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Boundedness properties of fractional integral operators associated to non-doubling measures. (English) Zbl 1045.42006
Summary: The main purpose of this paper is to investigate the behavior of 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. We start by analyzing the images of the Lebesgue spaces associated to the measure. The Lipschitz spaces, defined in terms of the metric, also play a basic role. For a Euclidean space equipped with one of these measures, we also consider the so-called regular BMO space introduced by X. Tolsa. We show that it contains the image of a Lebesgue space in the appropriate limit case and also that the image of the regular BMO space is contained in a suitable Lipschitz space.

##### MSC:
 42B20 Singular and oscillatory integrals, several variables 26A33 Fractional derivatives and integrals (real functions) 47B38 Operators on function spaces (general) 47G10 Integral operators