## Carlos Perez :
Poincare-BMO inequalities and weighted estimates for operators

It is well known that one of the key points in
the regularity theory of elliptic Partial Differential Equations are
the Poincare inequalities or the improved version so called
Poincare-Sobolev inequalities. In the first part of this course
I will concentrate on Poincare type inequalities. We plan to show that
these basic
estimates are intimately related to certain other basic spaces
in Analysis such as the BMO spaces of John-Nirenberg or the
Lipchitz spaces. The main techniques we will come from Calderon-Zygmund theory
from Harmonic Analysis, and in particular the good-lambda of Burkholder
and Gundy will play a main role.
As a sample, we will derive as a corollary inequalities such as the one due to
Fabes, Kenig and Serapioni important in the case of degenerate elliptic
equations:

( 1/w(B) \int_B |f(x)-f_B|^p w(x) dx )^1/p <= C r(B)
( 1/w(B) \int_B |\nabla f(x)|^2 w(x) dx )^1/2.
where w is an A_2 weight, and for some p>2. The unweighted estimate
is well known with p=2n/(n-2), n>2.

To simplify our presentation we will present the results and
techniques in R^n and with the metric associated to cubes;
however, we will point out possible extensions and
difficulties when extending the main issues to general Spaces of
Homogeneus Type. To work within this context is interesting since
we can go beyond the study of the standard gradient and to
consider differential operators such as Hormander Laplacian X
or the Baouendi-Grushin operator.

This work presented is in collaboration with B.Franchi and R. Wheeden
and with P. MacManus.

In the second part of the course I will concentrate on certain aspects
of the theory of the two weight problem for certain operators such as
fractional integration, classical singular integrals or commutators of
singular integrals with BMO functions. More precisely if T one of these
operators we look look for "reasonable" conditions for which either
the following inequality

\int_R^n |Tf(x)|^p w(x) dx <= C \int_R^n |f(x)|^p v(x) dx,
or the correpsonding weak version holds. In some cases there are
necesssary and sufficcient conditions due to Sawyer, which are not easy to
handle
in practice. We look for conditions more geometrical and close in
some sense to the usual A_p conditions.

We will show some known results in the case of fractional integration
and singular integrals. In the last case the problem becomes difficult and
we will sketch some recent joint work with D.
Cruz-Uribe where some sufficient conditions have been
obtained.

**See picture of this text in TeX (27 kB).**