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

Abstract

To cite this article: Luís V. Pessoa (2015): On the structure of polyharmonic Bergman-type spaces over the unit disk, Complex Variables and Let ( j, k) be a pair of nonnegative integers. We deduce explicit representations for the projections onto spaces of square integrable ( j, k)-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 ( j, k)-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 j ≥ k, we establish an isometry between a subspace of the true polyharmonic Bergman space of order k with codimension 4 j, and the true polyharmonic Bergman space of order k + j.