Under new 'rough radial structure' assumptions, ball averages on exponentially growing groups satisfy weak-type L(logL)^c maximal inequalities, and non-elementary hyperbolic groups satisfy the optimal weak-type (1,1) inequality.
Anker,La forme exacte de l’estimation fondamentale de Harish-Chandra, C
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Hardy-Littlewood maximal operator on spaces of exponential volume growth
Under new 'rough radial structure' assumptions, ball averages on exponentially growing groups satisfy weak-type L(logL)^c maximal inequalities, and non-elementary hyperbolic groups satisfy the optimal weak-type (1,1) inequality.