Książka - szczegóły

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