Spaces H^1 and BMO on ax+b-groups

TL;DR

This paper develops Hardy and BMO spaces on ax+b-groups, proving their properties, duality, and boundedness of singular integrals, extending harmonic analysis tools to exponential growth Lie groups.

Contribution

It introduces Hardy and BMO spaces on ax+b-groups, establishes their duality, and analyzes boundedness of singular integrals in this non-Euclidean setting.

Findings

BMO functions satisfy John-Nirenberg inequality.

BMO is the dual of H^1 on ax+b-groups.

Singular integrals satisfying Hormander condition are bounded from H^1 to L^1 and from L^{} to BMO.

Abstract

Let S be the semidirect product of R^d and R^+ endowed with the Riemannian symmetric space metric and the right Haar measure: this is a Lie group of exponential growth. In this paper we define an Hardy space H^1 and a BMO space in this context. We prove that the functions in BMO satisfy the John-Nirenberg inequality and that BMO may be identified with the dual space of H^1. We then prove that singular integral operators which satisfy a suitable integral Hormander condition are bounded from H^1 to L^1 and from L^{\infty} to BMO. We also study the real interpolation between H^1, BMO and the L^p spaces.