On the structure of polyharmonic Bergman-type spaces over the unit disk

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 .

Keywords:

• polyharmonic Bergman space
• true poly-Bergman space
• true polyharmonic Bergman space
• harmonic component
• singular integral operator
• Beurling–Ahlfors transform

AMS Subject Classifications:

• 31B30
• 46E22
• 32A25
• 31A10

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.