Littlewood's one-circle problem, revisited
Let \(U\) be the open unit disc in the plane, and let \(f\) be a continuous, bounded function on \(U\). Suppose that, for each \(x\in U\), there exists \textit{one} radius \(r(x)\leq 1-|x|\) such that \[ f(x)=\frac{1}{2\pi}\int_{0}^{2\pi}f\big(x+r(x)(\cos t,\sin t)\big)\,dt. \] Littlewood's one-circle problem is to decide whether \(f\) is necessarily harmonic. In an earlier paper [J. Lond. Math. Soc., II. Ser. 50, No. 2, 349--360 (1994; Zbl 0804.31001)], the author and N. Nadirashvili gave an example to show that \(f\) need not be harmonic. In the present paper, a simpler construction is given, which depends on the work of \textit{M. Talagrand} [Bull. Sci. Math., II. Ser. 104, 225--231 (1980; Zbl 0437.28004)].
scientific article; zbMATH DE number 5363193