On the polarizability and capacitance of the cube

An efficient integral equation based solver is constructed for the electrostatic problem on domains with cuboidal inclusions. It can be used to compute the polarizability of a dielectric cube in a dielectric background medium at virtually every permittivity ratio for which it exists. For example, polarizabilities accurate to between five and ten digits are obtained (as complex limits) for negative permittivity ratios in minutes on a standard workstation. In passing, the capacitance of the unit cube is determined with unprecedented accuracy. With full rigor, we develop a natural mathematical framework suited for the study of the polarizability of Lipschitz domains. Several aspects of polarizabilities and their representing measures are clarified, including limiting behavior both when approaching the support of the measure and when deforming smooth domains into a non-smooth domain. The success of the mathematical theory is achieved through symmetrization arguments for layer potentials.Comment: 33 pages, 7 figure

On M-ideals and o-O type spaces

We consider pairs of Banach spaces (M0,M)(M0,M) such that M0M0 is defined in terms of a little-oo condition, and MM is defined by the corresponding big-OO condition. The construction is general and pairs include function spaces of vanishing and bounded mean oscillation, vanishing weighted and weighted spaces of functions or their derivatives, Möbius invariant spaces of analytic functions, Lipschitz-Hölder spaces, etc. It has previously been shown that the bidual M∗∗0M0∗∗ of M0M0 is isometrically isomorphic with MM. The main result of this paper is that M0M0 is an M-ideal in MM. This has several useful consequences: M0M0 has Pełczýnskis properties (u) and (V), M0M0 is proximinal in MM, and M∗0M0∗ is a strongly unique predual of MM, while M0M0 itself never is a strongly unique predual

Nehari's theorem for convex domain Hankel and Toeplitz operators in several variables

We prove Nehari's theorem for integral Hankel and Toeplitz operators on simple convex polytopes in several variables. A special case of the theorem, generalizing the boundedness criterion of the Hankel and Toeplitz operators on the Paley--Wiener space, reads as follows. Let Ξ = (0, 1)d be a d-dimensional cube, and for a distribution f on 2Ξ, consider the Hankel operator Γf (g)(x) = ʃΞ f(x + y)g(y)dy, x ∈ Ξ Then Γf extends to a bounded operator on L2(Ξ) if and only if there is a bounded function b on Rd whose Fourier transform coincides with f on 2Ξ. This special case has an immediate application in matrix extension theory: every finite multi-level block Toeplitz matrix can be boundedly extended to an infinite multi-level block Toeplitz matrix. In particular, block Toeplitz operators with blocks which are themselves Toeplitz, can be extended to bounded infinite block Toeplitz operators with Toeplitz blocks

Rectangular summation of multiple Fourier series and multi-parametric capacity

We consider the class of multiple Fourier series associated with functions in the Dirichlet space of the polydisc. We prove that every such series is summable with respect to unrestricted rectangular partial sums, everywhere except for a set of zero multi-parametric logarithmic capacity. Conversely, given a compact set in the torus of zero capacity, we construct a Fourier series in the class which diverges on this set, in the sense of Pringsheim. We also prove that the multi-parametric logarithmic capacity characterizes the exceptional sets for the radial variation and radial limits of Dirichlet space functions. As a by-product of the methods of proof, the results also hold in the vector-valued setting

A mean counting function for Dirichlet series and compact composition operators

We introduce a mean counting function for Dirichlet series, which plays the same role in the function theory of Hardy spaces of Dirichlet series as the Nevanlinna counting function does in the classical theory. The existence of the mean counting function is related to Jessen and Tornehave's resolution of the Lagrange mean motion problem. We use the mean counting function to describe all compact composition operators with Dirichlet series symbols on the Hardy--Hilbert space of Dirichlet series, thus resolving a problem which has been open since the bounded composition operators were described by Gordon and Hedenmalm. The main result is that such a composition operator is compact if and only if the mean counting function of its symbol satisfies a decay condition at the boundary of a half-plane.Comment: This paper has been accepted for publication in Advances in Mathematic

Plasmonic eigenvalue problem for corners: limiting absorption principle and absolute continuity in the essential spectrum

We consider the plasmonic eigenvalue problem for a general 2D domain with a curvilinear corner, studying the spectral theory of the Neumann--Poincare operator of the boundary. A limiting absorption principle is proved, valid when the spectral parameter approaches the essential spectrum. Putting the principle into use, it is proved that the corner produces absolutely continuous spectrum of multiplicity 1. The embedded eigenvalues are discrete. In particular, there is no singular continuous spectrum