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The generalized Neumann-Poincaré operator and its spectrum

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Partyka D.

Instytut Matematyczny Polskiej Akademii Nauk

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

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1 Partyka D.

1 1997

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The generalized Neumann-Poincaré operator and its spectrum