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

TL;DR

This paper introduces a new scale of Carleson norms characterizing BMO functions, providing a unified, interpolation-free proof of the boundedness of paraproducts with BMO symbols, extending to Banach space-valued functions with the UMD property.

Contribution

It develops a novel scale of Carleson norms for BMO, offers a new proof for paraproduct boundedness, and extends results to Banach space-valued functions without interpolation.

Findings

Introduces L^q Carleson norms characterizing BMO.

Provides a new, interpolation-free proof of paraproduct boundedness.

Extends results to Banach space-valued functions with UMD property.

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 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.