Area operators on large Bergman spaces

Abstract

We completely characterize those positive Borel measures \(\mu\) on the open unit disk \(\mathbb{D}\) for which the area operator \(A_{\mu}\colon A^p_\varphi\rightarrow L^q(\mathbb{T})\) is bounded. Here, the indices \(0<p,q<\infty\) are arbitrary and \(\varphi\) belongs to a certain class \(\mathcal{W}_{0}\) of exponentially decreasing weights. Accordingly, the proofs require techniques adapted to such weights, like tent spaces, Carleson measures for \(A^p_\varphi\)-spaces, Kahane–Khintchine inequalities, and decompositions of the unit disc by \((\rho,r)\)-lattices, which differ from the conventional decompositions into subsets with essentially constant hyperbolic radii.