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Product rule and chain rule estimates for the Hajłasz gradient on doubling metric measure spaces

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Gatto A., Segovia C.

Colloquium Mathematicum

2006

tom 105

nr 1

19-24

We use the Calderón Maximal Function to prove the Kato-Ponce Product Rule Estimate and the Christ-Weinstein Chain Rule Estimate for the Hajłasz gradient on doubling measure metric spaces.

Boundedness properties of fractional integral operators associated to non-doubling measures

100%

García-Cuerva J., Gatto A.

Studia Mathematica

2004

tom 162

nr 3

245-261

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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Studia Mathematica

2 Gatto A.

1 García-Cuerva J.

1 Segovia C.

1 2006

1 2004

Wyniki wyszukiwania

Product rule and chain rule estimates for the Hajłasz gradient on doubling metric measure spaces

Boundedness properties of fractional integral operators associated to non-doubling measures