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Register a new userAn embedding relation for bounded mean oscillation on rectangles
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Authors
Beno^it F. Sehba
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In the two-parameter setting, we say a function belongs to the mean little $BMO$, if its mean over any interval and with respect to any of the two variables has uniformly bounded mean oscillation. This space has been recently introduced by S. Pott and the author in relation with the multiplier algebra of the product $BMO$ of Chang-Fefferman. We prove that the Cotlar-Sadosky space of functions of bounded mean oscillation $bmo(mathbb{T}^N)$ is a strict subspace of the mean little $BMO$.
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We establish a connection between the function space BMO and the theory of quasisymmetric mappings on emph{spaces of homogeneous type} $widetilde{X} :=(X,rho,mu)$. The connection is that the logarithm of the generalised Jacobian of an $eta$-quasisymmetric mapping $f: widetilde{X} rightarrow widetilde{X}$ is always in $rm{BMO}(widetilde{X})$. In the course of proving this result, we first show that on $widetilde{X}$, the logarithm of a reverse-H{o}lder weight $w$ is in $rm{BMO}(widetilde{X})$, and that the above-mentioned connection holds on metric measure spaces $widehat{X} :=(X,d,mu)$. Furthermore, we construct a large class of spaces $(X,rho,mu)$ to which our results apply. Among the key ingredients of the proofs are suitable generalisations to $(X,rho,mu)$ from the Euclidean or metric measure space settings of the Calder{o}n--Zygmund decomposition, the Vitali Covering Theorem, the Radon--Nikodym Theorem, a lemma which controls the distortion of sets under an $eta$-quasisymmetric mapping, and a result of Heinonen and Koskela which shows that the volume derivative of an $eta$-quasisymmetric mapping is a reverse-H{o}lder weight.
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