1 Introduction

Let $w\in L^{1}(\mathbb D)$ be a nonnegative superharmonic function in the unit disc $\mathbb D=\{z\in \mathbb C:|z|<1\}$ . The weighted Dirichlet space $D_{w}$ is the set of holomorphic functions f in $\mathbb D$ , for which

$$ \begin{align*}\int_{\mathbb D}|f'(z)|^{2}w(z)dA(z)<+\infty, \end{align*} $$

where A denotes the area measure in $\mathbb C$ . For $w\equiv 1,$ we obtain the classical Dirichlet space $\mathcal {D}$ and for $w_{p}(z)=(1-|z|^{2})^{p}$ , $p\in (0,1)$ , we obtain the well-known Dirichlet spaces $D_{p}$ having radial weights. Another well-studied type of weighted Dirichlet spaces was initiated by Richter [Reference Richter12] and corresponds to the case that w is a positive harmonic function on $\mathbb D$ . The superharmonically weighted Dirichlet spaces $D_{w}$ have been introduced and studied by Aleman [Reference Aleman1]. We refer the reader to [Reference Arcozzi, Rochberg, Sawyer and Wick2, Reference El-Fallah, Kellay, Mashreghi and Ransford9] and the references therein, for more information on Dirichlet spaces.

A sequence $(z_{n})\subset \mathbb D$ is said to be a zero set for $D_{w}$ if there exists $f\in D_{w}$ such that f is not identically zero and $f(z)=0$ if and only if $z=z_{n}$ for some $n\in \mathbb N$ . It is well known (see, e.g., [Reference Aleman1, Reference Bao, Göğüş and Pouliasis5]) that $D_{w}$ is contained in the Hardy–Hilbert space $H^{2}$ of holomorphic functions in $\mathbb D$ satisfying

$$ \begin{align*}\int_{\mathbb D}|f'(z)|^{2}\log\frac{1}{|z|}dA(z)<+\infty.\end{align*} $$

It follows (see, e.g., [Reference Duren8]) that every zero set $(z_{n})$ for $D_{w}$ is a Blaschke sequence; that is, $(z_{n})$ satisfies the Blaschke condition

$$ \begin{align*}\sum_{n=1}^{\infty}(1-|z_{n}|)<+\infty.\end{align*} $$

It is well known that there exist nonnegative superharmonic weights $w\in L^{1}(\mathbb D)$ , for which the family of zero sets of $D_{w}$ is strictly contained in the family of Blaschke sequences. For example (see [Reference Caughran7]), for the classical Dirichlet space $\mathcal {D}$ , there exists a Blaschke sequence $(z_{n})\subset \mathbb D$ , which converges “very tangentially” at some point in the unit circle, that is not a zero set for $\mathcal {D}$ . The same result holds for the Dirichlet spaces $D_{p}$ , $p\in (0,1)$ (see [Reference Pau and Peláez11, p. 1984] and the references therein).

In [Reference Bahajji-El Idrissi and El-Fallah4], Bahajji-El Idrissi and El-Fallah posed the following problem.

Earlier, in [Reference Guillot10], Guillot studied the above problem when w is a positive harmonic function. In particular, he considered weights that are Poisson integrals

$$ \begin{align*}P_{\nu}(z)=\int_{\partial\mathbb D}\frac{1-|z|^{2}}{|\zeta-z|^{2}}d\nu(\zeta),\end{align*} $$

where $\nu $ is a finite positive Borel measure on the unit circle $\partial \mathbb D$ having the form

$$ \begin{align*}\nu=\sum_{n=1}^{\infty}c_{n}\delta_{\zeta_{n}},\end{align*} $$

where $c_{n}>0$ , $(\zeta _{n})\subset \partial \mathbb D,$ and $\delta _{a}$ is the Dirac measure at $a\in \mathbb C$ . The Dirichlet space corresponding to $w=P_{\nu }$ is denoted by $D_{\nu }$ . Guillot proved that every Blaschke sequence is a zero set for $D_{\nu }$ if and only if

(1.1) $$ \begin{align} \int_{\partial\mathbb D}\log\left(\int_{\partial\mathbb D}\frac{1}{|\zeta-\eta|^{2}}d\nu(\zeta)\right)ds(\eta)<+\infty, \end{align} $$

where s is the normalized Lebesgue measure on $\partial \mathbb D$ . Recently, in [Reference Bao, Mashreghi, Wang and Wulan6], Bao, Mashreghi, Wang, and Wulan completely solved Problem 1.1 for harmonic weights, proving that the above equivalence holds for every finite positive Borel measure $\nu $ . Moreover, they considered weights that are Green potentials

$$ \begin{align*}U_{\mu}(z)=\int_{\mathbb D}\log\Big|\frac{1-\overline{w}z}{z-w}\Big|d\mu(w),\qquad z\in\mathbb D,\end{align*} $$

where $\mu $ is a positive Borel measure on the unit disc $\mathbb D$ satisfying

(1.2) $$ \begin{align} \int_{\mathbb D}(1-|z|^{2})d\mu(z)<+\infty. \end{align} $$

The Green potential $U_{\mu }\not \equiv +\infty $ exactly when (1.2) holds (see, e.g., [Reference Armitage and Gardiner3, p. 98]). The Dirichlet space corresponding to $w=U_{\mu }$ is denoted by $D_{\mu }$ . From the Riesz representation theorem (see, e.g., [Reference Armitage and Gardiner3, pp. 105–106]), every positive superharmonic function w in $\mathbb D$ is equal to the sum $U_{\mu }+P_{\nu }$ , for a positive Borel measure $\mu $ in $\mathbb D$ satisfying (1.2) and a finite positive Borel measure on $\partial \mathbb D$ . Therefore, the examination of the weights $U_{\mu }$ and $P_{\nu }$ covers all positive superharmonic functions.

