Functional Analysis and Applications Seminar 

15/03/2013, 14:30 — 15:30 — Room P3.10, Mathematics Building
Luís Pessoa, Instituto Superior Técnico, UTL and CEAF

The Essential Boundary on Polyanalytic Functions. Some Differences Between the Analytic and the Polyanalytic Cases.

I will begin to explain how the action of the compression of the Beurling transform on the Bergman space can give a transparent view of the structure of poly-Bergman spaces on domains Möbius Equivalent to a Disk. The existence of exact Dhzuraev formulas is an important tool. Note that the usual representations of poly-Bergman projections by means of two-dimensional singular integral operators are strongly dependent on the smoothness of the boundary. In the second part of the talk, I will consider a bounded domain without constrains on the boundary. A Fredholm symbolic calculus is constructed for poly-Toeplitz operators with continuous symbol and I will explain how such symbol requires the notion of j-essential boundary. The symbol calculus is well known for Bergman-Toeplitz operators, in which setting the removal boundary is a subset of the boundary having zero logarithmic capacity. Some surprising differences between the analytical and the poly-analytical case will be presented.