Fatou's Theorem for Symmetric Spaces: I

Abstract

and Fatou's theorem is the assertion that, as r tends to 1, h tends to f almost everywhere. If the disc and the circle are viewed as homogeneous spaces of the group SL(2, R), the setting for this theorem may be generalized as follows. Let G be any connected non-compact semi-simple Lie group with finite center, and let K be a maximal compact subgroup. G/K is the symmetric space of G, and a complex-valued function on GIK is harmonic if it is annihilated by every G-invariant differential operator on G/K. In [3] Furstenberg exhibited a Poisson integral formula for the bounded harmonic functions on GIK, and he generalized his results to positive harmonic functions as part of [4]. Furstenberg knew that the boundary (analogous to the circle) was a homogeneous space of G (and actually of K), and he wrote the Poisson kernel as Radon-Nikodym derivatives of the action of G on the K-invariant measure on the boundary. Moore [9] identified the boundary explicitly, and a concrete formula for the kernel followed from calculations of Harish-Chandra in [5]. Now a symmetric space admits polar coordinates, in which the radial direction is indexed by a cone in a euclidean space and the other coordinate is indexed by the boundary. A theorem of Fatou type would say that, as the radial coordinate tends to co in some fashion, the Poisson integral of an integrable function on the boundary tends to the function at almost every point of the boundary. Theorems of this sort are known in several special cases. In addition to