BMO is the intersection of two translates of dyadic BMO

Let T be the unite circle on $R^2$. Denote by BMO(T) the classical BMO space and denote by BMO_D(T) the usual dyadic BMO space on T. We prove that, BMO(T) is the intersction of BMO_D(T) and a translate of BMO_D(T).Comment: 4 page

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

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