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 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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