The Tb-theorem on non-homogeneous spaces that proves a conjecture of Vitushkin

By Fedor Nazarov, Sergei Treil, Alexander Volberg. 2014

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This article was written in 1999, and was posted as a preprint in CRM (Barcelona) preprint series n 0 519 in 2000. However, recently CRM (Barcelona) erased all preprints dated before 2006 from its site, and this paper became inacessible. It has certain importance though, as the reader shall see. Meanwhile this paper in bits and pieces appeared in several book formats, namely in Volberg’s lecture notes [Vo], in Doudziak’s book [Du], and in Tolsa’s book [To]. Formally this paper is a proof of the (qualitative version of the) Vitushkin conjecture. The last section is concerned with the quantitative version. This quantitative version turns out to be very important. It allowed Xavier Tolsa to close the subject concerning Vtushkin’s conjectures: namely, using the quantitative nonhomogeneous Tb theorem proved in the present paper, he proved the semiadditivity of analytic capacity. Another “theorem”, which is implicitly contained in this paper, is the statement that any non-vanishing L 2 -function is accretive in the sense that if one has a finite measure µ on the complex plane C that is Ahlfors at almost every point (i.e. for µ-almost every x ∈ C there exists a constant M > 0 such that µ(B(x,r)) 6 Mr for every r > 0) then �

Subfields: Applied Mathematics, Mathematical Physics

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