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INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1995

THE SMALLEST POSITIVE EIGENVALUE OF A QUASISYMMETRIC AUTOMORPHISM

OF THE UNIT CIRCLE

D A R I U S Z P A R T Y K A

Department of Mathematics, Maria Curie-Sk lodowska University Pl. M. Curie-Sk lodowskiej 1, 20-031 Lublin, Poland

E-mail: partyka@plumcs11.bitnet

Abstract. This paper provides sufficient conditions on a quasisymmetric automorphism γ of the unit circle which guarantee the existence of the smallest positive eigenvalue of γ. They are expressed by means of a regular quasiconformal Teichm¨ uller self-mapping ϕ of the unit disc ∆. In particular, the norm of the generalized harmonic conjugation operator A

γ

: H → H is determined by the maximal dilatation of ϕ. A characterization of all eigenvalues of a quasisymmetric automorphism γ in terms of the smallest positive eigenvalue of some other quasisymmetric automorphism σ is given.

1. Introduction. Let us denote by Q

T

(K), 1 ≤ K < ∞, the class of all homeomorphic self-mappings of the unit circle T = {z ∈ C : |z| = 1} which admit a K-quasiconformal extension to the unit disc ∆ = {z ∈ C : |z| < 1} and let Q

T

= S

1≤K<∞

Q

T

(K). For any homeomorphism γ ∈ Q

T

we set K(γ) = inf{K : γ ∈ Q

T

(K)}. Due to J. G. Krzy˙z, cf. [K], the class Q

T

coincides with the class of all quasisymmetric automorphisms of the unit circle T, i.e. all sense-preserving homeomorphisms γ : T → T satisfying

k

−1

≤ |γ(I

1

)|/|γ(I

2

)| ≤ k

for each pair of adjacent closed arcs I

1

, I

2

⊂ T of equal length: 0 < |I

1

| =

|I

2

| ≤ π where the constant k depends on γ only. Let us denote by L

pT

, 1 ≤ p < ∞, the space of all functions f : T → R, p-integrable on T, i.e. kf k

p

=

1991 Mathematics Subject Classification: 30C62, 45C05.

Key words and phrases: quasiconformal mappings, Teichm¨ uller mappings, harmonic conju- gation operator, eigenvalues and spectral values of a linear operator, quasisymmetric automor- phisms.

Research supported in part by KBN Grant #PB2-11-70-9101.

The paper is in final form and no version of it will be published elsewhere.

[303]

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( R

T

|f (z)|

p

|dz|)

1/p

< ∞ and let L

T

= {f ∈ L

1T

: kf k

= sup ess

z∈T

|f (z)| < ∞}.

The space L

2T

is a real Hilbert space with the inner product (f, g) = R

T

f (z)g(z)|dz|, f, g ∈ L

2T

.

With any function f ∈ L

1T

we can associate an analytic function f

: ∆ → C given by the formula

f

(z) = 1 2π

R

T

f (u) u + z u − z |du|

= 1 2π

R

T

f (u)|du| + 1 π

X

n=1



R

T

f (u)u

n

|du|



z

n

, z ∈ ∆.

The space H = {f ∈ L

1T

: R

|f

0

|

2

dS < ∞ and f

(0) = 0}, where f

0

= (f

)

0

, equipped with the inner product (f, g)

H

= Re R

f

0

g

0

dS, f, g ∈ H, is a real Hilbert space, isometric with the space e L

2T

= {f ∈ L

2T

: f

(0) = 0}, cf. [P1, Theorem 1.2]. In the paper [P2] a linear homeomorphism A

γ

of the Hilbert space H onto itself was associated with every quasisymmetric automorphism γ ∈ Q

T

. If γ ∈ Q

T

is sufficiently regular then the operator A

γ

has a nice form, cf. [P2, Theorem 1.4], given by means of a singular integral

A

γ

(f )(z) = 1

π Re P.V. R

T

f (u)

γ(z) − γ(u) dγ(u) − a

γ

(f ) for a.e. z ∈ T and f ∈ H where

a

γ

(f ) = 1 2π

R

T

 1

π Re P.V. R

T

f (u)

