Inner functions as improving multipliers and zero sets of Besov-type spaces

We denote the unit disk $\{z\in \mathbb{C}:|z|<1\}$ by $\mathbb{D}$ and its boundary $\{z\in \mathbb{C}:|z|=1\}$ by $\partial \mathbb{D}$. Let $H(\mathbb{D})$ be the space of all analytic functions in $\mathbb{D}$. An analytic function in the unit disc $\mathbb{D}$ is called an inner function if it is bounded and the modulus equals 1 almost everywhere on the boundary $\partial \mathbb{D}$.

and μ is a finite positive Borel measure in $[0,2\pi )$, which is singular with respect to Lebesgue measure. Let $\sigma (\theta )$ denote the singular set or boundary spectrum of inner function θ. From [1] we know that $\sigma (\theta )\subseteq \partial \mathbb{D}$ is the smallest closed set such that θ is analytic across $\partial \mathbb{D}\mathrm{\setminus }\sigma (\theta )$, and $\sigma (\theta )$ consists of the accumulation points of zeros of θ and the closed support of the associated singular measure. See [2–6] for more information on inner functions.

Since a nontrivial inner function θ is extremely oscillatory near $\sigma (\theta )$, the same should happen to the product fθ, where $f\in H(\mathbb{D})$ is smooth in some sense on $\partial \mathbb{D}$. But sometimes the product fθ inherits the nice properties of f, and it is possible that fθ has an added smoothness. In order to analyze this phenomenon, Dyakonov introduced the following notion in [7].

Suppose X and Y are two classes of analytic functions on $\mathbb{D}$, and $X\subseteq Y$. Let θ be an inner function, θ is said to be $(X,Y)$-improving, if every function $f\in X$ satisfying $f\theta \in Y$ must actually satisfy $f\theta \in X$.

In this paper, we study the inner functions in the Möbius invariant Besov-type spaces $F(p,p-2,s)$ as improving multipliers.

${\phi }_a(z)=\frac{a-z}{1-\overline{a}z}$, $dA(z)=\frac{1}{\pi }dxdy$. It is easy to check that $F(p,p-2,s)$ is a Möbius invariant Besov-type space. From [8], when $0<s<1$, $F(2,0,s)={\mathcal{Q}}_s$, for more information on ${\mathcal{Q}}_s$ spaces, we refer to [9] and [10]. When $s=1$, $F(2,0,s)=BMOA$, the space of analytic functions in the Hardy space $H^1(\mathbb{D})$ whose boundary functions have bounded mean oscillation. When $s>1$, $F(2,0,s)=\mathcal{B}$, the Bloch space (see [11]).

The following result was proved by Peláez in [[12], Theorem 1].

In this paper, we extend Theorem A from ${\mathcal{Q}}_s$ spaces to a more general space $F(p,p-2,s)$, $0<s<1$.

Notice the fact that $F(p,p-2,s)\subset A{B}_p^{\frac{1-s}{p}}$. The proof of Corollary 1 is similar to that of Theorem 1 and thus is omitted.

Remark 1 From Theorem 1, we know that any inner function $\theta \in F(p,p-2,s)$ is $(F(p,p-2,s),F(p,p-2,t))$-improving, when $p>1$ and $0<s<t<1$. Conversely, by using the following results (Theorem 2 and Proposition 1) on zero sets of Besov-type spaces, we will prove that there exists an inner function θ which is $(F(p,p-2,s),F(p,p-2,t))$-improving, but θ does not belong to $F(p,p-2,s)$.

In order to state Theorem 2, we need a few notions.

Here $\varrho (a_j,a_k)=|{\phi }_{a_j}(a_k)|$ denotes the pseudo-hyperbolic metric in $\mathbb{D}$. We also say that ${\{a_k\}}_{k=1}^{\mathrm{\infty }}$ is an interpolating sequence or an uniformly separated sequence. A finite union of interpolating sequences is usually called a Carleson-Newman sequence. Similarly, a Carleson-Newman Blaschke product is a finite product of interpolating Blaschke products.

where $\eta \in \partial \mathbb{D}$.

We say that $Z=\{z_n\}\subset \mathbb{D}$ is a zero set of an analytic function space X defined on $\mathbb{D}$ if there is a $f\in X$ that vanishes on Z and nowhere else. Although the study of zero sets for analytic function spaces is a difficult problem, there are some excellent papers related to this question. We may refer to Carleson [14, 15], Caughran [16], Shapiro and Shields [17], Taylor and Williams [18]. Recently, Pau and Peláez obtained a theorem on zero sets of Dirichlet spaces ${\mathcal{D}}_s=A{B}_2^{\frac{1-s}{2}}$ in [[19], Theorem 1]. In the next theorem, we characterize the zero sets of $A{B}_p^t$ spaces which generalizes the Pau and Peláez’s result in [19]. We characterize the Carleson-Newman sequences that are zero sets in $A{B}_p^t$ spaces.

Using Theorem 2, we can deduce the following result.

Proposition 1 Suppose $p>1$, $0<t<1$, $0<pt<1$. There exists a Carleson-Newman sequence ${\{a_k\}}_{k=1}^{\mathrm{\infty }}$ which is not an $A{B}_p^t$-zero set and with 1 as unique accumulation point.

The proof of Proposition 1 is similar to that of Theorem 2 of [19] and is omitted here.

Applying Proposition 1, we can prove the following result whose proof (as well as the proof of Theorem 2) is given in Section 4.

Corollary 2 Let $p>1$ and $0<s<t<1$. Then there exists an inner function θ which is $(F(p,p-2,s),F(p,p-2,t))$-improving, but θ does not belong to $F(p,p-2,s)$.

Throughout this paper, for two functions f and g, $f\asymp g$ means that $g\lesssim f\lesssim g$, that is, there are positive constants $C_1$ and $C_2$ depending only on the index $p,s,t,\dots$ , such that $C_{1}g\le f\le C_{2}g$.

To prove Theorem 1 we need some auxiliary results. Lemmas 2.1 and 2.2 should be known to some experts, but we cannot find a reference. For the completeness of the paper, we give proofs below.

 □

can be defined as a norm of $BMOA$ space.

Lemma 2.2 Let $1<p<\mathrm{\infty }$, $0<s<1$. Then $F(p,p-2,s)\subseteq BMOA$.

That is, $f\in BMOA$. □

The following three lemmas will be used in the proof of Lemma 2.6. Their proofs can be found in [[22], Corollary 2.4], [[23], Lemma 2.5] or [[24], Lemma 1] and [[25], Lemma 2.1], respectively.

Lemma 2.6 plays an important role in the proof of Theorem 1, which generalizes Theorem 6 of [12], with a different proof motivated by [21] and [19].

Hence, we have $f\theta \in F(p,p-2,s)$. □

The following well-known results will also be used in the proof of Theorem 1.

Lemma 2.7 ([[22], Theorem 1.4])

Lemma 2.8 ([[27], Lemma 21])

if and only if ${\{a_k\}}_{k=1}^{\mathrm{\infty }}$ is a finite union of uniformly separated sequences.

Lemma 2.9 ([[28], Theorem 1])

In this section, we borrow the idea in [19] to study the zero set of $A{B}_p^t$ spaces. Following the proof of Lemma 2.6, we have the next result.

When $p=q$, $\sigma =1$ or $\sigma =\frac{1}{p}$ in Theorem 1 of [13], we get the following lemma.