# Dyadic Sets, Maximal Functions and Applications on $ax b$ -Groups - Mathematics > Classical Analysis and ODEs

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Abstract: Let $S$ be the Lie group $\mathrm{R}^n\ltimes \mathrm{R}^+$ endowed with theleft-invariant Riemannian symmetric space structure and the right Haar measure$ho$, which is a Lie group of exponential growth. Hebisch and Steger inMath. Z. 2452003, 37-61 proved that any integrable function on $S, ho$admits a Calder\-on-Zygmund decomposition which involves a particular familyof sets, called Calder\-on-Zygmund sets. In this paper, we first show theexistence of a dyadic grid in the group $S$, which has {nice} propertiessimilar to the classical Euclidean dyadic cubes. Using the properties of thedyadic grid we shall prove a Fefferman-Stein type inequality, involving thedyadic maximal Hardy-Littlewood function and the dyadic sharp dyadic function.As a consequence, we obtain a complex interpolation theorem involving the Hardyspace $H^1$ and the $BMO$ space introduced in Collect. Math. 602009,277-295.

Autor: Liguang Liu, Maria Vallarino, Dachun Yang

Fuente: https://arxiv.org/