SpacesH 1 andBMO onax +b-groups

Abstract

LetS be the group ℝ d ⋉ ℝ+ endowed with the Riemannian symmetric space metricd and the right Haar measure ρ The space (S, d, ρ) is a Lie group of exponential growth. In this paper we define an Hardy spaceH 1 and aBMO space in this context. We prove that the functions inBMO satisfy the John-Nirenberg inequality and thatBMO may be identified with the dual space ofH 1. We then prove that singular integral operators whose kernels satisfy a suitable integral Hormander condition are bounded fromH 1 toL 1 and fromL ∞ toBMO. We also study the real interpolation betweenH1,BMO and theL p spaces.