On Carleson Outer Measures on Probability Spaces

Abstract The main goal of this work is to outline a framework that makes it possible to understand the real-variable roots of the notion of Carleson measure in the setting of a complete measure space of finite measure. The main difficulties in outlining such a framework are the following ones. Firstly, in this setting no metric is given in the space under consideration, contrary to the classical cases of the unit disc or the Euclidean half-spaces. Secondly, while in these classical settings one has, so to speak, both a “boundary” and an “interior”, in our setting one is only given a measure space which is not the boundary of a space given a priori . Hence in order to set up the desired framework, we have to identify a “space” of which the given measure space may be considered, in some sense, a “boundary”, and we have to do it without relying on a metric structure. In our more general setting, we recapture the root of the link between the properties of Carleson measures on the unit disc and the boundedness properties of the Hardy-Littlewood maximal function, that has been highlighted in 1967 by L. Hörmander (essentially in the context of a space of homogeneous type) and in 1971 by C. Fefferman and E.M. Stein (in the context of Euclidean upper half-spaces).