THE LOCAL NON-HOMOGENEOUS Tb THEOREM FOR VECTOR-VALUED FUNCTIONS

We extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg to the setting of singular integrals with operator-valued kernel that act on vector-valued functions. Here, ‘vector-valued’ means ‘taking values in a function lattice with the UMD (unconditional martingale differences) prop...

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Bibliographic Details
Published in:Glasgow mathematical journal 2015-01, Vol.57 (1), p.17-82
Main Authors: Hytonen, Tuomas P, Vahakangas, Antti V
Format: Article
Language:English
Subjects:
Functions (mathematics)
Integrals
Kernels
Lattices
Martingales
Mathematical analysis
Mathematical functions
Theorems
Vector space
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Summary:We extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg to the setting of singular integrals with operator-valued kernel that act on vector-valued functions. Here, ‘vector-valued’ means ‘taking values in a function lattice with the UMD (unconditional martingale differences) property’. A similar extension (but for general UMD spaces rather than UMD lattices) of Nazarov-Treil-Volberg's global non-homogeneous Tb theorem was achieved earlier by the first author, and it has found applications in the work of Mayboroda and Volberg on square-functions and rectifiability. Our local version requires several elaborations of the previous techniques, and raises new questions about the limits of the vector-valued theory.
ISSN:0017-0895
1469-509X
DOI:10.1017/S0017089514000123