Abstract
Paper I: Perfekt, K.M. and Putinar, M., Spectral bounds for the NeumannPoincaré operator on planar domains with corners, to appear in J. Anal. Math., (2012).
The boundary double layer potential, or the NeumannPoincaré operator, is studied on the Sobolev space of order $1/2$ along the boundary, coinciding with the space of charges giving rise to double layer potentials with finite energy in the whole space. Poincaré's program of studying the spectrum of the boundary double layer potential is developed in complete generality, on closed Lipschitz hypersurfaces in Euclidean space. Furthermore, the NeumannPoincaré operator is realized as a singular integral transform bearing similarities to the BeurlingAhlfors transform in 2D. As an application, bounds for the spectrum of the NeumannPoincaré operator are derived from recent results in quasiconformal mapping theory, in the case of planar curves with corners.
Paper II: Perfekt, K.M., Duality and distance formulas in spaces defined by means of oscillation, Arkiv för Matematik, (2012), pp. 117 (Online First).
For the classical space of functions with bounded mean oscillation, it is well known that VMO** = BMO and there are many characterizations of the distance from a function f in BMO to VMO. When considering the Bloch space, results in the same vein are available with respect to the little Bloch space. In this paper such duality results and distance formulas are obtained by pure functional analysis. Applications include general Möbius invariant spaces such as Q_Kspaces, weighted spaces, LipschitzHölder spaces and rectangular BMO of several variables.
Paper III: Aleman, A. and Perfekt, K.M., Hankel forms and embedding theorems in weighted Dirichlet spaces, Int. Math. Res. Not., 2012 (2012), pp. 44354448.
We show that for a fixed operatorvalued analytic function g as its symbol, the boundedness of a bilinear Hankeltype form, defined on appropriate cartesian products of dual weighted Dirichlet spaces of Schatten classvalued functions, is equivalent to corresponding Carleson embedding estimates.
The boundary double layer potential, or the NeumannPoincaré operator, is studied on the Sobolev space of order $1/2$ along the boundary, coinciding with the space of charges giving rise to double layer potentials with finite energy in the whole space. Poincaré's program of studying the spectrum of the boundary double layer potential is developed in complete generality, on closed Lipschitz hypersurfaces in Euclidean space. Furthermore, the NeumannPoincaré operator is realized as a singular integral transform bearing similarities to the BeurlingAhlfors transform in 2D. As an application, bounds for the spectrum of the NeumannPoincaré operator are derived from recent results in quasiconformal mapping theory, in the case of planar curves with corners.
Paper II: Perfekt, K.M., Duality and distance formulas in spaces defined by means of oscillation, Arkiv för Matematik, (2012), pp. 117 (Online First).
For the classical space of functions with bounded mean oscillation, it is well known that VMO** = BMO and there are many characterizations of the distance from a function f in BMO to VMO. When considering the Bloch space, results in the same vein are available with respect to the little Bloch space. In this paper such duality results and distance formulas are obtained by pure functional analysis. Applications include general Möbius invariant spaces such as Q_Kspaces, weighted spaces, LipschitzHölder spaces and rectangular BMO of several variables.
Paper III: Aleman, A. and Perfekt, K.M., Hankel forms and embedding theorems in weighted Dirichlet spaces, Int. Math. Res. Not., 2012 (2012), pp. 44354448.
We show that for a fixed operatorvalued analytic function g as its symbol, the boundedness of a bilinear Hankeltype form, defined on appropriate cartesian products of dual weighted Dirichlet spaces of Schatten classvalued functions, is equivalent to corresponding Carleson embedding estimates.
Original language  English 

Qualification  Doctor 
Awarding Institution 

Supervisors/Advisors 

Award date  2013 May 17 
Publisher  
Publication status  Published  2013 
Bibliographical note
Defence detailsDate: 20130517
Time: 13:15
Place: Lund, Matematikcentrum, Sölvegatan 18, sal MH:C
External reviewer(s)
Name: Rochberg, Richard
Title: Professor
Affiliation: Washington University, St. Louis, USA

Subject classification (UKÄ)
 Mathematics
Free keywords
 NeumannPoincare operator
 layer potential
 Lipschitz
 spectrum
 bidual
 predual
 distance
 BMO
 Hankel form
 Carleson embedding
 vectorvalued.