The case of a finite positive Borel measure $\mu $ in $\mathbb D$ is trivial. If $\mu (\mathbb D)<+\infty $ , then all bounded holomorphic functions are contained in $D_{\mu }$ (see, e.g., [Reference Aleman1, Proposition 3.1, p. 83]); in particular, all Blaschke products are contained in $D_{\mu }$ and therefore all Blaschke sequences are zero sets for $D_{\mu }$ . In what follows, we will consider infinite measures $\mu $ . In [Reference Bao, Mashreghi, Wang and Wulan6], the authors showed that a sufficient condition for all Blaschke sequences to be zero sets for $D_{\mu }$ is that

(1.3) $$ \begin{align} \int_{\partial\mathbb D}\log(B_{\mu}(\zeta))ds(\zeta)<+\infty, \end{align} $$

where

$$ \begin{align*}B_{\mu}(\zeta)=\frac{1}{2\pi}\int_{\mathbb D}\frac{1-|z|^{2}}{|\zeta-z|^{2}}d\mu(z),\qquad\zeta\in\partial\mathbb D,\end{align*} $$

is the balayage of $\mu $ . Also, assuming that one of the conditions

1. C1) there exists $C>0$ , such that

   $$ \begin{align*}\mathrm{{supp}}(\mu)\subset\left\{z\in\mathbb D:\frac{|1-z|^{4}}{1-|z|}<C\right\},\end{align*} $$

2. C2) there exist $r\in (0,1)$ and an ordered Blaschke sequence $(a_{n})\subset \mathbb D$ , such that

   $$ \begin{align*}\mathrm{{supp}}(\mu)\subset\bigcup_{n=1}^{+\infty}\left\{z\in\mathbb D:\left|\frac{z-a_{n}}{1-\overline{a_{n}}z}\right|<r\right\},\end{align*} $$

for the support of the measure $\mu $ holds, they proved that (1.3) is a necessary and sufficient condition for every Blaschke sequence to be a zero set for $D_{\mu }$ .

In this article, our purpose is to enlarge the class of infinite Borel measures $\mu $ on $\mathbb D$ for which every Blaschke sequence is a zero set for $D_{\mu }$ . This will be done by considering a class of measures satisfying a condition weaker than (C1) and show that, for these measures, condition (1.3) holds.

Let $k\in \mathbb N$ and $E\subset \partial \mathbb D$ . Following [Reference Caughran7, p. 315], a set $S\subset \mathbb D$ is said to have finite degree of contact k at E if there exists a constant $M>0$ such that

$$ \begin{align*}\mathrm{{dist}}(z,E)^{k}\leq M(1-|z|^{2}),\end{align*} $$

for all $z\in S$ . In this terminology, condition (C1) says that the support of the measure $\mu $ is contained in an open set that has finite degree of contact $4$ at $\{1\}$ .

Let $\mu $ be an infinite positive Borel measure in $\mathbb D$ , satisfying condition (1.2). From (1.2), it follows that $\mu $ is finite on each compact subset of $\mathbb D$ . We will denote by $D(z,r)$ the open disc in $\mathbb C$ centered at $z\in \mathbb C$ with radius $r>0$ . Let

$$ \begin{align*}E_{\mu}=\{\zeta\in\partial\mathbb D:\mu(D(\zeta,\varepsilon))=+\infty,\,\forall\varepsilon>0\}.\end{align*} $$

We note that $E_{\mu }$ is not empty. Suppose to the contrary that $E_{\mu }=\varnothing $ . Then, for every $\zeta \in \partial \mathbb D$ , there exists $\varepsilon _{\zeta }>0$ , such that $\mu (D(\zeta ,\varepsilon _{\zeta }))<+\infty $ . Since $\partial \mathbb D$ is compact, it will be covered by finitely many discs having finite $\mu $ -measure. Since the remaining of $\mathbb D$ is compact, we conclude that $\mu $ is finite, contradiction. An infinite Borel measure $\mu $ in $\mathbb D$ is said to have a finite degree of contact at $\partial \mathbb D$ if there exist $k\in \mathbb N$ and an open set $G\subset \mathbb D$ , such that

1. 1) $\mu (\mathbb D\setminus G)<+\infty $ , and

2. 2) G has finite degree of contact k at $E_{\mu }$ .

We will need one more definition before stating the main result. A closed subset $E\subset \partial \mathbb D$ is said to be a Carleson set if

$$ \begin{align*}\int_{\partial\mathbb D}\log\left(\frac{1}{\mathrm{{dist}}(\zeta,E)}\right)ds(\zeta)<+\infty.\end{align*} $$

Theorem 1.2 Let $\mu $ be a positive infinite Borel measure in $\mathbb D$ satisfying condition (1.2). Suppose that $\mu $ has a finite degree of contact at $\partial \mathbb D$ and $E_{\mu }$ is a Carleson set. Then,

$$ \begin{align*}\int_{\partial\mathbb D}\log(B_{\mu}(\zeta))ds(\zeta)<+\infty\end{align*} $$

and all Blaschke sequences are zero sets for $D_{\mu }$ .

As Guillot has pointed out in [Reference Guillot10, p. 270], the proof of Theorem 1.2 will be based on the existence of outer functions vanishing in a certain order on the set $E_{\mu }$ . We refer the reader to [Reference Duren8] for the definition and information about outer functions. Theorem 1.2 will be proved in Section $2$ . In Section $3,$ we show that for measures $\mu $ having finite degree of contact $2$ at $\partial \mathbb D$ , the set $E_{\mu }$ has zero length. The final section contains some remarks and questions.