On a modification of the Poisson integral operator

Given a quasisymmetric automorphism γ of the unit circle T we define and study a modification Pγ of the classical Poisson integral operator in the case of the unit disk D. The modification is done by means of the generalized Fourier coefficients of γ. For a Lebesgue’s integrable complexvalued function f on T, Pγ [f ] is a complex-valued harmonic function in D and it coincides with the classical Poisson integral of f provided γ is the identity mapping on T. Our considerations are motivated by the problem of spectral values and eigenvalues of a Jordan curve. As an application we establish a relationship between the operator Pγ , the maximal dilatation of a regular quasiconformal Teichmüller extension of γ to D and the smallest positive eigenvalue of γ. Introduction. A number of important problems in the potential theory of the complex plane C can be reduced to a linear integral equation of Fredholm type with the Neumann–Poincaré kernel k or its transposition. This kernel is assigned to a rectifiable and sufficiently smooth Jordan curve Γ ⊂ C by the formula (0.1) k(ζ, z) := − 1 π ∂ ∂~nζ log |ζ − z| , ζ, z ∈ Γ, ζ 6= z, 2000 Mathematics Subject Classification. Primary 30C62, 30C75.


Introduction.
A number of important problems in the potential theory of the complex plane C can be reduced to a linear integral equation of Fredholm type with the Neumann-Poincaré kernel k or its transposition.This kernel is assigned to a rectifiable and sufficiently smooth Jordan curve Γ ⊂ C by the formula For details the reader is referred to e.g.[3], [5], [32].A very short but essential survey of basic problems can be found in [30].Let us recall that a real number λ is called a Fredholm eigenvalue of Γ if it is an eigenvalue of the kernel k, i.e. if there exists a real-valued function µ integrable on Γ and non-constant almost everywhere (a.e. for brevity), which satisfies the homogeneous integral equation The theory of Fredholm eigenvalues of a Jordan curve has been intensively studied by a number of eminent mathematicians like Ahlfors, Bergman and Schiffer, next by Schober and Springer, and lately by Krushkal, Krzyż and Kühnau.Two Krzyż's ideas seem to be especially important in the contemporary theory of eigenvalues of a Jordan curve.First of all he observed in [11] and [10] that every Fredholm eigenvalue λ of a sufficiently regular Jordan curve Γ can be expressed equivalently by a pair of continuous functions F : cl(Ω) → C and F * : cl(Ω * ) → C which are analytic in the domains Ω and Ω * ∞ complementary to Γ, and satisfy the following boundary assumptions on Γ: Im F = Im F * and (1 − λ) Re F = (1 + λ) Re F * .
The notation cl(A) means the closure of a set A ⊂ Ĉ := C ∪ {∞} in the spherical metric.It is worth noting here that all Fredholm eigenvalues of Γ can be represented without using the Neumann-Poincaré kernel.This idea appears implicitly also in the works of Kühnau; cf.e.g.[17].Having disposed of the equalities (0. where γ := H −1 * • H : T → T is so-called welding homeomorphism of Γ.This way the eigenvalue problem for a Jordan curve Γ can be reduced to a new problem of studying G, G * and λ satisfying the equalities (0.4) for a given homeomorphic self-mapping γ of the unit circle.The author, encouraged by Krzyż, pursued this line of research in several works by introducing and studying so-called eigenvalues of an automorphism of the unit circle; cf.e.g.[14], [24], [21], [22], [23] and [25].Therefore the author is much indebted to professor Jan Krzyż for introducing him to the theory of Fredholm eigenvalues of a Jordan curve.
The definition of an eigenvalue of a quasisymmetric automorphism γ of the unit circle is recalled in Section 3.This is done by applying the harmonic conjugation operator and certain operator B γ assigned to γ.Then the smallest positive eigenvalue of γ is described by the Poisson integral modified by γ; cf.Theorem 3.1.This modification P γ is defined by means of the generalized Fourier coefficients of γ in Section 1. Then the basic properties of the operator P γ are studied.In Section 2 we recall the definition of the operator B γ and prove an important relationship between the operators P γ and B γ ; cf.Theorem 2.1.Then we characterize the norm of B γ (f ) by means of the Dirichlet integral D[P γ [f ]] (Corollary 2.2), where f is the abstract class of a real-valued function f ∈ H 1/2 with respect to the equivalence relation .The class H 1/2 consists of all Lebesgue's integrable complex-valued functions on T whose harmonic extensions to D have finite Dirichlet's integral.The relation identifies any complex-valued and Lebesque's measureable functions on T whose difference is a constant function a.e. in T.
