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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Subspaces of $ {\rm BMO}({\bf R}\sp n)$

Author: Michael Frazier
Journal: Trans. Amer. Math. Soc. 290 (1985), 101-125
MSC: Primary 42B20; Secondary 42B30, 46E99, 47G05
MathSciNet review: 787957
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Abstract: We consider subspaces of $ {\text{BMO}}({{\mathbf{R}}^n})$ generated by one singular integral transform. We show that the averages along $ {x_j}$-lines of the $ j$ th Riesz transform of $ g \in {\text{BMO}} \cap {L^2}({{\mathbf{R}}^n})$ or $ g \in {L^\infty }({{\mathbf{R}}^n})$ satisfy a certain strong regularity property. One consquence of this result is that such functions satisfy a uniform doubling condition on a.e. $ {x_j}$-line. We give an example to show, however, that the restrictions to $ {x_j}$-lines of the Riesz transform of $ g \in {\text{BMO}} \cap {L^2}({{\mathbf{R}}^n})$ do not necessarily have uniformly bounded $ {\text{BMO}}$ norm. Also, for a Calderón-Zygmund singular integral operator $ K$ with real and odd kernel, we show that $ K({\text{BMO}_c}) \subseteq \overline {{L^\infty } + K(L_c^\infty )} $, where $ L_c^\infty $ and $ {\text{BMO}_c}$ are the spaces of $ {L^\infty }$ or $ {\text{BMO}}$ functions of compact support, respectively, and the closure is taken in $ {\text{BMO}}$ norm.

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Keywords: $ {\text{BMO}}$, singular integral operator
Article copyright: © Copyright 1985 American Mathematical Society

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