The vector-valued non-homogeneous Tb theorem

Abstract

The paper gives a Banach space -valued extension of the Tb theorem of Nazarov, Treil and Volberg (2003) concerning the boundedness of singular integral operators with respect to a measure, which only satisfies an upper control on the size of balls. Under the same assumptions as in their result, such operators are shown to be bounded on the Bochner spaces of functions with values in a Banach space with the unconditionality property of martingale differences (UMD). The new proof deals directly with all Lebesgue exponents p in the range 1<p<infinity, and relies on delicate estimates for the non-homogenous "Haar" functions, as well as McConnell's (1989) decoupling inequality for tangent martingale differences.

Substantial revision with stronger forms of the main theorems. 44 pages

References in corpus (1)

• R-boundedness of smooth operator-valued functions

Cited by in corpus (10)

• Sharp weighted estimates for dyadic shifts and the $A_2$ conjecture
• Boundedness of the square function and rectifiability
• The sharp weighted bound for general Calderon-Zygmund operators
• A sharp estimate of weighted dyadic shifts of complexity 0 and 1
• A Representation Theorem for Singular Integral Operators on Spaces of Homogeneous Type
• Local Tb theorems and Hardy inequalities
• Non-homogeneous T1 theorem for bi-parameter singular integrals
• Pseudo-localisation of singular integrals in L^p
• Positive operators and maximal operators in a filtered measure space
• On the Rademacher maximal function