Planar Beurling Transform and Bergman Type Spaces

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

“…Note that the true poly-Bergman spaces were first introduced by Vasilevski in [24] and since then they appeared in several studies (see e.g. [14][15][16][18][19][20][21]). In Theorem 4.4, we establish one-to-one bounded operators from the harmonic Bergman space onto the true polyharmonic Bergman spaces.…”

“…We know (see [16,Proposition 3.4]) that N ( j),k and N j,(k) are k-dimensional and jdimensional, respectively, and N ( j),(k) is one-dimensional. The projections of L 2 (D) onto the finite dimensional spaces N j,k , N ( j),k , N j, (k) and N ( j),(k) are denoted by P j,k , P ( j),k , P j,(k) and P …”

“…It is clear that a Hilbert basis for the poly-Bergman space comes by means of a Hilbert basis for the Bergman space and by isomorphisms of spaces with finite codimension on the Bergman space onto the true poly-Bergman spaces. The following result is used several times in the paper and restates [16,Theorem 2.1]. Indeed, we want to replace the second lines in (2.8) and (2.9) of [16, Theorem 2.1] by (24) and (26) below, respectively.…”

“…Remark, see e.g. [14][15][16][17][18], that some structural properties of poly-Bergman spaces can be described based on special properties of S U . The equalities in (1), or the analogous ones in the upper half-plane, constitute also an important step in several papers on operator theory, see e.g.…”

“…Based on the (two-sided) compression of the Beurling-Ahlfors transform to disks, half planes or disk complements (see e.g. [14][15][16][17][18]), we obtain examples of power partial isometries which have several important properties related to the theory of polyanalytic and polyharmonic functions, and hence can be considered as natural examples of power partial isometries of mathematical analysis and its applications.…”