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Commutators in the Two-Weight Setting
Let be the vector of Riesz transforms on , and let
be two weights on , . The
two-weight norm inequality for the commutator is shown to be equivalent to the function
being in a BMO space adapted to and . This is a common extension
of a result of Coifman-Rochberg-Weiss in the case of both and
being Lebesgue measure, and Bloom in the case of dimension one.Comment: v3: suggestions from two referees incorporate
The Dirichlet space: A Survey
In this paper we survey many results on the Dirichlet space of analytic
functions. Our focus is more on the classical Dirichlet space on the disc and
not the potential generalizations to other domains or several variables.
Additionally, we focus mainly on certain function theoretic properties of the
Dirichlet space and omit covering the interesting connections between this
space and operator theory. The results discussed in this survey show what is
known about the Dirichlet space and compares it with the related results for
the Hardy space.Comment: 35 pages, typoes corrected, some open problems adde
Multiparameter Riesz Commutators
It is shown that product BMO of Chang and Fefferman, defined on the product
of Euclidean spaces can be characterized by the multiparameter commutators of
Riesz transforms. This extends a classical one-parameter result of Coifman,
Rochberg, and Weiss, and at the same time extends the work of Lacey and
Ferguson and Lacey and Terwilleger on multiparameter commutators with Hilbert
transforms. The method of proof requires the real-variable methods throughout,
which is new in the multi-parameter context.Comment: 38 Pages. References updated. To appear in American J Mat
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