Articles on Potential Theory, Functional Analysis and Hankel Forms

Sammanfattning: Paper I: Perfekt, K.-M. and Putinar, M., Spectral bounds for the Neumann-Poincaré operator on planar domains with corners, to appear in J. Anal. Math., (2012). The boundary double layer potential, or the Neumann-Poincaré 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 Neumann-Poincaré operator is realized as a singular integral transform bearing similarities to the Beurling-Ahlfors transform in 2D. As an application, bounds for the spectrum of the Neumann-Poincaré operator are derived from recent results in quasi-conformal 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. 1-17 (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_K-spaces, weighted spaces, Lipschitz-Hölder 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. 4435-4448. We show that for a fixed operator-valued analytic function g as its symbol, the boundedness of a bilinear Hankel-type form, defined on appropriate cartesian products of dual weighted Dirichlet spaces of Schatten class-valued functions, is equivalent to corresponding Carleson embedding estimates.