A Littlewood-Type Theorem for Harmonic Functions in Euclidean Half-Spaces

Abstract

In 1927 J.E. Littlewood proved that, for bounded harmonic functions on the unit disc, the Fatou-type result, on the existence of almost everywhere boundary values through approach regions that are nontangential, will fail for any system of tangential approach regions that has the following two additional properties: It consists of curves ending at the various boundary points, and it is rotationally invariant. However, in 1984 A. Nagel and E.M. Stein, elaborating results of Rudin (1979) and Nagel, Rudin and Shapiro (1982), proved the existence of rotationally invariant systems of tangential sequences along which a Fatou-type result holds, i.e., along which every bounded harmonic function converges a.e. to its nontangential boundary values. Moreover, they extended their result to higher-dimensional Euclidean half-spaces. The Nagel-Stein result has prompted the question of formulating and proving a Littlewood type theorem that can also be applied to tangential approach regions which are sequential. In this paper we prove a Littlewood-type theorem for bounded harmonic functions in higher-dimensional Euclidean half-spaces, for systems of tangential approach regions which includes the sequential ones. This is the first result of this kind, apart from a recent result of ours whose setting is the unit disc. Indeed, in all the other results of Littlewood type, the tangential approach regions were required to be curvilinear or at least to possess a certain topological property that excluded the possibility that they could be sequential.