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

José García-Cuerva ¹   ; A. Eduardo Gatto ²

¹ Departamento de Matemáticas C-XV, Universidad Autónoma 28049, Madrid, Spain
² Department of Mathematics DePaul University Chicago, IL 60614, U.S.A.

Studia Mathematica, Tome 162 (2004) no. 3, pp. 245-261

Résumé

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 $\mathop {\rm BMO}\nolimits $ 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 $\mathop {\rm BMO}\nolimits $ space is contained in a suitable Lipschitz space.

Détail

DOI : 10.4064/sm162-3-5

Keywords: main purpose paper investigate behavior fractional integral operators associated measure metric space satisfying just mild growth condition namely measure each ball controlled fixed power its radius allows particular non doubling measures turns out condition enough build theory contains classical results based lebesgue measure euclidean space their known extensions doubling measures start analyzing images lebesgue spaces associated measure lipschitz spaces defined terms metric play basic role euclidean space equipped these measures consider so called regular mathop bmo nolimits space introduced tolsa contains image lebesgue space appropriate limit image regular mathop bmo nolimits space contained suitable lipschitz space