A class of pseudoconvex domains in C n generalizing the Hartogs triangle is considered. The L p boundedness of the Bergman projection associated to these domains is established, for a restricted range of p depending on the "fatness" of domains. This range of p is shown to be sharp.

Regularity and irregularity of the Bergman projection on L p spaces is established on a natural family of bounded, pseudoconvex domains. The family is parameterized by a real variable γ. A surprising consequence of the analysis is that, whenever γ is irrational, the Bergman projection is bounded only for p = 2.

The Bergman theory of domains $\{ |{z_{1} |^{\gamma}} < |{z_{2}} | < 1 \}$ in $\mathbb{C}^2$ is studied for certain values of $\gamma$, including all positive integers. For such $\gamma$, we obtain a closed form expression for the Bergman kernel, $\mathbb{B}_{\gamma}$. With these formulas, we make new observations relating to the Lu Qi-Keng problem and analyze the boundary behavior of $\mathbb{B}_{\gamma}(z,z)$.Comment: 13 page

We obtain sharp ranges of L p -boundedness for domains in a wide class of Reinhardt domains representable as sub-level sets of monomials, by expressing them as quotients of simpler domains. We prove a general transformation law relating L pboundedness on a domain and its quotient by a finite group. The range of p for which the Bergman projection is L p -bounded on our class of Reinhardt domains is found to shrink as the complexity of the domain increases. This is a ``preproof'' accepted article for Canadian Journal of Mathematics This version may be subject to change during the production process.

Expected duality and approximation properties are shown to fail on Bergman spaces of domains in C n , via examples. When the domain admits an operator satisfying certain mapping properties, positive duality and approximation results are proved. Such operators are constructed on generalized Hartogs triangles. On a general bounded Reinhardt domain, norm convergence of Laurent series of Bergman functions is shown. This extends a classical result on Hardy spaces of the unit disc.