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

Abstract

Let S be the Lie group \({{\mathbb R}^n\ltimes {\mathbb R}}\), where \({{\mathbb R}}\) acts on \({{\mathbb R}^n}\) by dilations, endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure ρ, 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, ρ) admits a Calderón–Zygmund decomposition which involves a particular family of sets, called Calderón–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].