γ(z) − γ(u) dγ(u)



|dz|

is a normalization constant. If γ(z) = z, z ∈ T, then A

γ

becomes the usual harmonic conjugation operator A, see [G]. Moreover, A

2γ

= −I for any γ ∈ Q

T

, where I is the identity operator. Thus A

γ

may be called a generalized harmonic conjugation operator. For basic properties of the operator A

γ

we refer to [P2]. Now we quote two properties essential for our considerations. Namely, AA

γ

: H → H is a symmetric operator and

(1.1) A

γ

= B

γ

AB

γ−1

where B

γ

is a linear homeomorphism of the space (H, k · k

H

) onto itself such that (1.2) B

γ

(f ) = f ◦ γ − (f ◦ γ)

(0) on T

for every continuous function f ∈ H. This shows that the operator A

γ

is related

to the Neumann–Poincar´ e integral operator of a Jordan curve Γ. More precisely,

eigenvalues of the Neumann–Poincar´ e kernel k, cf. [BS], [S], correspond to eigen-

values of the symmetric operator AA

γ

: H → H where γ is a welding homeo-

morphism of a sufficiently smooth Jordan curve Γ, cf. [P1], [KP]. This justifies

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introducing the notion of an eigenvalue and a spectral value of a quasisymmetric automorphism of the unit circle, by means of the spectrum of the operator AA

γ

, cf. [P3], or equivalently, by means of the spectrum of the operator R

γ

, cf. [P1], [KP], because R

γ

= I + AA

γ

, cf. [P2, (2.4)]. For the reader’s convenience we quote Definition 1.1. A real number λ is said to be an eigenvalue of a quasisym- metric automorphism γ ∈ Q

T

if there exists a function f ∈ H with the norm kf k

H

= 1 such that

(1.3) (λ + 1)A(f ) = (λ − 1)A

γ

(f ).

The function f is said to be an eigenfunction of γ associated with the eigenvalue λ.

The set of all eigenvalues of γ ∈ Q

T

is denoted by Λ

γ

.

Definition 1.2. A real number λ is said to be a spectral value or an ap- proximate eigenvalue of a quasisymmetric automorphism γ ∈ Q

T

if there exist functions f

n

∈ H, kf

n

k

H

= 1, n = 1, 2, . . . such that

(1.4) k(λ + 1)A(f

n

) − (λ − 1)A

γ

(f

n

)k

H

→ 0 as n → ∞.

The set of all spectral values of γ ∈ Q

T

is denoted by Λ

γ

. From [P2, Theo- rem 2.2] we are able to infer the following basic properties of the spectra Λ

γ

and Λ

γ

:

(i) Λ

γ

= ∅ iff γ = Q

T

(1) ; (ii) Λ

γ

⊂ Λ

γ

;

(iii) if λ ∈ Λ

γ

then |λ| ≥

K(γ)+1K(γ)−1

;

(iv) for every ν, η ∈ Q

T

(1) Λ

γ

= Λ

ν◦γ◦η

and Λ

γ

= Λ

ν◦γ◦η

; (v) Λ

γ

= Λ

γ−1

= Λ

¯iT◦γ◦¯iT

and Λ

γ

= Λ

γ−1

= Λ

¯i

T◦γ◦¯iT

; (vi) if λ ∈ Λ

γ

then −λ ∈ Λ

γ

and if λ ∈ Λ

γ

then −λ ∈ Λ

γ

where i

T

(z) = z, z ∈ T. For the proof of these properties cf. [P3]. A natural question appears when is the inequality (iii) sharp, i.e. when

(1.5) inf{|λ| : λ ∈ Λ

γ

} = K(γ) + 1 K(γ) − 1 .

This is strictly related to the problem of determining the norm kA

γ

k which al-

ways does not exceed K(γ), cf. [P2, Theorem 1.3]. Namely, kA

γ

k = K(γ) iff the

equality (1.5) holds. In Section 2 of this paper we establish Theorem 2.2 giving

a sufficient condition on a quasisymmetric automorphism γ ∈ Q

T

which guaran-

tees the existence of the smallest positive eigenvalue λ ∈ Λ

γ

satisfying (1.5). In

particular, this implies the equality kA

γ

k = K(γ). As a consequence we obtain

Corollary 2.3 which characterizes every positive eigenvalue λ ∈ Λ

γ

as the smallest

positive eigenvalue of some other quasisymmetric automorphism σ ∈ Q

T

.

