The Banach space -valued BMO, Carleson's condition, and paraproducts

Details The Banach space -valued BMO, Carleson's condition, and paraproducts The Banach space -valued BMO, Carleson's condition, and paraproducts

2008-11-20

arxiv.org/abs/0811.3333v1

Mathematics

Functional Analysis

14 pages, submitted

Scientific paper

We define a scale of L^q Carleson norms, all of which characterize the membership of a function in BMO. The phenomenon is analogous to the John-Nirenberg inequality, but on the level of Carleson measures. The classical Carleson condition corresponds to the L^2 case in our theory. The result is applied to give a new proof for the L^p-boundedness of paraproducts with a BMO symbol. A novel feature of the argument is that all p are covered at once in a completely interpolation-free manner. This is achieved by using the L^1 Carleson norm, and indicates the usefulness of this notion. Our approach is chosen so that all these results extend in a natural way to the case of X-valued functions, where X is a Banach space with the UMD property.

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