01/02/2008, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
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 and . This will permit to decompose the
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 for integer . The talk is partially based on
joint work with Yu. I. Karlovich.