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2. Main results and proofs. In what follows we need the following Lemma 2.1 If ϕ ∈ Q

is a quasiconformal extension of a quasisymmetric automorphism γ ∈ Q

T

and there exist functions f, g ∈ H and a constant c ∈ C satisfying the equality

(2.1) Re g

(z) = Re f

◦ ϕ(z) + c, z ∈ ∆, then g = B

γ

(f ).

P r o o f. By [P1, Theorem 1.2] C

T

∩ H is a dense subset of the space (H, k · k

H

) where the class C

T

consists of all continuous real–valued functions on the unit circle T. Thus there exist functions f

n

, h

n

∈ C

T

∩ H, n ∈ N, approximating the functions f , g in (H, k · k

H

), respectively, i.e.

(2.2) kf

n

− f k

H

→ 0 and kh

n

− gk

H

→ 0 as n → ∞.

Since a harmonic function minimizes the Dirichlet integral within the class of real continuous functions on the closed unit disc ∆ with given boundary values and absolutely continuous on a.e. chord of ∆, parallel to the coordinate axes, setting g

n

= B

γ

(f

n

) = f

n

◦ γ − (f

n

◦ γ)

(0), n ∈ N, we obtain by (2.1) that for any n ∈ N (2.3) R

|(g

n

)

0

− g

0

|

2

dS

= lim

m→∞

R

|(g

n

)

0

− (h

m

)

0

|

2

dS = lim

m→∞

R

|((f

n

◦ γ)

− (h

m

)

)

0

|

2

dS

= 4 lim

m→∞

R

|∂ Re((f

n

◦ γ)

− (h

m

)

)|

2

dS

≤ 4 lim

m→∞

R

|∂ Re((f

n

)

◦ ϕ − (h

m

)

)|

2

dS = 4 R

|∂ Re((f

n

)

◦ ϕ − g

)|

2

dS

= 4 R

|∂ Re((f

n

)

◦ ϕ − f

◦ ϕ)|

2

dS = 4 R

|∂ Re((f

n

− f )

◦ ϕ)|

2

dS where

∂ = 1 2

 ∂

∂x − i ∂

∂y



, ∂ = 1 2

 ∂

∂x + i ∂

∂y



are formal derivatives. This and the K-quasi-invariance of the Dirichlet integral, cf. [A], lead to

4 R

|∂ Re((f

n

− f )

◦ ϕ)|

2

dS ≤ 4K(γ) R

|∂ Re(f

n

− f )

|

2

dS

= K(γ) R

|(f

n

− f )

0

|

2

dS

= K(γ)kf

n

− f k

2

H

→ 0 as n → ∞.

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Hence by (2.3)

(2.4) kB

γ

(f

n

) − gk

H

→ 0 as n → ∞.

On the other hand, by the continuity of the operator B

γ

: (H, k · k

H

) → (H, k · k

H

) and (2.2) we have

kB

γ

(f

n

) − B

γ

(f )k

H

→ 0 as n → ∞.

This together with (2.4) yields g = B

γ

(f ) which ends the proof.

The following main theorem involves the notion of a regular quasiconformal Teichm¨ uller mapping. We recall that a quasiconformal self-mapping ϕ of the unit disc ∆ is said to be a regular Teichm¨ uller mapping if there exists an analytic function ψ : ∆ → C and a constant k, 0 ≤ k < 1, such that the complex dilatation of ϕ has the form

(2.5) ∂ϕ

∂ϕ = k ψ

|ψ| .

Theorem 2.2. If f ∈ H, kf k

H

> 0, and ϕ is a regular quasiconformal Te- ichm¨ uller extension of an automorphism γ ∈ Q

T

to ∆ with the complex dilatation

(2.6) ∂ϕ

∂ϕ = − 1 λ

f

0

f

0

where λ, |λ| > 1, is a real constant then λ ∈ Λ

γ

and |λ| is the smallest positive eigenvalue of γ, i.e.

