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

2008/11/20 by Hytönen, Tuomas, Weis, Lutz
#42B20 #42B25 #42B35 #46E40 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.0811.3333

Abstract

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 L² 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 L¹ 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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