On a Weak $L^1$ Property of Maximal Operators on Non-Compact Semisimple Lie Groups

Abstract

We shall give a simple proof of the weak type $L^1$ 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 $L^1$ property.