A Liouville property for spherical averages in the plane

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 $\limsup_{|x|\to\infty} r(x)/|x|=4$.