The author would like to express his sincere thanks to the referee for his kind and helpful comments.

The Poisson integral modified by a quasisymmetric automorphism of the unit circle.
Let us denote by Hom + (T) the class of all sense-preserving homeomorphic self-mappings of T. For K ≥ 1 let Q(T, K) be the class of all γ ∈ Hom + (T) which admit a K-quasiconformal extension to D. Homeomorphisms belonging to the class Q(T) := K≥1 Q(T, K) were called by Krzyż as quasisymmetric automorphisms of the unit circle; cf.[12] and [13].He noticed that each f ∈ Q(T) can be described by a similar condition to the well-known Beurling-Ahlfors quasisymmetricity condition; cf.[1].For other characterizations of the class Q(T) see [33] and [27].
Let L 0 (T) stand for the class of all Lebesgue's measurable functions f : T → C. We denote by L 1 (T) the class of all f ∈ L 0 (T) which are integrable on T with respect to the Lebesgue arc-length measure, i.e.
T |f (z)||dz| < +∞.Let P[f ] be the Poisson integral of a function f ∈ L 1 (T), i.e. (1.1) where It is well known that P[f ] is a complex-valued harmonic function on D.Moreover, if the function f is continuous, then the function P[f ] is the unique solution to the Dirichlet problem for the boundary function f , which means that for every z ∈ T, Given a function f ∈ L 0 and m, n ∈ Z we define provided the respective functions are integrable on T. If m = 1, then (1.3) takes the form of and so f (1, n) is just the nth Fourier coefficient of the function f .This justifies to call f (m, n) the (m, n)-generalized Fourier coefficient of the function f .If f satisfies the following condition In particular, if f ∈ Hom + (T), then all generalized Fourier coefficients f (m, n), m, n ∈ Z, are well defined.
In [26] the following result was proved.
Here and subsequently, we write ∞ n=−∞ c n := lim m→∞ m n=−m c n for any sequence Z n → c n ∈ C, provided the limit exists.
Note that the inequalities (1.6) look similarly to the Grunsky inequalities for holomorphic functions in the classes Σ(k), 0 ≤ k ≤ 1; cf.[28,Sect. 3.1 and 9.4].Due to the works of R. Kühnau [18], [19], [20] and Y. Shen [31] we know that the inequalities (1.6) can be improved. Let where ∂F ∂x + i ∂F ∂y are so-called the formal derivatives of F .If F : D → C is a harmonic mapping in D given by the series expansion with coefficients a n ∈ C, n ∈ Z, then integrating by substitution we obtain and consequently, cf. [26, (1.2)].Using Theorem A we can modify the Poisson integral P[f ] as follows.From (1.1) and (1.8) it follows that (1.9) Applying now Theorem A for an arbitrarily fixed K ≥ 1 and γ ∈ Q(T, K) and the sequence Z n → λ n := f (n) we know that for each n ∈ Z \ {0} the sequence N p → p m=−p γ(m, n)λ m is convergent as p → ∞ and we may define Moreover, by the second inequality in (1.6), (1.12) This means that the operator H 1/2 f → P γ [f ] is well defined by the formula (1.13) and by (1.8), (1.14) We call P γ the Poisson integral operator modified by γ.Applying the operator P γ and the equalities (1.9), (1.11) and (1.14), we can rewrite the inequalities (1.6) in the following shorter form.
Then by (1.10) the sequence Z n → λ n := f (n) satisfies the condition (1.5).Theorem A now shows that Combining this with (1.9) and (1.14), we obtain the inequalities (1.15), which is the desired conclusion.
Remark 1.2.By the definition of the operator P γ we can infer directly its following properties valid for any γ ∈ Q(T): (i) If γ is the identity mapping on T, then P γ = P, i.e. the mapping For the proof we apply (1.3), (1.11) and (1.13) to arbitrarily fixed µ, ν ∈ C and f, g ∈ H 1/2 .If γ is the identity mapping on T, then from (1.3) we conclude that and γ(m, m) = 1 as m ∈ Z, hence and by (1.11) that and finally by (1.1) and (1.13) that the property (i) holds.
