CHAPTER 2

Quadrature Domains

Let μ be a finite measure with compact support. In this paper, a measure means

a positive Borel measure in C. Let λ be the two-dimensional Lebesgue measure. If

a bounded open set Ω in C satisfies μ(C \ Ω) = 0 and

sdμ ≤

Ω

sdλ

for every subharmonic function s in Ω, which is integrable on Ω with respect to

λ, we call Ω a quadrature domain of μ. The domain may not exist. If it exists, it

is uniquely determined, except for a set of Lebesgue measure zero. That is, as a

measure, λ|Ω is uniquely determined. However, the notion of a quadrature domain

is more precise than that of a measure. There are smallest quadrature domains for

many measures. If there exists a smallest quadrature domain of μ, we say that μ

defines the smallest quadrature domain and denote it by Ω(μ). If μ and μ satisfy

μ ≤ μ and define the smallest quadrature domains Ω(μ) and Ω(μ ), respectively,

then Ω(μ) ⊂ Ω(μ ), which we call the fundamental inclusion relation for quadrature

domains.

It is known that if μ 0 is singular with respect to λ, then there exists a

smallest quadrature domain Ω(μ) of μ. Another typical case is as follows: Let D

be a bounded connected open set and let ν be a finite measure on D satisfying

ν(D) = ν 0. We write D for λ|D and set μ = D + ν. Then there exists a

smallest quadrature domain Ω(D + ν) of D + ν. The same is true for a bounded

connected open set D and a finite measure ν such that ν(D) 0 and ν|(C \ D)

is singular with respect to λ. For the definition and the existence of the smallest

quadrature domain, we refer to [S1] and [GS].

In this paper, we need to treat a measure which may not be finite, nor define

the smallest quadrature domain. For the sake of simplicity, we mainly treat the

case in which μ = D + ν, where D is an open set and ν is a finite measure such

that ν|(C \ D) is singular with respect to λ.

Assume first that D is bounded. In this case, D+ν can be expressed as D0 +ν0,

where D0 denotes the union of some connected components of D and ν0 = D\D0+ν

defines the smallest quadrature domain Ω(ν0), and D0 and Ω(ν0) are disjoint. If

D0 is not empty, there is not a smallest quadrature domain of D + ν. Actually, for

every open set U satisfying λ((D0 \ U) ∪ (U \ D0)) = 0, U ∪ Ω(ν0) is a quadrature

domain of D + ν. Since D0 and Ω(ν0) are uniquely determined by D and ν, we set

Ω(D +ν) = D0 ∪Ω(ν0) and say that D +ν defines the quadrature domain Ω(D +ν).

The fundamental inclusion relation for quadrature domains is as follows: If D ⊂ D

and ν ≤ ν , then Ω(D + ν) ⊂ Ω(D + ν ), where D denotes a bounded open set

and ν denotes a finite measure such that ν |(C \ D ) is singular with respect to λ.

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