This is an announcement for the paper "The vector-valued non-homogeneous Tb theorem" by Tuomas Hytonen.

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 (\mu), 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 (L^p(\mu;X)) of functions with values in (X) --- a Banach space with the unconditionality property of martingale differences (UMD) and a certain maximal function property, which holds for all typical examples of UMD spaces. The new proof deals directly with all (p\in(1,\infty)) and relies on delicate estimates for the non-homogenous ``Haar'' functions, as well as McConnell's (1989) decoupling inequality for tangent martingale differences.

Archive classification: math.FA math.CA

Mathematics Subject Classification: 42B20;42B25; 46B09; 46E40; 60G46

Remarks: 40 pages, submitted for publication

The source file(s), nonhomog.tex: 143196 bytes, is(are) stored in gzipped form as 0809.3097.gz with size 39kb. The corresponding postcript file has gzipped size 209kb.

Submitted from: tuomas.hytonen@helsinki.fi

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