Bounded Functions in Möbius Invariant Dirichlet Spaces

Bounded Analytic Functions in the Dirichlet Space

In this paper we study the Hilbert space of analytic functions with finite Dirichlet integral in a connected open set C2 in the complex plane. We show that every such function can be represented as a quotient of two bounded analytic functions, each of which has a finite Dirichlet integral. This has several consequences for the structure of invariant subspaces of the algebra of multiplication op...

Invariant Percolation and Harmonic Dirichlet Functions

The main goal of this paper is to answer question 1.10 and settle conjecture 1.11 of BenjaminiLyons-Schramm [BLS99] relating harmonic Dirichlet functions on a graph to those on the infinite clusters in the uniqueness phase of Bernoulli percolation. We extend the result to more general invariant percolations, including the Random-Cluster model. We prove the existence of the nonuniqueness phase f...

Möbius-invariant natural neighbor interpolation

Shift-invariant spaces from rotation-covariant functions

We consider shift-invariant multiresolution spaces generated by rotation-covariant functions ρ in R2. To construct corresponding scaling and wavelet functions, ρ has to be localized with an appropriate multiplier, such that the localized version is an element of L2(R2). We consider several classes of multipliers and show a new method to improve regularity and decay properties of the correspondi...

Compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions

We characterize compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions on metric spaces, not necessarily compact, with Lipschitz involutions and determine their spectra.