- Open Access
Inner functions as improving multipliers and zero sets of Besov-type spaces
© Lou and Qian; licensee Springer. 2014
Received: 18 March 2014
Accepted: 16 July 2014
Published: 21 August 2014
Assume X and Y are two spaces of analytic functions in the unit disk with . Let θ be an inner function. If every function satisfying must actually satisfy , then θ is said to be -improving. In this paper, we characterize the inner functions in the Möbius invariant Besov-type spaces as improving multipliers for and . Our result generalizes Peláez’s result on spaces () (Peláez in J. Funct. Anal. 255:1403-1418, 2008).
We denote the unit disk by and its boundary by . Let be the space of all analytic functions in . An analytic function in the unit disc is called an inner function if it is bounded and the modulus equals 1 almost everywhere on the boundary .
and μ is a finite positive Borel measure in , which is singular with respect to Lebesgue measure. Let denote the singular set or boundary spectrum of inner function θ. From  we know that is the smallest closed set such that θ is analytic across , and 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 , the same should happen to the product fθ, where is smooth in some sense on . 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 .
Suppose X and Y are two classes of analytic functions on , and . Let θ be an inner function, θ is said to be -improving, if every function satisfying must actually satisfy .
In this paper, we study the inner functions in the Möbius invariant Besov-type spaces as improving multipliers.
, . It is easy to check that is a Möbius invariant Besov-type space. From , when , , for more information on spaces, we refer to  and . When , , the space of analytic functions in the Hardy space whose boundary functions have bounded mean oscillation. When , , the Bloch space (see ).
The following result was proved by Peláez in [, Theorem 1].
θ is -improving;
θ is -improving.
In this paper, we extend Theorem A from spaces to a more general space , .
θ is -improving;
θ is -improving.
θ is -improving;
θ is -improving.
Notice the fact that . 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 is -improving, when and . 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 -improving, but θ does not belong to .
In order to state Theorem 2, we need a few notions.
Here denotes the pseudo-hyperbolic metric in . We also say that 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.
We say that is a zero set of an analytic function space X defined on if there is a 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 , Shapiro and Shields , Taylor and Williams . Recently, Pau and Peláez obtained a theorem on zero sets of Dirichlet spaces in [, Theorem 1]. In the next theorem, we characterize the zero sets of spaces which generalizes the Pau and Peláez’s result in . We characterize the Carleson-Newman sequences that are zero sets in spaces.
is an -zero set;
- (2)there exists an outer function such that
- (3)there exists an outer function such that
- (4)there exists an outer function such that
Using Theorem 2, we can deduce the following result.
Proposition 1 Suppose , , . There exists a Carleson-Newman sequence which is not an -zero set and with 1 as unique accumulation point.
The proof of Proposition 1 is similar to that of Theorem 2 of  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 and . Then there exists an inner function θ which is -improving, but θ does not belong to .
Throughout this paper, for two functions f and g, means that , that is, there are positive constants and depending only on the index , such that .
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 space.
Lemma 2.2 Let , . Then .
That is, . □
Hence, we have . □
The following well-known results will also be used in the proof of Theorem 1.
Lemma 2.7 ([, Theorem 1.4])
Lemma 2.8 ([, Lemma 21])
if and only if is a finite union of uniformly separated sequences.
Lemma 2.9 ([, Theorem 1])
, for every ϵ, ;
, for some ϵ, .
3 Proof of Theorem 1
- (1)⇒ (2). For inner functions , by Lemma 2.7, θ is a Blaschke product with zeros , and
⇒ (1), (3) ⇒ (2). Their proofs are obvious.
⇒ (3). Let θ be -improving. If the inner function , then θ is a Carleson-Newman Blaschke product. For , if , then any Carleson-Newman Blaschke product is -improving by Corollary 1 of . Therefore, . Notice that θ is -improving, we have . Thus, θ is -improving. The proof of Theorem 1 is completed.
4 Proofs of Theorem 2 and Corollary 2
In this section, we borrow the idea in  to study the zero set of spaces. Following the proof of Lemma 2.6, we have the next result.
When , or in Theorem 1 of , we get the following lemma.
⇒ (1). It is obvious from Lemma 4.1.
- (3)⇒ (2). By the Poisson integral formula
The desired result follows.
Bearing in mind Lemma 4.2, we have (3).
That is, has the generalized sub-mean-value property.
(2) ⇔ (4). The proof is similar to that of (2) ⇔ (3) and thus is omitted. The theorem is proved. □
Lemma 4.3 Let and . Suppose , where and are inner-outer factors. Let I be an inner function dividing . Then .
The desired result follows. □
Proof of Corollary 2 Taking the same sequence as in Proposition 1. Let be the associated Blaschke product. Notice the fact that , it implies that is not a -zero set. We deduce that does not belong to . Combining this with Lemma 4.3, the proof of the rest is similar to that of Theorem 2 of  and thus is omitted. □
This work was supported by NSF of China (No. 11171203) and NSF of Guangdong Province (Grant No. 10151503101000025 and No. S2011010004511).
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