Dyadic Sets, Maximal Functions and Applications on $ax+b$ --Groups

Details Dyadic Sets, Maximal Functions and Applications on $ax+b$ --Groups Dyadic Sets, Maximal Functions and Applications on $ax+b$ --Groups

2010-03-02

arxiv.org/abs/1003.0580v2

Mathematics

Classical Analysis and ODEs

Math. Z. (to appear)

Scientific paper

Let $S$ be the Lie group $\mathrm{R}^n\ltimes \mathrm{R}^+$ 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(2003), 37--61] proved that any integrable function on $(S,\rho)$ admits a Calder\'on--Zygmund decomposition which involves a particular family of sets, called Calder\'on--Zygmund sets. In this paper, we first 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 shall prove a Fefferman--Stein type inequality, involving the dyadic maximal Hardy--Littlewood function and the dyadic sharp dyadic function. As a consequence, we obtain a complex interpolation theorem involving the Hardy space $H^1$ and the $BMO$ space introduced in [Collect. Math. 60(2009), 277--295].

Affiliated with

Also associated with

No associations

Rating

Dyadic Sets, Maximal Functions and Applications on $ax+b$ --Groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Dyadic Sets, Maximal Functions and Applications on $ax+b$ --Groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Dyadic Sets, Maximal Functions and Applications on $ax+b$ --Groups will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-663965