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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A remark on Carleson's characterization of BMO


Author: Akihito Uchiyama
Journal: Proc. Amer. Math. Soc. 79 (1980), 35-41
MSC: Primary 42B30; Secondary 46E99
MathSciNet review: 560579
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Abstract: L. Carleson showed that if $ \varphi \in$   BMO$ (R ^n)$ and supp $ \varphi $ is compact, then $ \varphi $ can be written in the form $ \varphi (x) = \Sigma _{k = 1}^\infty \smallint {P_{{t_k}(y)}}(x - y){b_k}(y)dy + {b_0}(x)$ where $ \Sigma _{k = 0}^\infty {\left\Vert {{b_k}} \right\Vert _\infty } \leqslant C{\left\Vert \varphi \right\Vert _{{\text{BMO}}}},{t_k}(y) > 0$ and $ {P_t}(x) = {C_n}t{(\vert x{\vert^2} + {t^2})^{ - (n + 1)/2}}$ is the Poisson kernel. We show that we can take $ {b_2} = {b_3} = \cdots = 0$. This can be generalized on the space of homogeneous type with certain assumptions.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0560579-3
PII: S 0002-9939(1980)0560579-3
Keywords: BMO, Poisson kernel, space of homogeneous type
Article copyright: © Copyright 1980 American Mathematical Society