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.