(2.7) |λ| = min{|µ| : µ ∈ Λ

γ

} = min{|µ| : µ ∈ Λ

γ

}.

Moreover ,

(2.8) kA

γ

k = K(γ) = |λ| + 1

|λ| − 1 and

(2.9) (λ + 1)A(f ) = (λ − 1)A

γ

(f ).

P r o o f. Let G be a complex function in the unit disc ∆ such that

(2.10) G ◦ ϕ = f

− λf

.

Differentiating both sides of this equality with respect to z and z we get the simultaneous equations

(∂G) ◦ ϕ∂ϕ + (∂G) ◦ ϕ∂ϕ = −λf

0

, (∂G) ◦ ϕ∂ϕ + (∂G) ◦ ϕ∂ϕ = f

0

.

Since ∂ϕ∂ϕ − ∂ϕ∂ϕ = ∂ϕ∂ϕ − ∂ϕ∂ϕ = |∂ϕ|

2

− |∂ϕ|

2

> 0 a.e. in ∆, (2.6) implies

that ∂G = 0 a.e. in ∆. This way G is an analytic function in ∆, cf. [A]. Moreover,

by the quasi–invariance of the Dirichlet integral we derive from the equality (2.10)

that

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R

|G

0

|

2

dS = 4 R

|∂ Re((f

− λf

) ◦ ϕ

−1

)|

2

dS

≤ 4K(ϕ

−1

)|1 − λ|

2

R

|∂ Re f

|

2

dS = K(ϕ)|1 − λ|

2

R

|f

0

|

2

dS < ∞.

Thus there exists a function g ∈ H such that G(z) = g

(z) + G(0), z ∈ ∆, and by the equality (2.10) we get on ∆

Re g

◦ ϕ + Re G(0) = (1 − λ) Re f

, Im g

◦ ϕ + Im G(0) = −(1 + λ) Im f

.

Hence, by the definition of the harmonic conjugation operator A and Lemma 2.1 we obtain

B

γ

(g) = (1 − λ)f and B

γ

(A(g)) = −(1 + λ)A(f ).

This gives by (1.1)

(1 + λ)A(f ) = (λ − 1)B

γ

AB

γ−1

(f ) = (λ − 1)A

γ

(f )

which proves the equality (2.9). This means that λ ∈ Λ

γ

. It follows from the assumption of Lemma 2.1 that ϕ is a K-quasiconformal regular Teichm¨ uller ex- tension of the automorphism γ on ∆ with K = (|λ| + 1)/(|λ| − 1). Thus

(2.11) kA

γ

k ≤ K(γ) ≤ K.

If λ < −1 then by the property (vi) |λ| = −λ ∈ Λ

γ

as well. Therefore there exists h ∈ H, khk

H

= 1, satisfying

(|λ| + 1)A(h) = (|λ| − 1)A

γ

(h).

Hence

kA

γ

(h)k

H

= kAA

γ

(h)k

H

= |λ| + 1

|λ| − 1 = K

because A

2

= −I and A is an isometry of the space (H, k · k

H

) onto itself, cf.

[P2, Theorem 1.3]. Thus kA

γ

k ≥ K. This together with (2.11) gives the equality (2.8) which yields the equality (2.7) because of the properties (ii) and (iii). This completes the proof.

R e m a r k. This result seems to be closely related to that in [K¨ u, p. 302].

The equality (2.8) states additionally that the mapping ϕ in Theorem 2.2 is the extremal quasiconformal extension of γ to the unit disc ∆, i.e. an extension with the smallest maximal dilatation. This way we have proved, by the way, Strebel’s theorem, cf. [St1], [St2], [L], in the case when the function ψ in (2.5) is a square of some analytic function, square integrable on ∆.

The smallest positive eigenvalue of a quasisymmetric automorphism γ of the unit circle plays a particularly important role among other eigenvalues of γ.

Namely, every positive eigenvalue λ ∈ Λ

γ

can be expressed as the smallest positive

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eigenvalue of some other quasisymmetric automorphism σ ∈ Q

T

. This interesting fact is the subject of the following corollary to Theorem 2.2.

