Almost Everywhere Convergence of Inverse Spherical Transforms on Noncompact Symmetric Spaces

LetGbe a connected noncompact semisimple Lie group with finite center and real rank one. Fix a maximal subgroupK. We considerKbi-invariant functionsfonGand their spherical transformf(λ)=∫G f(g) ϕλ(g) dg,whereϕλdenote the elementary spherical functions onG/Kandλ⩾0. We consider the maximal operatorsS*f(t)=SupR>1 ∫R1 f(λ) λ(a(t)) |c(λ)|−2 dλand prove thatS* maps boundedlyKLKs(G)→Ls(G)+L2(G) for 2n/(n+1)<s⩽2 wheren=dim(G/K). The result is sharp and it implies a.e. convergence properties of the inverse spherical transforms.