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