Corollary 2.3. If λ ∈ Λ

γ

, λ > 0 is any eigenvalue of an automorphism γ ∈ Q

T

then there exists an automorphism σ ∈ Q

T

such that

(2.12) |λ| = min{|µ| : µ ∈ Λ

σ

} = min{|µ| : µ ∈ Λ

σ

} and

(2.13) K(σ) = kA

σ

k = λ + 1

λ − 1 .

Moreover , the automorphism σ ∈ Q

T

(K) admits a K(σ)-quasiconformal exten- sion ϕ to the unit disc ∆ with a complex dilatation

(2.14) ∂ϕ

∂ϕ = − 1 λ

f

0

f

0

where

(2.15) (λ + 1)A(f ) = (λ − 1)A

σ

(f ).

P r o o f. Assume λ ∈ Λ

γ

, λ > 0 is an eigenvalue of an automorphism γ ∈ Q

T

. Then there exists a function f ∈ H, kf k

H

= 1 satisfying (1.3). It follows from the mapping theorem, cf. [LV; p. 194], also [B1], [B2], [ LK] that there exists a solution ϕ of the Beltrami equation (2.14) being a K-quasiconformal self-mapping of ∆ with K = (λ + 1)/(λ − 1). It is well known that ϕ has a continuous extension to T as a quasisymmetric automorphism σ ∈ Q

T

, cf. [LV]. Then the assumptions of Theorem 2.2 are satisfied for the automorphism σ ∈ Q

T

which satisfies the equalities (2.12), (2.13) and (2.15). This ends the proof.

References

[A] L. V. A h l f o r s, Lectures on Quasiconformal Mappings, D. Van Nostrand, Princeton, 1966.

[BS] S. B e r g m a n and M. S c h i f f e r, Kernel functions and conformal mapping , Compositio Math. 8 (1951), 205–249.

[B1] B. B o j a r s k i, Homeomorphic solution of Beltrami systems, Dokl. Akad. Nauk SSSR 102 (1955), 661–664 (in Russian).

[B2] —, Generalized solutions of a system of differential equations of the first order and elliptic type with discontinuous coefficients, Mat. Sb. N.S. 43 (1957), 451–503 (in Russian).

[G] J. B. G a r n e t t, Bounded Analytic Functions, Academic Press, New York, 1981.

[K] J. G. K r z y ˙z, Quasicircles and harmonic measure, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), 19–24.

[KP] J. G. K r z y ˙z and D. P a r t y k a, Generalized Neumann–Poincar´ e operator, chord-arc curves and Fredholm eigenvalues, Complex Variables, Theory Appl. 21 (1993), 253–

263.

[K¨ u] R. K ¨ u h n a u, Wann sind die Grunskyschen Koeffizientenbedingungen hinreichend f¨ ur

Q-quasikonforme Fortsetzbarkeit? , Comment. Math. Helv. 61 (1986), 290–307.

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[L] O. L e h t o, Univalent Functions and Teichm¨ uller Spaces, Graduate Texts in Math. 109 Springer, New York, 1987.

[LV] O. L e h t o and K. I. V i r t a n e n, Quasiconformal Mappings in the Plane, 2nd ed., Grundlehren Math. Wiss. 126, Springer, New York, 1973.

[ LK] J. L a w r y n o w i c z and J. G. K r z y ˙z, Quasiconformal Mappings in the Plane: Para- metrical Methods, Lecture Notes in Math. 978, Springer, Berlin, 1983.

[P1] D. P a r t y k a, Spectral values of a quasicircle, Complex Variables Theory Appl., to appear.

[P2] —, Generalized harmonic conjugation operator , Proceedings of the Fourth Finnish–

Polish Summer School in Complex Analysis at Jyv¨ askyl¨ a, Ber. Univ. Jyv¨ askyl¨ a Math.

Inst. 55 (1993), 143–155.

[P3] —, Spectral values and eigenvalues of a quasicircle, preprint.

[S] M. S c h i f f e r, The Fredholm eigenvalues of plane domains, Pacific J. Math. 7 (1957), 1187–1225.

[St1] K. S t r e b e l, Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreises I , Comment. Math. Helv. 36 (1962), 306–323.

[St2] —, Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheits-

kreises II , ibid. 39 (1964), 77–89.

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