A Beurling-Carleson set which is a uniqueness set for a given weighted space of analytic functions

Abstract:

Let $p = \big ( p(n) \big )_{n \geq 0}$ be a sequence of positive real numbers. We define $B_{p}$ as the space of functions $f$ which are analytic in the unit disc $\mathbb {D}$, continuous on $\overline {\mathbb {D}}$ and such that \[ \big \| f \big \|_{p} := \sum _{n=0}^{+\infty } | \hat {f}(n) | p(n) < +\infty , \] where $\hat {f}(n)$ is the $n^{\textrm {th}}$ Fourier coefficient of the restriction of $f$ to the unit circle $\mathbb {T}$. Let $E$ be a closed subset of $\mathbb {T}$. We say that $E$ is a Beurling-Carleson set if \[ \int _{0}^{2\pi } \log ^{+} \frac {1}{d(e^{it},E)} \mathrm {d} t < +\infty , \] where $d(e^{it},E)$ denotes the distance between $e^{it}$ and $E$. In 1980, A. Atzmon asked whether there exists a sequence $p$ of positive real numbers such that $\displaystyle \lim _{n \rightarrow +\infty } \frac {p(n)}{n^{k}} = +\infty$ for all $k \geq 0$ and that has the following property: for every Beurling-Carleson set $E$, there exists a non-zero function in $B_{p}$ that vanishes on $E$. In this note, we give a negative answer to this question.