Quadrature Domains for the Bergman Space in Several Complex Variables
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Quadrature Domains for the Bergman Space in Several Complex Variables Alan R. Legg1
Received: 10 April 2017 / Revised: 9 September 2017 / Accepted: 22 September 2017 © Springer-Verlag GmbH Germany 2017
Abstract We make use of the Bergman kernel function to study quadrature domains whose quadrature identities hold for L 2 holomorphic functions of several complex variables. We generalize some mapping properties of planar quadrature domains and point out some differences from the planar case. We then show that every smooth bounded convex domain in Cn is biholomorphic to a quadrature domain. Finally, the possibility of continuous deformations within the class of planar quadrature domains is examined. Keywords Quadrature domain · Bergman kernel · Several complex variables · Biholomorphic mapping Mathematics Subject Classification 32A10 · 32A25 · 32H02
1 Introduction This article examines the properties of quadrature domains which can be analyzed from the viewpoint of complex analysis. A major goal is to generalize elegant planar
Communicated by Dmitry Khavinson. Research partially supported by the National Science Foundation Analysis and Cyber-enabled Discovery and Innovation Programs Grant DMS 1001701, and by the National Science Foundation Grant DMS 1464150.
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Alan R. Legg [email protected] Indiana University-Purdue University Fort Wayne, Fort Wayne, USA
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A. R. Legg
phenomena to several dimensions, and so attention will be given to the relationship between one and several variables. A classical quadrature domain, following Aharanov and Shapiro [1], is a domain Ω ⊂ C such that evaluation of integrals of functions in the class AL 1 (Ω) (holomorphic functions which are integrable) is a finite computation involving point evaluations : Ω is a quadrature domain if there exist finitely many points z 1 , . . . , z K of Ω, and finitely 1 many complex constants {c jk }| J,K j=1,k=1 such that, for every f ∈ AL (Ω), Ω
f dA =
c jk f ( j) (z k ).
(1.1)
j≤J,k≤K
If Ω is such a domain, the relation (1.1) is called its ‘quadrature identity,’ and the points z k are called ‘quadrature nodes.’ The definition can be modified by changing the test class of functions for which the quadrature identity must hold. In this article, we will do just that and consider quadrature domains using the Bergman space H 2 of square-integrable holomorphic functions as the test class. Furthermore, we will ∂α expand the definition to include domains in Cn , using holomorphic derivatives ∂z α in our quadrature identities, α standing for multi-indices. After some background on the planar case, we will briefly outline our approach to the multi-dimensional case in this article. In the plane, Aharanov and Shapiro originally found a satisfying relationship between the quadrature identity of a quadrature domain and the so-called ‘Schwarz function’ of a domain. Developing this line of thought led to a nice list of properties enjoyed by quadrature domains; in [1], they showed among other things that a bounded simply connected quadrature domain
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