Extremal functions and invariant subspaces in Dirichlet spaces

Abstract

Let μ be a positive finite Borel measure on the unit circle of the complex plane, and let D(μ) be the associated Dirichlet space. The Beurling-Deny capacity associated with D(μ) is denoted by cμ. Brown-Shields conjecture for D(μ) says that a function f∈D(μ) is cyclic for D(μ), meaning that {pf:pis a polynomial} is dense in D(μ), if and only if f is outer and the boundary zeros set of f is of cμ− capacity zero. It was recently proved, using Banach algebras techniques, that Brown-Shields conjecture is true for D(μ) whenever the support of μ is countable. In this paper, we produce a new class of measures for which Brown-Shields conjecture holds. This class includes the canonical measures on Cantor sets. For these measures, we also give an explicit description of all invariant subspaces for the shift operator, i.e. the operator of multiplication by the independent variable. Our method is based on a study of the behavior of extremal functions for Dirichlet spaces. We prove that if the support of μ is a Carleson set, then every outer extremal function φ can be explicitly recovered from the restriction of its boundary values |φ⁎| on the support of μ. This allows us to give estimates of φ for a large class of measures.