On a Weak L₁ Property of Maximal Operators on Non-CompactSemisimple Lie Groups

We shall give a simple proof of the weak type L₁ inequality for the K-bi-invariant Hardy-Littlewood maximal functions on non-compact real rank one semisimple Lie groups. For higher rank groups we do under an assumption which holds for the most parts. And on SU(n, n + k) we introduce a maximal operator defined by the characteristic function supported on a cube, and show that the operator also satisfies the weak L₁ property.