Spaces H 1 and BMO on ax+b groups

Let S be the group R^dx R+ endowed with the Riemannian symmetric space metric d and the right Haar measure de. The space (S, d, dr) is a Lie group of exponential growth. In this paper we define an Hardy space H1 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 H1. We then prove that singular integral operators whose kernels satisfy a suitable integral H¨ormander condition are bounded from H1 to L1 and from L1 to BMO. We also study the real interpolation between H1, BMO and the Lp spaces.