalen, 348(3), 565-575. doi:10.1007/s00208-010-0488-4
Hansen, W., and Nikolov, N. (2010). One-radius results for supermedian functions on R-d, d <= 2. Mathematische Annalen 348, 565-575.
Hansen, W., & Nikolov, N., 2010. One-radius results for supermedian functions on R-d, d <= 2. Mathematische Annalen, 348(3), p 565-575.
W. Hansen and N. Nikolov, “One-radius results for supermedian functions on R-d, d <= 2”, Mathematische Annalen, vol. 348, 2010, pp. 565-575.
Hansen, W., Nikolov, N.: One-radius results for supermedian functions on R-d, d <= 2. Mathematische Annalen. 348, 565-575 (2010).
Hansen, Wolfhard, and Nikolov, Nikolai. “One-radius results for supermedian functions on R-d, d <= 2”. Mathematische Annalen 348.3 (2010): 565-575.

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