Dyadic sets, maximal functions and applications on ax plus b-groups

Let S be the Lie group R-n X R by dilations, endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure rho, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245: 37-61, 2003] proved that any integrable function on (S, rho) admits a Caldern-Zygmund decomposition which involves a particular family of sets, called Caldern-Zygmund sets. In this paper, we show the existence of a dyadic grid in the group S, which has nice properties similar to the classical Euclidean dyadic cubes. Using the properties of the dyadic grid, we prove a Fefferman-Stein type inequality, involving the dyadic Hardy-Littlewood maximal function and the dyadic sharp function. As a consequence, we obtain a complex interpolation theorem involving the Hardy space H (1) and the space BMO introduced in [Collect. Math. 60: 277-295, 2009].