Havin-Mazya type uniqueness theorem for Dirichlet spaces

Abstract

Let μ be a positive finite Borel measure on the unit circle. The associated Dirichlet space $\mathcal{D}(\mu )$ consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson integral of μ. We give a sufficient condition on a Borel subset E of the unit circle which ensures that E is a uniqueness set for $\mathcal{D}(\mu )$. We also give some examples of positive Borel measures μ and uniqueness sets for $\mathcal{D}(\mu )$.

Introduction

Let μ be a positive finite Borel measure on the unit circle $\mathbb{T}$, the harmonically weighted Dirichlet space ${\mathcal{D}}^h(\mu )$ is the set of all functions $f\in L^2(\mathbb{T})$ for which

$${\mathcal{D}}_{\mu }(f)=\underset{\mathbb{T}}{\int }{\mathcal{D}}_{\xi }(f)d\mu (\xi )<\infty ,$$

where ${\mathcal{D}}_{\xi }(f)$ is the local Dirichlet integral of f at $\xi \in \mathbb{T}$ given by

$${\mathcal{D}}_{\xi }(f):=\underset{\mathbb{T}}{\int }|\frac{f(\zeta )-f(\xi )}{\zeta -\xi }{|}^2\frac{|d\zeta |}{2\pi }.$$

The space ${\mathcal{D}}^h(\mu )$ is endowed with the norm

$${\Vert f\Vert }_{\mu }^2:={\Vert f\Vert }_{L^2(\mathbb{T})}^2+{\mathcal{D}}_{\mu }(f).$$

The analytic weighted Dirichlet space $\mathcal{D}(\mu )$ is defined by

$$\mathcal{D}(\mu )=\{f\in {\mathcal{D}}^h(\mu )\text{ : }\hat{f}(n)=0,n<0\}.$$

Let $\mathbb{D}$ be the open unit disc in the complex plane. Let $H^2$ denote the Hardy space of analytic functions on $\mathbb{D}$. As usual, we use the identification $H^2=\{f\in L^2(\mathbb{T})\text{ : }\hat{f}(n)=0,n<0\}$. Since $\mathcal{D}(\mu )\subset H^2$, every function $f\in \mathcal{D}(\mu )$ has non-tangential limits almost everywhere on $\mathbb{T}$. We denote by $f(\zeta )$ the non-tangential limit of f at $\zeta \in \mathbb{T}$ if it exists. Note that by Douglas Formula the Dirichlet-type space $\mathcal{D}(\mu )$ is the set of analytic functions $f\in H^2$, such that

$$\underset{\mathbb{D}}{\int }|f^{\prime }(z){|}^2{P}_{\mu }(z)dA(z)<\infty ,$$

where $dA(z)=dxdy/\pi$ stands for the normalized area measure in $\mathbb{D}$ and $P_{\mu }$ is the Poisson integral of μ given by

$$P_{\mu }(z):=\underset{\mathbb{T}}{\int }\frac{1-|z{|}^2}{|\zeta -z{|}^2}d\mu (\zeta ),z\in \mathbb{D}.$$

For a proof of this fact see [4, Theorem 7.2.5] and for numerous results on the Dirichlet-type space and operators acting thereon see [4], [6], [13], [14], [15], [16].

The capacity associated with $\mathcal{D}(\mu )$ is denoted by $c_{\mu }$ and is given by

$$c_{\mu }(E):=\inf \{{\Vert f\Vert }_{\mu }^2\text{ : }f\in {\mathcal{D}}^h(\mu )\text{, }|f|\ge 1\text{ a.e. on a neighborhood of }E\}.$$

Since the $L^2$ norm is dominated by the Dirichlet norm ${\Vert .\Vert }_{\mu }$, it is obvious that $c_{\mu }$–capacity 0 implies Lebesgue measure 0. We say that a property holds $c_{\mu }$-quasi-everywhere ($c_{\mu }$-q.e.) if it holds everywhere outside a set of zero $c_{\mu }$ capacity. Note that $c_{\mu }$-q.e implies a.e. and we have

$$c_{\mu }(E)=\inf \{{\Vert f\Vert }_{\mu }^2\text{ : }f\in {\mathcal{D}}^h(\mu )\text{, }|f|\ge 1c_{\mu }\text{-q.e. on }E\}.$$

