A Liouville property for spherical averages in the plane

Hansen W (2001)
Mathematische Annalen 319(3): 539-551.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
It is shown that any continuous bounded function f on R-2 such that f(x) = 1/(2 pi) integral (2 pi)(0) f(x + r(x)e(it))dt, x is an element of R-2, is constant provided r is a strictly positive real function on R-2 satisfying [GRAPHICS] The proof is based on a minimum principle exploiting that lim(\x\-->infinity) ln\x\ = infinity and on a study of (sigma, r)-stable sets, i.e., sets A such that the circle of radius r(x) centered at x is contained in A whenever x is an element of A. The latter reveals that there is no disjoint pair of non-empty closed (sigma, r)-stable subsets in R-2 unless lim sup(\x\-->infinity) r(x)/\x\ greater than or equal to 3 (taking spheres this holds for any R-d, d greater than or equal to 2). A counterexample is given where lim sup(\x\-->infinity) r(x)/\x\ = 4.
Erscheinungsjahr
2001
Zeitschriftentitel
Mathematische Annalen
Band
319
Ausgabe
3
Seite(n)
539-551
ISSN
0025-5831
Page URI
https://pub.uni-bielefeld.de/record/1617667

Zitieren

Hansen W. A Liouville property for spherical averages in the plane. Mathematische Annalen. 2001;319(3):539-551.
Hansen, W. (2001). A Liouville property for spherical averages in the plane. Mathematische Annalen, 319(3), 539-551. https://doi.org/10.1007/PL00004448
Hansen, W. (2001). A Liouville property for spherical averages in the plane. Mathematische Annalen 319, 539-551.
Hansen, W., 2001. A Liouville property for spherical averages in the plane. Mathematische Annalen, 319(3), p 539-551.
W. Hansen, “A Liouville property for spherical averages in the plane”, Mathematische Annalen, vol. 319, 2001, pp. 539-551.
Hansen, W.: A Liouville property for spherical averages in the plane. Mathematische Annalen. 319, 539-551 (2001).
Hansen, Wolfhard. “A Liouville property for spherical averages in the plane”. Mathematische Annalen 319.3 (2001): 539-551.

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