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The local Dirichlet integral and applications

Omar El-Fallah, Karim Kellay, Houssame Mahzouli

TL;DR

The paper develops a comprehensive framework for the local Dirichlet integral in distance-function models within harmonic Dirichlet spaces $\mathcal{D}(\mu)$. It derives sharp asymptotics for $\mathcal{D}_\zeta(f_{\omega,E})$ under $\mathcal{L}_1$ and $\mathcal{L}_2$-set hypotheses on $E$, connecting growth rules to multiplier membership and cyclicity. By leveraging Carleson-measure characterizations (ARS) and Richter–Sundberg formulas, it provides explicit criteria for when distance functions belong to $\mathcal{D}$ and to the multiplier algebra $\mathcal{M}(\mathcal{D})$, with detailed Cantor-set examples. The work also advances the understanding of cyclicity and polarity in $\mathcal{D}(\mu)$, offering sufficient conditions for polar sets and proving cyclicity results in broad geometric settings, thereby extending Brown–Shields-type phenomena beyond the classical Dirichlet space. Overall, the results yield practical tools to identify multipliers, cyclic vectors, and polar sets in variable-parameter Dirichlet spaces, with implications for interpolation and invariant-subspace theory.

Abstract

We study the local Dirichlet integral of distance functions and their behavior within the harmonic Dirichlet space. We provide estimates for the local Dirichlet integral of distance functions, which allow us to study their membership in the algebra of multipliers of the Dirichlet space. We give sufficient condition for a closed subset of the unit circle to be polar and we also examine cyclicity in the harmonic Dirichlet spaces.

The local Dirichlet integral and applications

TL;DR

The paper develops a comprehensive framework for the local Dirichlet integral in distance-function models within harmonic Dirichlet spaces . It derives sharp asymptotics for under and -set hypotheses on , connecting growth rules to multiplier membership and cyclicity. By leveraging Carleson-measure characterizations (ARS) and Richter–Sundberg formulas, it provides explicit criteria for when distance functions belong to and to the multiplier algebra , with detailed Cantor-set examples. The work also advances the understanding of cyclicity and polarity in , offering sufficient conditions for polar sets and proving cyclicity results in broad geometric settings, thereby extending Brown–Shields-type phenomena beyond the classical Dirichlet space. Overall, the results yield practical tools to identify multipliers, cyclic vectors, and polar sets in variable-parameter Dirichlet spaces, with implications for interpolation and invariant-subspace theory.

Abstract

We study the local Dirichlet integral of distance functions and their behavior within the harmonic Dirichlet space. We provide estimates for the local Dirichlet integral of distance functions, which allow us to study their membership in the algebra of multipliers of the Dirichlet space. We give sufficient condition for a closed subset of the unit circle to be polar and we also examine cyclicity in the harmonic Dirichlet spaces.
Paper Structure (17 sections, 30 theorems, 149 equations)

This paper contains 17 sections, 30 theorems, 149 equations.

Table of Contents

  1. Introduction and main results
  2. Estimates of the local Dirichlet integral of distance functions
  3. Multipliers in the Dirichlet space
  4. Cyclicity in Dirichlet spaces
  5. A final word on notation.
  6. Preliminary estimates of the local Dirichlet integral
  7. First estimates of the local Dirichlet integral
  8. Minimal growth and $\mathcal{L}$-sets
  9. Remarks and examples
  10. Proof of Theorem \ref{['estimationlocal']}
  11. Membership to $\mathcal{D}(\mu)$
  12. Multipliers in the Classical Dirichlet space
  13. Multipliers from the Local Dirichlet Integral
  14. Remarks and Examples
  15. Approximation in Dirichlet spaces
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\omega$ be an increasing function such that ${\omega(t^\gamma)}$ is concave for some $\gamma >2$, and ${t \omega'(t)}/{\omega(t)}$ is increasing. Let $E$ be a Carleson set of $\mathbb{T}$. Then

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 38 more