Abstract.
It is shown that any continuous bounded function f on \({\mathbb R}^2\) such that
\[f(x)=\frac{1}{(2\pi)} \int_0^{2\pi} f(x+r(x)e^{it}) dt\; , \]
\(x\in{\mathbb R}^2\), is constant provided r is a strictly positive real function on \({\mathbb R}^2\) satisfying
\[ \limsup_{|x|\to\infty} (r(x)-|x|)<+\infty\;. \]
The proof is based on a minimum principle exploiting that \(\lim_{|x|\to\infty} \ln |x| = \infty\) 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 \(x\in A\). The latter reveals that there is no disjoint pair of non-empty closed \((\sigma,r)\)-stable subsets in \({mathbb R}^2\) unless \(\limsup_{|x|\to\infty} r(x)/|x|\ge 3\) (taking spheres this holds for any \({\mathbb R}^d\), \(d\ge2\)). A counterexample is given where \(\limsup_{|x|\to\infty} r(x)/|x|=4\).
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Received November 24, 1999 / Published online December 8, 2000
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Hansen, W. A Liouville property for spherical averages in the plane. Math Ann 319, 539–551 (2001). https://doi.org/10.1007/PL00004448
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DOI: https://doi.org/10.1007/PL00004448