Abstract. Let S be the group R ⋉ R endowed with the Riemannian symmetric space metric d 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 space H 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. We then prove tha...

We investigate a scale of dyadic operator-valued BMO spaces, corresponding to the different yet equivalent characterizations of dyadic BMO in the scalar case. In the language of operator spaces, we investigate different operator space structures on the scalar dyadic BMO space which arise naturally from the different characterisations of scalar BMO. We also give sharp dimensional growth estimate...

Let T be the unit circle on R. Denote by BMO(T) the classical BMO space and denote by BMOD(T) the usual dyadic BMO space on T. Then, for suitably chosen δ ∈ R, we have ‖φ‖BMO(T) ⋍ ‖φ‖BMOD(T) + ‖φ(· − 2δπ)‖BMOD(T) ,∀φ ∈ BMO(T) To cite this article: C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Let T be the unit circle on R2. Denote by BMO(T) the classical BMO space and denote by BMOD(T) the usual dyadic BMO space on T. Then, for suitably chosen δ ∈R, we have ‖φ‖BMO(T) ‖φ‖BMOD(T) + ‖φ(· − 2δπ)‖BMOD(T), ∀φ ∈ BMO(T). To cite this article: T. Mei, C. R. Acad. Sci. Paris, Ser. I 336 (2003).  2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All ri...

Given a Radon measure μ on R, which may be non doubling, we introduce a space of type BMO with respect to this measure. It is shown that many properties which hold for the classical space BMO(μ) when μ is a doubling measure remain valid for the space of type BMO introduced in this paper, without assuming μ doubling. For instance, Calderón-Zygmund operators which are bounded on L(μ) are also bou...

We show that Zorboska’s criterion for compactness of Toeplitz operators with BMO symbols on the Bergman space of the unit disc holds, by a different proof, for the Segal-Bargmann space of Gaussian square-integrable entire functions on Cn. We establish some basic properties of BMO for p ≥ 1 and complete the characterization of bounded and compact Toeplitz operators with BMO symbols. Via the Barg...

For every cube $$Q \subset \mathbb {R}^n$$ we let $$X_Q$$ be a quasi-Banach function space over Q such that $$\Vert \chi _Q\Vert _{X_Q} \simeq 1$$ , and for $$X= \{X_Q\}$$ define $$\begin{aligned} \Vert f\Vert _{{{\,\textrm{BMO}\,}}_X}&:=\sup _Q \,\Vert f-{\textstyle \frac{1}{|Q|}\int _Qf} _{X_Q},\\ _{{{\,\textrm{BMO}\,}}_X^*}&:=\sup \,\inf _c\, f-c\Vert _{X_Q}. \end{aligned}$$ We study necessa...

In this paper we establish mapping properties of bilinear Coifman-Meyer multipliers acting on the product spaces H1(Rn)×bmo(Rn) and Lp(Rn)×bmo(Rn), with 1<p<∞. As application these results, obtain some related Kato-Ponce-type inequalities involving endpoint space bmo(Rn), also study pointwise a function in bmo(Rn) functions H1(Rn), h1(Rn) Lp(Rn),