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Singular and fractional integrals associated to non doubling measures
Jose Garcia-Cuerva (Universidad Autonoma de Madrid, Spain)
(with A. Eduardo Gatto, DePaul University in Chicago)
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Abstract
In these talks I shall investigate the behaviour of singular
and fractional integral operators associated to a measure on a metric
space satisfying just a mild growth condition, namely that the
measure of each ball is controlled by a fixed power of its radius.
This allows, in particular, non--doubling measures. It turns out
that this condition is enough to build up a theory that contains
the classical results based upon the Lebesgue measure on
Euclidean space and their known extensions for doubling measures.
I start by analyzing the images of the Lebesgue spaces associated
to the measure. The Lipschitz spaces, defined in terms of the
metric, play a basic role too. For a Euclidean space equipped with
one of these measures, I also consider the so-called \lq\lq
regular\rq\rq $\bmo$ space introduced by X. Tolsa. I show that it
contains the image of a Lebesgue space in the appropriate limit
case and also that the image of the space \lq\lq regular\rq\rq
$\bmo$ is contained in the adequate Lipschitz space.
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