A Liouville property for spherical averages in the plane
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.
319
3
539-551
539-551
Springer