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Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues

100%

Krushkal S.

Open Mathematics

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2007

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tom 5

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nr 3

551-580

EN

The Grunsky and Teichmüller norms ϰ(f) and k(f) of a holomorphic univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to $$\widehat{\mathbb{C}}$$ are related by ϰ(f) ≤ k(f). In 1985, Jürgen Moser conjectured that any univalent function in the disk Δ* = {z: |z| > 1} can be approximated locally uniformly by functions with ϰ(f) < k(f). This conjecture has been recently proved by R. Kühnau and the author. In this paper, we prove that approximation is possible in a stronger sense, namely, in the norm on the space of Schwarzian derivatives. Applications of this result to Fredholm eigenvalues are given. We also solve the old Kühnau problem on an exact lower bound in the inverse inequality estimating k(f) by ϰ(f), and in the related Ahlfors inequality.

2

The harmonic and quasiconformal extension operators

80%

Partyka D., Sakan K.-I., Zając J.

Banach Center Publications

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1999

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tom 48

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nr 1

141-177

EN

Different aspects of the boundary value problem for quasiconformal mappings and Teichmüller spaces are expressed in a unified form by the use of the trace and extension operators. Moreover, some new results on harmonic and quasiconformal extensions are included.

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The generalized Neumann-Poincaré operator and its spectrum

64%

Partyka D.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Introduction..........................................................................................................................................................................5 Preliminaries. Complex harmonic functions..........................................................................................................................7 I. Spectral values and eigenvalues of a Jordan curve........................................................................................................19 1.1. On a boundary integral..............................................................................................................................................20 1.2. The generalized Cauchy singular integral operator $C_𝕍$.......................................................................................23 1.3. The Hilbert transformation $T_Ω$.............................................................................................................................28 1.4. The boundary space Ḣ²(∂Ω)......................................................................................................................................31 1.5. The generalized Neumann-Poincaré operator $N_𝕍$...............................................................................................36 II. Quasisymmetric automorphisms of the unit circle...........................................................................................................41 2.1. The Douady-Earle extension $E_γ$..........................................................................................................................42 2.2. On an approximation of the Hersch-Pfluger distortion function $Φ_K$......................................................................46 2.3. On the maximal dilatation of the Douady-Earle extension..........................................................................................48 2.4. The Hilbert space H...................................................................................................................................................54 2.5. The linear operator $B_γ$.........................................................................................................................................60 III. The generalized harmonic conjugation operator............................................................................................................64 3.1. The generalized harmonic conjugation operator $A_γ$.............................................................................................64 3.2. Spectral values and eigenvalues of a quasisymmetric automorphism of the unit circle..............................................73 3.3. The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle..............................................80 3.4. Limiting properties of spectral values and eigenvalues of a quasisymmetric automorphism of the unit circle............84 IV. Spectral values of a quasicircle.....................................................................................................................................90 4.1. Characterizations of the boundary space Ḣ²(∂Ω).......................................................................................................91 4.2. Spaces symmetric with respect to a Jordan curve.....................................................................................................93 4.3. Plemelj's formula for a quasicircle..............................................................................................................................96 4.4. The main spectral theorem for quasicircles.............................................................................................................103 4.5. Spectral values and eigenvalues of a quasicircle....................................................................................................108 Appendix. The inner completion of pseudo-normed spaces............................................................................................114 References......................................................................................................................................................................117 List of symbols.................................................................................................................................................................122 Index................................................................................................................................................................................124

Ograniczanie wyników

1 Banach Center Publications

1 Open Mathematics

2 Partyka D.

1 Krushkal S.

1 Sakan K.-I.

1 Zając J.

1 2007

1 1999

1 1997

Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues

The harmonic and quasiconformal extension operators

The generalized Neumann-Poincaré operator and its spectrum