Projection theorems for hitting probabilities and a theorem of Littlewood

Littlewood (Proc. London Math. Soc. (2), 28 1928, 383–394) showed that a positive superharmonic function u on the unit disc has radial limits a.e. Using techniques due to Doob this result is extended to all rank one symmetric spaces. In addition simplifications are obtained of Doob's (Ann. Inst. Fourier (Grenoble), 15 1965, 113–135) proof of normal convergence a.e. of a positive superharmonic function on a half space. The symmetric space analogue of this half space result is also obtained. The methods used are shown to fail for the potential theory on $\text{R}$ⁿ associated with Δu = αu (α > 4 0). It is an open question as to whether Littlewood's theorem holds in this context.