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Transactions of the American Mathematical Society

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$ {\rm BMO}(\rho)$ and Carleson measures


Author: Wayne Stewart Smith
Journal: Trans. Amer. Math. Soc. 287 (1985), 107-126
MSC: Primary 42B30; Secondary 46E15
DOI: https://doi.org/10.1090/S0002-9947-1985-0766209-9
MathSciNet review: 766209
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Abstract: This paper concerns certain generalizations of $ {\text{BMO}}$, the space of functions of bounded mean oscillation. Let $ \rho $ be a positive nondecreasing function on $ (0,\infty )$ with $ \rho (0 + ) = 0$. A locally integrable function on $ {{\mathbf{R}}^m}$ is said to belong to $ {\text{BMO}}(\rho)$ if its mean oscillation over any cube $ Q$ is $ O(\rho (l(Q)))$, where $ l(Q)$ is the edge length of $ Q$.

Carleson measures are known to be closely related to $ {\text{BMO}}$. Generalizations of these measures are shown to be similarly related to the spaces $ {\text{BMO}}(\rho)$. For a cube $ Q$ in $ {{\mathbf{R}}^m},\;\vert Q\vert$ denotes its volume and $ R(Q)$ is the set $ \{ (x,y) \in {\mathbf{R}}_ + ^{m + 1}:x \in Q,\;0 < y < l(Q)\} $. A measure $ \mu $ on $ {\mathbf{R}}_ + ^{m + 1}$ is called a $ \rho $-Carleson measure if $ \vert\mu \vert(R(Q)) = O(\rho (l(Q))\vert Q\vert)$, for all cubes $ Q$.

L. Carleson proved that a compactly supported function in $ {\text{BMO}}$ can be represented as the sum of a bounded function and the balyage, or sweep, of some Carleson measure. A generalization of this theorem involving $ {\text{BMO}}(\rho )$ and $ \rho $-Carleson measures is proved for a broad class of growth functions, and this is used to represent $ {\text{BMO}}(\rho )$ as a dual space. The proof of the theorem is based on a proof of J. Garnett and P. Jones of Carleson's theorem. Another characterization of $ {\text{BMO}}(\rho )$ using $ \rho $-Carleson measures is a corollary. This result generalizes a characterization of $ {\text{BMO}}$ due to C. Fefferman. Finally, an atomic decomposition of the predual of $ {\text{BMO}}(\rho )$ is given.


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DOI: https://doi.org/10.1090/S0002-9947-1985-0766209-9
Article copyright: © Copyright 1985 American Mathematical Society

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