Problème de Fatou ponctuel et dérivabilité des mesures. (Pointwise Fatou problem and differentiability of measures)
Let \({\mathcal M}\) be the class of signed Radon measures \(\mu\) on \(R^{\nu}\) such that \(\int_{R^{\nu}}(1+\| x\|)^{-\nu -1}d| \mu | (x)<+\infty.\) For \(\mu\in {\mathcal M}\), let \(P_{\mu}\) be the corresponding half-space Poisson integral on \(D=R^{\nu}\times (0,+\infty)\). The pointwise Fatou problem of the paper's title is to find necessary and sufficient conditions for \(P_{\mu}\) to have a non-tangential limit at the origin. A measure \(\mu\) is said to be differentiable at the origin 0 if there is a real number L such that \(r^{-\nu}\mu (rB)\to L\) as \(r\to 0+\) for every open ball B in \(R^{\nu}\). A harmonic function u on D is said to satisfy condition (\({\mathcal H})\) (resp. (\({\mathcal H}'))\) if the Green potential of the Riesz measure associated with the subharmonic function \(| u|\) is bounded on \(V\cap D\) for some neighbourhood V of 0 (resp. bounded on \(\{\) \(0\}\times (0,1])\). The authors show that if \(\mu\in {\mathcal M}\) and \(\mu\) is differentiable at 0 and if \(P_{\mu}\) satisfies condition (\({\mathcal H}'),then\) \(P_{\mu}\) has a non-tangential limit at (0,0). Corollaries of this theorem include recent results of \textit{W. Ramey} and \textit{D. Ullrich} [Trans. Am. Math. Soc. 305, 207-220 (1988; Zbl 0646.31006)]. A partial converse, which also implies recent results, is as follows: if \(\mu\in {\mathcal M}\) and if \(P_{\mu}\) satisfies condition (\({\mathcal H})\) and has a non-tangential limit at (0,0), then \(\mu\) is differentiable at 0. The paper closes with some indications of generalizations in which integrals of a wide class of kernels are studied.
scientific article; zbMATH DE number 4212422