Functional Analysis and Applications Seminar 

01/02/2008, 15:00 — 16:00 — Room P3.10, Mathematics Building
Luís Pessoa, Instituto Superior Técnico, UT Lisboa

Sequences of analytical type spaces and related Calderon-Zygmund operators

We will present explicit representations of Bergman kind projections in terms of the Calderon-Zygmund operators on the unit disk $S$ and $S^\ast$. This will permit to decompose the $L^2$ space of unit disk with area measure, in orthogonal spaces of analytical type (or of anti-analytical type). The mentioned Calderon-Zygmund operators will play the role of unitary operators between such analytical type spaces. The dependence on boundary regularity of equalities between Bergman kind projections and singular integral operators, is far way of being understood. We will present easy examples of domains that do not admit Dzhuraev's formulas, exemplify how inside variations of the domain can bring some light to the problem and establish explicit forms of the Bergman and anti-Bergman projections for some open sectors. On the other hand, inside approximation of a domain also permits to get formulas for the poly kernel functions of the upper half plane, to establish corresponding new proofs of Dzhuraev's formulas and to clarify its existence for open sectors. By simple remarks we will achieve known and important properties of Calderon-Zygmund operators $S_j$ for integer $j$. The talk is partially based on joint work with Yu. I. Karlovich.