From (1.11) and (1.2) we see that for every n ∈ Z, Hence and by (1.13) we deduce the property (ii).From (1.3) and (1.2) we see that for every m, n ∈ Z, as well as Hence and by (1.11) we have as n ∈ Z. Combining this with (1.13) we conclude that for every z ∈ D, which yields the property (iii).
2. Relationships between operators P γ and B γ .In [24] and [23] the operator B γ was assigned to every γ ∈ Q(T).We recall now its construction.For all f, g ∈ L 0 (T) the notation f g means that f − g equals a constant function a.e. in T. It is clear that is an equivalence relation in the class L 0 (T).Let [f / ] stand for the abstract class of f ∈ L 0 (T) with respect to .Consider the class (2.1) Here and subsequently, we set Re X := {Re f : f ∈ X} for any family X of complex-valued functions.It can be verified in the standard way that (H, • H ) is a real Hilbert space, where cf. [24, Sect.2.4].We adopt the usual notation C(T) for the class of all complex-valued continuous functions on T. From (2.1), (2.2) and (1.9) it follows that the set Moreover, it may be concluded from [24, (2.5.1) and Theorems 2.5.3 and 2.4.3] that the inequalities hold for all K ≥ 1, f ∈ C(T) and γ ∈ Q(T, K).Then there exists the unique linear continuous operator As a matter of fact, B γ is a linear homeomorphism of the space (H, • H ) onto itself; cf.[24, Corollary 2.5.4].Various properties of spectral values and eigenvalues of a quasisymmetric automorphism γ of the unit circle were obtained by means of the operator B γ and its norm; cf.[24] and [23].Note that the operator B γ is defined implicitly by the condition (2.3).From the famous Beurling-Ahlfors result [1] we know that a quasisymmetric automorphism γ of T does not have to be an absolutely continuous function.Moreover, γ can be even purely singular.Therefore, in such a case the composite mapping f • γ is not Lebesgue's measurable function in general.
In consequence, f • γ / ∈ L 0 (T) for certain f ∈ H 1/2 , and so the family Re C(T) ∩ H 1/2 cannot be replaced by Re H 1/2 in (2.3).This means that defining the operator B γ directly by composition of functions fails for a singular γ ∈ Q(T).This problem was overcome in [24, Sect.2.5], where the following result was stated: Theorem 2.1.For every γ ∈ Q(T), Proof.Fix γ ∈ Q(T) and f ∈ Re H 1/2 .Then γ ∈ Q(T, K) for certain K ≥ 1 and the condition (1.10) holds.Since the function f is real-valued, we conclude from (1.2) that From (1.10) it follows that the sequence Z n → λ n := f (n) satisfies the condition (1.5).For every p ∈ N we define (2.8) By (2.7) each function f p , p ∈ N, is real-valued on T.Moreover, by (1.9) we have and so Then setting we conclude from (1.9) that Hence f − f p ∈ H 1/2 as p ∈ N, and by (1.10) we also have Thus setting f := [f / ], we see that f − f p ∈ H as p ∈ N, and by (2.13) and (2.2) we obtain Since B γ is a linear and continuous operator in (H, • H ), we conclude from (2.14) that On the other hand, where by (1.11), (2.17 By (2.13), Corollary 1.1 and Remark 1.2 we have Write f := [f / ] and choose g ∈ B γ (f ).By (2.11), (2.15) and (2.2) we have Combining this with (1.8), ( , we obtain To be more precise, Finally, . This implies the property (2.6), which is the desired conclusion.
Corollary 2.2.For every γ ∈ Q(T), (2.20) Proof.By Theorem 2.1 and (2.2) we conclude that for every 3. The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle.The operator P γ seems to be a more convenient tool for studying spectral values and eigenvalues problems as compared to the operator B γ .Applying Theorem 2.1 and Corollary 2.2, we can rewrite a number of known so far results in this subject by means of the operator P γ .
As an example we will show Theorem 3.1 which is related to the following Krzyż result on quasiconformal reflection; cf.[10].
