Direct Cauchy theorem and Fourier integral in Widom domains
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PETER YUDITSKII In the study of mathematics, there is a grave injustice: we put in so much effort, but we get such miserable results... Larry Zalcman (from a private conversation) Abstract. We derive the Fourier integral associated with the complex Martin function in the Denjoy domain of the Widom type with the Direct Cauchy Theorem (DCT). As an application we study canonical systems and corresponding transfer matrices generated by reflectionless Weyl–Titchmarsh functions in such domains. The DCT property appears to be crucial in many aspects of the underlying theory.
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Introduction
We develop here some specific aspects of the general de Branges theory [4] (see also [25, 26]) which deal with the function theory in infinitely connected domains [12] and spectral properties of random and almost-periodic operators [22]. Recall that a Denjoy domain is an open subset of the complex plane C whose complement is a subset of the real axis R. Let E be a closed unbounded subset of the positive half axis, E = R+ \ (aj , bj ). j≥1
We assume that the Denjoy domain = C \ E is regular in the sense of the potential theory [10]. By G(λ, λ0 ) we denote the Green function of the domain with singularity at λ0 ∈ . The complex Green function is defined by λ0 (λ) = eiθλ0 (λ) ,
θλ0 (λ) = − G(λ, λ0 ) + iG(λ, λ0 ),
where G(λ, λ0 ) is the function harmonically conjugated to G(λ, λ0 ), G(λ∗ , λ0 ) = 0 for a normalization point λ∗ ∈ R− . The complex Green function is multivalued in . Let π1 () be the fundamental group of . It is generated by simple 411 ´ JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 141 (2020) DOI 10.1007/s11854-020-0122-7
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loops {γ(j) }j≥1 , γ(j) starts and ends at λ∗ ∈ R− and goes through the gap (aj , bj ). To be extended by continuity along γ(j) , the complex Green function obeys the following identity: λ0 (γ(j) (λ)) = e2πiω(λ0 ,Ej ) λ0 (λ), where ω(λ0 , Ej ) = ω(λ0 , Ej , ) is the harmonic measure of the set Ej = E ∩ [0, aj ] computed at λ0 . By π1 ()∗ we denote the group of characters of the group π1 () (see, e.g., [13]), α : π1 () → R/Z,
α(γ1 γ2 ) = α(γ1 ) + α(γ2 ),
γj ∈ π1 ().
Note that π1 () is a discrete group and π1 ()∗ is a compact Abelian group. We say that F(λ) is character automorphic with a certain character α ∈ π1 ()∗ if (1.1)
F(γ(λ)) = e2πiα(γ) F(λ).
Note that |F(λ)| is single valued in the domain. ∞ For a fixed character α, by H (α) we denote the collections of bounded analytic multivalued functions F(λ) such that (1.1) holds [12]. More generally, the Hardy p (α), p ∈ [1, ∞), are formed by functions which obey (1.1); in addition, spaces H |F(λ)|p possesses a harmonic majorant in the domain. Theorem 1.1 (Widom [32, 12]). The following two statements are equivalent: ∞ (α) contains a non-constant function for all α ∈ π1 ()∗ . • H • Let {cj } be the collection of critical points of G(λ, λ∗ ), i.e., ∇G(cj , λ∗ ) = 0. Then (1.2) G(cj, λ∗ ) < ∞. j
If (1.2) holds, we say that is of Widom type or a Widom domain. In the Widom domain the harmonic measure ω(λ∗ , dξ)
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