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

TL;DR

This paper proves the qualitative Vitushkin conjecture using a non-homogeneous $Tb$ theorem, with significant implications for analytic capacity and Calderón-Zygmund operators, and discusses a quantitative version crucial for recent advances.

Contribution

It provides a proof of the Vitushkin conjecture and introduces a quantitative non-homogeneous $Tb$ theorem, advancing the understanding of analytic capacity and singular integral operators.

Findings

Proof of the qualitative Vitushkin conjecture.

Development of a quantitative non-homogeneous $Tb$ theorem.

Establishment of an all-or-nothing principle for Calderón-Zygmund operators.

Abstract

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 erased all preprints dated before 2006 from its site, and this paper became inacessible. It has certain importance though, as the reader shall see. 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 $\mu$ on the complex plane ${\mathbb C}$ that is Ahlfors at almost every point (i.e. for $\mu$-almost every $x\in {\mathbb C}$ there exists a constant $M>0$ such that $\mu(B(x,r))\le Mr$ for every $r>0$) then any one-dimensional antisymmetric Calder\'on-Zygmund operator $K$ (e.g. a Cauchy integral type operator) satisfies the following "all-or-nothing" princple: if there exists at least one function $\phi\in L^2(\mu)$ such that $\phi(x)\ne 0$ for $\mu$-almost every $x\in {\mathbb C}$ and such that {\it the maximal singular operator} $K^*\phi\in L^2(\mu)$, then there exists an everywhere positive weight $w(x)$, such that $K$ acts from $L^2(\mu)$ to $L^2(wd\mu)$.