The Bergman projection on a general bounded, smooth pseudoconvex domain in two complex variables need not be globally regular, that is, need not preserve the class of all functions that are smooth up to the boundary. In this article the construction of the worm domains is reviewed, with emphasis on those features relevant to their role as counterexamples to global regularity. Prior results, and...

‎In this paper we characterize Bergman spaces with‎ ‎respect to double integral of the functions $|f(z)‎ ‎-f(w)|/|z-w|$,‎ ‎$|f(z)‎ -‎f(w)|/rho(z,w)$ and $|f(z)‎ ‎-f(w)|/beta(z,w)$,‎ ‎where $rho$ and $beta$ are the pseudo-hyperbolic and hyperbolic metrics‎. ‎We prove some necessary and sufficient conditions that implies a function to be...

In this paper we introduce the family of spaces RM(p,q), 1≤p,q≤+∞. They are holomorphic functions in unit disc with average radial integrability. This contains classical Hardy (when p=∞) and Bergman p=q). We characterize inclusion between RM(p1,q1) RM(p2,q2) depending on parameters. For 1<p,q<∞, our main result provides a characterization dual RM(p,q) by means boundedness projection. show that ...

Let $$\Omega \subset {\mathbb {C}}^n$$ be a smooth bounded pseudoconvex domain and $$A^2 (\Omega )$$ denote its Bergman space. $$P:L^2(\Omega )\longrightarrow A^2(\Omega the projection. For measurable $$\varphi :\Omega \longrightarrow \Omega $$ , projected composition operator is defined by $$(K_\varphi f)(z) = P(f \circ \varphi )(z), z \in f\in A^2 ).$$ In 1994, Rochberg studied boundedness of...