On weak and strong sharp weighted estimates for square functions revisited

Categories: Uncategorized

Weak and strong estimate of square function and Monge–Amp\`ere operators with drift

March 28, 2015 Leave a comment
Categories: Uncategorized

Lectures in Santander

September 2, 2014 Leave a comment
Categories: Uncategorized

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

Categories: Math

Random walks and Bellman functions for the weak estimate blow-ups with $A_1$ and $A_2$ weights

October 2, 2013 Leave a comment
Categories: Math

The A_1 conjecture revised

September 28, 2013 Leave a comment
Categories: Math

Nonhomogeneous harmonic analysis: 16 years of development

September 23, 2013 Leave a comment
Categories: Uncategorized

Get every new post delivered to your Inbox.