How to pull ourselves up by the hair (and to prove Vitushkin’s conjecture)

January 11, 2014 Leave a comment

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 $\C$ that is

Ahlfors at almost every point (i.e. for $\mu$-almost every $x\in\C$ 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 $K$ (i.e. a Cauchy integral type

operator) satisfies the following “all-or-nothing” princple:

if there exists at least one function $\f\in L^2(\mu)$ such that $\f(x)\ne 0$

for $\mu$-almost every $x\in\C$ and such that{\it the maximal singular operator} $K^*\f\in L^2(\mu)$, then there

exists an everywhere positive weight $w(x)$, such that $K$ acts from

$L^2(\mu)$ to $L^2(\mu,w)$.

The pdf file

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