The Dirichlet space D is the space of all analytic functions f on the open unit disc D such that f ′ is square integrable with respect to two-dimensional Lebesgue measure. In this paper we prove that the invariant subspaces of the Dirichlet shift are in 1-1 correspondence with the kernels of the Dirichlet-Hankel operators. We then apply this result to obtain information about the invariant subs...

In this paper we explore two constructions of the same family of metric measure spaces. The first construction was introduced by Laakso in 2000 where he used it as an example that Poincaré inequalities can hold on spaces of arbitrary Hausdorff dimension. This was proved using minimal generalized upper gradients. Following Cheeger’s work these upper gradients can be used to define a Sobolev spac...

[C] A. Selberg, Harmonic analysis and discon-tinuous groups in weakly symmetric Rieman-nian spaces with applications to Dirichlet series ,

This is a survey on reverse Carleson measures for various Hilbert spaces of analytic functions. These spaces include the Hardy, Bergman, certain harmonically weighted Dirichlet, Paley-Wiener, Fock, model (backward shift invariant), and de Branges-Rovnyak spaces. The reverse Carleson measure for backward shift invariant subspaces in the non-Hilbert situation is new.

[C] A. Selberg, Harmonic analysis and discon-tinuous groups in weakly symmetric Rieman-nian spaces with applications to Dirichlet series ,

Potential spaces and Dirichlet forms associated with Lévy processes subordinate to Brownian motion in R with generator f(−Δ) are investigated. Estimates for the related Rieszand Bessel-type kernels of order s are derived which include the classical case f(r) = r with 0 < α < 2 corresponding to α-stable Lévy processes. For general (tame) Bernstein functions f potential representations of the tra...

We prove that any Hunt process on a Hausdorr topological space associated with a Dirichlet form can be approximated by a Markov chain in a canonical way. This also gives a new and \more explicit" proof for the existence of Hunt processes associated with strictly quasi-regular Dirichlet forms on general state spaces.

We study the boundary behavior of functions in the Hardy spaces H p for ordinary Dirichlet series. Our main result, answering a question of H. Hedenmalm, shows that the classical F. Carlson theorem on integral means does not extend to the imaginary axis for functions in H ∞, i.e., for ordinary Dirichlet series in H∞ of the right half-plane. We discuss an important embedding problem for H , the ...

The main topic of this work is the definition and investigation of a nonlinear energy for maps with values in trees and graphs and the analysis of the corresponding nonlinear Dirichlet problem. The nonlinear energy is defined using a semigroup approach based on Markov kernels and the nonlinear Dirichlet problem is given as a minimizing problem of the nonlinear energy. Conditions for the existen...