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).