Harmonic Analysis/Functional Analysis
BMO is the intersection of two translates of dyadic BMO
Comptes Rendus. Mathématique, Volume 336 (2003) no. 12, pp. 1003-1006.

Let $𝕋$ be the unit circle on ${ℝ}^{2}$. Denote by BMO$\left(𝕋\right)$ the classical BMO space and denote by BMO${}_{𝒟}\left(𝕋\right)$ the usual dyadic BMO space on $𝕋$. Then, for suitably chosen $\delta \in ℝ,$ we have

 ${\parallel \varphi \parallel }_{\mathrm{BMO}\left(𝕋\right)}\backsimeq {\parallel \varphi \parallel }_{{\mathrm{BMO}}_{𝒟}\left(𝕋\right)}+{\parallel \varphi \left(·-2\delta \pi \right)\parallel }_{{\mathrm{BMO}}_{𝒟}\left(𝕋\right)},\forall \varphi \in \mathrm{BMO}\left(𝕋\right).$

Soit $𝕋$ le cercle unité dans ${ℝ}^{2}.$ On note BMO$\left(𝕋\right)$ l'espace BMO classique et l'on note BMO${}_{𝒟}\left(𝕋\right)$ l'espace BMO dyadique usuel sur $𝕋.$ Pour certaines valeurs de $\delta \in ℝ$, nous montrons que l'espace BMO$\left(𝕋\right)$ coı̈ncide avec l'intersection de BMO${}_{𝒟}\left(𝕋\right)$ et du translaté par δ de BMO${}_{𝒟}\left(𝕋\right)$, en d'autres termes que l'on a

 ${\parallel \varphi \parallel }_{\mathrm{BMO}\left(𝕋\right)}\backsimeq {\parallel \varphi \parallel }_{{\mathrm{BMO}}_{𝒟}\left(𝕋\right)}+{\parallel \varphi \left(·-2\delta \pi \right)\parallel }_{{\mathrm{BMO}}_{𝒟}\left(𝕋\right)},\forall \varphi \in \mathrm{BMO}\left(𝕋\right).$

Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00234-6
Mei, Tao 1

1 Mathematics Department, Texas A&M University, College Station, TX 77843, USA
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Mei, Tao. BMO is the intersection of two translates of dyadic BMO. Comptes Rendus. Mathématique, Volume 336 (2003) no. 12, pp. 1003-1006. doi : 10.1016/S1631-073X(03)00234-6. http://www.numdam.org/articles/10.1016/S1631-073X(03)00234-6/

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[2] Garnett, J.B. Bounded Analytic Functions, Pure Appl. Math., 96, Academic Press, New York, 1981

[3] T. Mei, Operator valued Hardy spaces, Preprint

[4] Petermichl, S. Dyadic shifts and a logarithmic estimate for Hankel operator with matrix symbol, C. R. Acad. Sci. Paris, Ser. I, Volume 330 (2000), pp. 455-460

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