We prove optimal estimates for the mapping properties of the Bergman projection on the Hartogs triangle in weighted L p spaces when p > 4 3 , where the weight is a power of the distance to the singular boundary point. For 1 < p ≤ 4 3 we show that no such weighted estimates are possible.2010 Mathematics Subject Classification. 32A25, 32A07.

We establish the L 2 theory for the Cauchy-Riemann equations on product domains provided that the Cauchy-Riemann operator has closed range on each factor. We deduce regularity of the canonical solution on ( p, 1)-forms in special Sobolev spaces represented as tensor products of Sobolev spaces on the factors of the product. This leads to regularity results for smooth data.

An L 2 version of the Serre duality on domains in complex manifolds involving duality of Hilbert space realizations of the ∂-operator is established. This duality is used to study the solution of the ∂-equation with prescribed support. Applications are given to ∂-closed extension of forms, as well to Bochner-Hartogs type extension of CR functions.2000 Mathematics Subject Classification. 32C37, 35N15, 32W05.

We show that the ∂-problem is globally regular on a domain in C n , which is the n-fold symmetric product of a smoothly bounded planar domain. Remmert-Stein type theorems are proved for proper holomorphic maps between equidimensional symmetric products and proper holomorphic maps from Cartesian products to symmetric products. It is shown that proper holomorphic maps between equidimensional symmetric products of smooth planar domains are smooth up to the boundary.

The regularity of the ∂ -problem on the domain {|z 1 | < |z 2 | < 1} in C 2 is studied using L 2 -methods. Estimates are obtained for the canonical solution in weighted L 2 -Sobolev spaces with a weight that is singular at the point (0, 0). In particular, the singularity of the Bergman projection for the Hartogs triangle is contained at the singular point and it does not propagate.

We study the approximation of J-holomorphic maps continuous to the boundary from a domain in into an almost complex manifold by maps Jholomorphic to the boundary, giving partial results in the non-integrable case. For the integrable case, we study arcs in complex manifolds and establish the existence of neighborhoods biholomorphic to open sets in Euclidean space for several classes of arcs. As an application we obtain C k approximation of holomorphic maps continuous to the boundary into complex manifolds by maps holomorphic to the boundary, provided the boundary is nice enough.2000 Mathematics Subject Classification. 32Q60,32Q65,32H02,30E10

Abstract. We consider proper holomorphic mappings of equidimensional pseudoconvex domains in complex Euclidean space, where both source and target can be represented as Cartesian products of smoothly bounded domains. It is shown that such mappings extend smoothly up to the closures of the domains, provided each factor of the source satisfies Condition R. It also shown that the number of smoothly bounded factors in the source and target must be the same, and the proper holomorphic map splits as product of proper mappings between the factor domains.

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.