Let S[f ] be the Schwarz integral of a function f ∈ L 1 (T), i.e. (3.2) satisfies the following properties and so the operator A is an isometry of the space (H, • H ) onto itself.Therefore, for each γ ∈ Q(T) the operator (3.5)A γ := B γ AB −1 γ , called the generalized harmonic conjugation operator, is a linear homeomorphism of the space (H, • H ) onto itself; cf.[24,Sect. 3.1].We recall that a real number λ is said to be an eigenvalue of γ ∈ Q(T) if there exists f ∈ H with f H = 0 such that (3.6) (λ + 1)A(f ) = (λ − 1)A γ (f ) ; cf. [21, Definition 1.1] and also [24,Sect. 3.2].Let Λ * γ be the set of all eigenvalues of γ ∈ Q(T).We recall that a quasiconformal self-mapping ψ of D is said to be a regular Teichmüller mapping if there exists a non-zero holomorphic function F in D and a constant k, 0 ≤ k < 1, such that the complex dilatation of ψ is of the form If Γ admits a Q-quasiconformal reflection Ψ , then ψ In particular, since Ψ is an extremal K-quasiconformal reflection, ψ is an extremal K-quasiconformal extension of γ to D. Moreover, by (3.1), the complex dilatation of ψ satisfies i.e. ψ is a regular Teichmüller mapping.From [24, Theorems 4.5.2 and 4.4.2] it follows that λ is an eigenvalue of γ.Therefore, Theorem B is deeply related to the following theorem.
Conversely, suppose that the property (ii) holds.Then by [23, Theorem 2.1 (⇒)] we see that the equality (3.12) holds for a certain f ∈ H such that f H = 1.The same conclusion can be deduced from [23, Theorem a.e. in D.
However, the proof of this statement exceeds the scope of this paper and will be published elsewhere.
Remark 3.3.The smallest positive eigenvalue λ of γ in Theorem 3.1 is strictly related to a number of important constants like, e.g.: the Schober constant, the Grunsky-Kühnau constant, as well as the supremum norms of the Neumann-Poincaré operator, the Hilbert transformation and the operator A γ .For the detailed exposition of this topic the reader is referred to [24,Sect. 4.4].By [24, Theorem 4.4.2], the set Λ * (Γ) of all eigenvalues of a quasicircle Γ ⊂ C coincides with the set Λ * γ , where γ is a welding homeomorphism of Γ.Therefore, Theorem 3.1 is closely related to various results involving these constants obtained by Kühnau ([15], [16], [17]), Schiffer ([29]) and Krushkal ([6], [7], [8,Sect. 2]); see also the survey article by Krushkal [9,p. 528] and the references given there.This relationship provides a motivation for the further study of the inequalities (1.6) in Theorem A.

Theorem B .
Let Γ ⊂ C be a quasicircle and let F : cl(Ω) → C and F * : cl(Ω * ) → C be continuous and locally univalent functions on cl(Ω) and cl(Ω * ), and analytic in the complementary domains Ω and Ω * ∞ of Γ, respectively.Assume that both the functions F and F * have finite Dirichlet integrals on Ω and Ω * , respectively.If the equalities (0.3) hold on Γ with a real constant λ, then |λ| > 1, Γ admits a unique extremal Kquasiconformal (i.e. with the smallest maximal dilatation K) reflection Ψ with K = (|λ| + 1)/(|λ| − 1), and the following equality holds (3.1) Since the kernel function is holomorphic with respect to z, S[f ] is a holomorphic function.Moreover, by (3.2) and (1.1) we see thatP[f ] = Re S[f ] as f ∈ Re L 1 (T),and hence Im S[f ] is a harmonic conjugate function to P[f ] and Im S[f ](0) = 0. Therefore, Re L 1 (T) f → Tr[Im S[f ]] is called the harmonic conjugation operator ; cf.e.g.[4, Sect.III.1].Since D[Re F ] = D[Im F ] for any holomorphic function F in D, we conclude from the definition of the space (H, • H ) that the operator A, defined by . in D. Under the assumptions of Theorem B, let H : cl(D) → cl(Ω) and H * : cl(D * ) → cl(Ω * ) be continuous mappings and conformal in D and D * , respectively.Then γ := H −1 * •H is a sense-preserving homeomorphic self-mapping of T. Let be the mapping defined by (z) := 1/z as z ∈ C \ {0} and (0) := ∞ , (∞) := 0 .