Abstract
Let be a pair of nonnegative integers. We deduce explicit representations for the projections onto spaces of square integrable
-polyanalytic functions in terms of the two-sided compression of the Beurling–Ahlfors transform to the unit disk. We show that the space of square integrable
-polyanalytic functions is a reproducing kernel Hilbert space and we deduce representations for one-to-one bounded operators from the Bergman space onto the true poly-Bergman spaces and from the harmonic Bergman space onto the true polyharmonic Bergman spaces. Moreover, we prove a decomposition theorem for polyharmonic functions in terms of their harmonic components. Finally, for positive integers
we establish an isometry between a subspace of the true polyharmonic Bergman space of order
with codimension
and the true polyharmonic Bergman space of order
.
Acknowledgements
I would like to express all my sincere gratitude to Ana Moura Santos for her helpful remarks.
Notes
No potential conflict of interest was reported by the author.