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∈(1,∞) 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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Communicated by Fulvio Ricci.
T. Hytönen was supported by the Academy of Finland (project 114374 “Vector-valued singular integrals”). L. Weis was supported by DFG grant WE 284711-2.
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Hytönen, T.P., Weis, L. The Banach Space-valued BMO, Carleson’s Condition, and Paraproducts. J Fourier Anal Appl 16, 495–513 (2010). https://doi.org/10.1007/s00041-009-9100-2
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DOI: https://doi.org/10.1007/s00041-009-9100-2