Mapping properties of the Bergman projections on elementary Reinhardt domains

The elementary Reinhardt domain associated to multi-index k = (k 1, …, k n ) ∈ ℤ n is defined by ℋ ( k ) : = { z ∈ D n : z k is defined and | z k | < 1 } . $$\mathcal{H}(\mathbf{k}):=\{z\in\mathbb{D}^n: z^{\mathbf{k}}\ \text{is defined and}\ |z^{\mathbf{k}}|<1\}.$$ In this paper, we study the mapping properties of the associated Bergman projection on L p spaces and L p Sobolev spaces of order ≥ 1.

Construction of Biholomorphic Convex Mappings of OrderαonBpn

Some sufficient conditions for biholomorphic convex mappings of orderαon the Reinhardt domainBpninCnare given; from that, criteria for biholomorphic convex mappings of orderαwith particular form become direct. As applications of these sufficient conditions, some concrete biholomorphic convex mappings of orderα onBpnare provided.

A New Roper-Suffridge Extension Operator on a Reinhardt Domain

We introduce a new Roper-Suffridge extension operator on the following Reinhardt domaingiven bywherefis a normalized locally biholomorphic function on the unit discD,pjare positive integer,ajare complex constants, andj=2,…,n. Some conditions forajare found under which the operator preserves almost starlike mappings of orderαand starlike mappings of orderα, respectively. In particular, our results reduce to many well-known results when allαj=0.