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On Veech’s conjecture for harmonic functions. (English) Zbl 0846.31003

Let \(r\) be some positive function on a domain \(U \subset \mathbb{R}^d\) such that the ball \(B(x,r (x))\) of center \(x\) and radius \(r(x)\) be contained in \(U\) for any \(x \in U\). Let \(f\) be a Lebesgue measurable real function defined on \(U\). Roughly speaking Veech’s Conjecture asserts that if \(f\) possesses the mean value property on the balls \(B(x,r (x))\) for any \(x \in U\) then \(f\) is harmonic. This conjecture has been proved for a continuous function \(f\) which is bounded by some harmonic function or in dimension one, assuming that \(f\) is nonnegative and that \(r\) is locally bounded away from zero.
It is shown in this paper that this last result fails in dimension greater than two and for any open ball \(U\) even assuming \(C^\infty\) regularity for \(f\) and \(r\)! The proof is based on the properties of the Markov chain given by the kernel \(P(x,A) = {|B (x,r (x)) \cap A |\over |B (x,r(x)) |}\).
Reviewer: J.Lacroix (Paris)

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
60J45 Probabilistic potential theory
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References:

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