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 | References | Similar Articles | Additional Information

Abstract: This paper concerns certain generalizations of , the space of functions of bounded mean oscillation. Let be a positive nondecreasing function on with . A locally integrable function on is said to belong to if its mean oscillation over any cube is , where is the edge length of .

Carleson measures are known to be closely related to . Generalizations of these measures are shown to be similarly related to the spaces . For a cube in denotes its volume and is the set . A measure on is called a -Carleson measure if , for all cubes .

L. Carleson proved that a compactly supported function in 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 and -Carleson measures is proved for a broad class of growth functions, and this is used to represent 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 using -Carleson measures is a corollary. This result generalizes a characterization of due to C. Fefferman. Finally, an atomic decomposition of the predual of is given.

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DOI:
https://doi.org/10.1090/S0002-9947-1985-0766209-9

Article copyright:
© Copyright 1985
American Mathematical Society