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On the cyclic behavior of singular inner functions in Besov and sequence spaces

Alberto Dayan, Daniel Seco

Abstract

We show the existence of singular inner functions that are cyclic in some Besov-type spaces of analytic functions over the unit disc. Our sufficient condition is stated only in terms of the modulus of smoothness of the underlying measure. Such singular inner functions are cyclic also in the space $\ell^p_A$ of holomorphic functions with coefficients in $\ell^p$. This can only happen for measures that place no mass on any Beurling-Carleson set.

On the cyclic behavior of singular inner functions in Besov and sequence spaces

Abstract

We show the existence of singular inner functions that are cyclic in some Besov-type spaces of analytic functions over the unit disc. Our sufficient condition is stated only in terms of the modulus of smoothness of the underlying measure. Such singular inner functions are cyclic also in the space of holomorphic functions with coefficients in . This can only happen for measures that place no mass on any Beurling-Carleson set.

Paper Structure

This paper contains 6 sections, 16 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Dyadic Martingales and Singular Measures
  3. Multipliers of $D^p_{p-1}$
  4. Proof of Theorem \ref{['theo:main']}
  5. Outer Functions in $D^p_{p-1}$
  6. Logarithmic Conditions

Key Result

Theorem 1.1

Let $S_\mu$ be a singular inner function such that for some $C>0$. Then $S_\mu$ is cyclic in $D^p_{p-1}$ (and hence in $\ell^p_A$) for all $p>2$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • Definition 1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 3.1
  • ...and 21 more