The Cauchy–Riemann equations on product domains

Abstract

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.

Spectrum of the complex Laplacian on product domains

“…The study of the ∂ and ∂-Neumann problems on product domains raises a series of interesting questions, which have been studied by many authors [8,9,10,1,3,12]. In [12], the method of separation of variables was used to compute the spectrum of the complex Laplacian = ∂∂ * + ∂ * ∂ on a polydisc, and for each eigenvalue, the corresponding eigenspace was identified.…”

“…Another approach, using an explicit solution of the d-equation on a product domain was given by Zucker in [18]. This can be extended to the ∂-equation (see [3]), or, in another direction, to abstractly defined products of "Hilbert Complexes" (see [2]). A crucial feature in this approach is the assumption that the d-or ∂-operator on the product satisfies a "Leibniz rule" in the strict operator sense (cf.…”

“…In practice, this means that some boundary-smoothness or completeness assumptions must be made on the factors Ω j in order to get a handle on the domain of the dor ∂-operator (via Friedrichs' lemma when the Ω j have boundaries). A version of Theorem 1.2 is proved in [3], where it is assumed that each of the factors Ω j has Lipschitz boundary. Consequently, using the results of [3], we cannot conclude, for example, that ∂ has closed range on the product domain in C 4 given as…”

“…A version of Theorem 1.2 is proved in [3], where it is assumed that each of the factors Ω j has Lipschitz boundary. Consequently, using the results of [3], we cannot conclude, for example, that ∂ has closed range on the product domain in C 4 given as…”

Weighted Bergman projections on the Hartogs triangle

“…In [CS11] and [CS13], Chakrabarti and Shaw focus on the ∂-equation and the corresponding Sobolev regularity on the product domains and the Hartogs triangle. In [Che13], the author shows the (ordinary) Bergman projection is L p bounded on the Hartogs triangle if and only if p ∈ 4 3 , 4 .…”