For more details see [8]. Furthermore every function $f\in \mathcal{D}(\mu )$ has non-tangential limits $c_{\mu }$-q.e. on $\mathbb{T}$ [3, Theorem 2.1.9]. Note also that if E is a closed subset of $\mathbb{T}$ such that $c_{\mu }(E)=0$, then there exists a function $f\in \mathcal{D}(\mu )$ uniformly continuous on $\mathbb{T}$ such that the zero set $\mathcal{Z}(f):=\{\zeta \in \mathbb{T}\text{ : }f(\zeta )=0\}=E$ [7, Theorem 1].

Recall that if $I\subset \mathbb{T}$ be the arc of length $|I|=1-\rho$ with midpoint $\zeta \in \mathbb{T}$, then

$$\frac{1}{c_{\mu }(I)}\asymp 1+\underset{0}{\overset{\rho }{\int }}\frac{dr}{(1-r)P_{\mu }(r\zeta )+{(1-r)}^2},$$

where the implied constants are absolute. For the proof see [5, Theorem 2].

Let E be a subset of $\mathbb{T}$, the set E is said to be a uniqueness set for $\mathcal{D}(\mu )$ if, for each $f\in \mathcal{D}(\mu )$ such that its non-tangential limit $f=0$ $c_{\mu }$–q.e on E, we have $f=0$.

It is well known that for the Hardy space the uniqueness sets coincide with the sets of positive length on $\mathbb{T}$. Note that if $d\mu =dm$ the normalized arc measure on $\mathbb{T}$, then the space $\mathcal{D}(\mu )$ coincides with the classical Dirichlet space $\mathcal{D}$ given by

$$\mathcal{D}=\{f\in H^2:\underset{\mathbb{D}}{\int }|f^{\prime }(z){|}^{2}dA(z)<\infty \}.$$

In this case, $c_m$ is comparable to the logarithmic capacity and $c_m(I)\asymp |\log |I|{|}^{-1}$ for every arc $I\subset \mathbb{T}$, [12, Theorem 14].

Khavin and Maz'ya proved in [10] that a Borel subset E of $\mathbb{T}$ is a uniqueness set for $\mathcal{D}$ if there exists a family of pairwise disjoint open arcs $(I_n)$ such that

$$\sum _n|I_n|\log \frac{|I_n|}{c_m(E\cap I_n)}=-\infty .$$

For other uniqueness results for $\mathcal{D}$ see also [1], [2], [9], [4], [11].

Our aim in this paper is to extend Khavin and Maz'ya uniqueness theorem to general Dirichlet spaces $\mathcal{D}(\mu )$.

Let $\gamma >1$ and let $I=(e^{ia},e^{ib})$. The arc γI is given by

$$\gamma I=(e^{i(a-(\gamma -1)\frac{b-a}{2})},e^{i(b+(\gamma -1)\frac{b-a}{2})}).$$

The main result of this paper is the following theorem.

Theorem 1.1

Let E be a Borel subset of $\mathbb{T}$. Suppose that there exists a family of open arcs $(I_n)$ such that $(\gamma I_n)$ are pairwise disjoint for some $\gamma >1$ and

$$\sum _n|I_n|\log \frac{|I_n|}{c_{\mu }(E\cap I_n)}=-\infty ;$$

then E is a uniqueness set for $\mathcal{D}(\mu )$.

The key of the proof of this theorem is an upper estimate of the average

$$\frac{1}{|I|}\underset{I}{\int }|f(\zeta )||d\zeta |,$$

for $f\in \mathcal{D}(\mu )$ vanishing on a set $E\subset \mathbb{T}$, in terms of capacity of $E\cap I$, for any open arc I.

The next section is devoted to the proof of Theorem 1.1. In section 3 we give some examples of positive Borel measures μ and uniqueness sets for $\mathcal{D}(\mu )$.

Throughout the paper, we use the following notations: $A\lesssim B$ means that there is an absolute constant C such that $A\le CB$ and $A\asymp B$ if both $A\lesssim B$ and $B\lesssim A$.