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Inner functions as improving multipliers and zero sets of Besov-type spaces
Journal of Inequalities and Applications volume 2014, Article number: 312 (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 .
It is well known that every inner function has a factorization , where , is a Blaschke product and is a singular inner function, that is,
where is a sequence of points in which satisfies the Blaschke condition
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.
Let , , , the space is the set of  such that
where g denotes the Green function given by
, . 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].
Theorem A Suppose that and θ is an inner function. Then the following conditions are equivalent:
θ is -improving;
θ is -improving.
In this paper, we extend Theorem A from spaces to a more general space , .
Theorem 1 Let , . Suppose that θ is an inner function. Then the following conditions are equivalent:
θ is -improving;
θ is -improving.
The proof of Theorem 1 is gave in Section 3. Theorem 1 can be farther generalized to a more general space , where space is the set of functions such that (see )
Corollary 1 Let , and . Suppose θ is an inner function. Then the following conditions are equivalent:
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.
A Blaschke product B with sequence of zeros is called interpolating if there exists a positive constant δ such that
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 recall that a function is called an outer function if and
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.
Theorem 2 Suppose , , and is a Carleson-Newman sequence. Then the following conditions are equivalent:
is an -zero set;
there exists an outer function such that
there exists an outer function such that
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.
Lemma 2.1 Let and . Then if and only if
Proof From Theorem 2.4 of  and making the change of variables , we have
For , denotes the Hardy space of with
From , we know that
can be defined as a norm of space.
Lemma 2.2 Let , . Then .
Proof From Lemma 2.4 of , we know and
Let . By Lemma 2.1, we get
That is, . □
Lemma 2.3 Let θ be an inner function and let , and such that . Then, for any and ,
Lemma 2.4 Let , , and . Then
Lemma 2.5 Let , and , with , . If , then
Lemma 2.6 Let , , and B be a Carleson-Newman Blaschke product with a sequence of zeros . Then if and only if
Proof Necessity. If , then it is easy to deduce that
We will prove that
From the sub-mean-value property of and the estimate
for (see [, p.69]). Here and afterwards
Since B is a Carleson-Newman Blaschke product with a sequence of zeros , we have
Since the function is decreasing in , we get
Combining this with (1) and using Lemma 2.3 yield
Sufficiency. Applying the p-triangle inequality gives
Since , , by Theorem 2.4 of , we have
We now estimate . Using the fact that and
and employing the p-triangle inequality again, we have
Applying Lemma 2.4 yields
Let , . Making the change of variables and using Lemma 2.5, we get
Hence, we have . □
The following well-known results will also be used in the proof of Theorem 1.
Lemma 2.7 ([, Theorem 1.4])
Let . Then an inner function belongs to the Möbius invariant Besov-type space for all if and only if it is the Blaschke product associated with a sequence which satisfies
Lemma 2.8 ([, Lemma 21])
Let be a sequence in . Then the measure is a Carleson measure, i.e.
if and only if is a finite union of uniformly separated sequences.
If θ is an inner function, for , define the level set of order ϵ of θ as
Lemma 2.9 ([, Theorem 1])
If and θ is an inner function, then the following conditions are equivalent:
, for every ϵ, ;
, for some ϵ, .
3 Proof of Theorem 1
⇒ (2). For inner functions , by Lemma 2.7, θ is a Blaschke product with zeros , and
which implies that
From Lemma 2.8, θ is a Carleson-Nemwman Blaschke product. Suppose that and . Lemma 2.9 gives
Applying Lemma 2.6 implies that . Hence, θ is -improving.
⇒ (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.
Lemma 4.1 Let , , . Suppose , B is a Carleson-Newman Blaschke product with sequence of zeros . Then if and only if
When , or in Theorem 1 of , we get the following lemma.
Lemma 4.2 Let , , . Then the following conditions are equivalent.
Proof of Theorem 2 (1) ⇒ (2). Since is an -zero set, there exists such that nowhere else. By , there exists a Blaschke product B, a singular inner function S and an outer function g, such that . Since , , B is a Carleson-Newman Blaschke product with zeros , from Corollary 3.1 of , we know that has the f-property (see  for the definition of the f-property). Then
Applying Lemma 4.1, we have (2).
⇒ (1). It is obvious from Lemma 4.1.
⇒ (2). By the Poisson integral formula
Applying Hölder’s inequality, we have
The desired result follows.
(2) ⇒ (3). By the p-triangle inequality, we obtain
It is obvious that
We next estimate . Since is a finite union of interpolating sequences, we can write
and there exist a positive integer n and , , such that
Then for fixed i, the pseudo-hyperbolic disks are pairwise disjoint. Notice that
has the generalized sub-mean-value property. Therefore,
Bearing in mind Lemma 4.2, we have (3).
Now, we prove that has the generalized sub-mean-value property. Since is subharmonic, using the sub-mean-value property, we have
From [, p.69 and Lemma 4.30], we know
for , , . Thus, it follows that
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 .
Proof Let . By the Möbius invariant property of , we have . From Corollary 3.1 of , we get
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. □
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This work was supported by NSF of China (No. 11171203) and NSF of Guangdong Province (Grant No. 10151503101000025 and No. S2011010004511).
The authors declare that they have no competing interests.
All authors contributed equally in the preparation of this article. All authors read and approved the final manuscript.
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Lou, Z., Qian, R. Inner functions as improving multipliers and zero sets of Besov-type spaces. J Inequal Appl 2014, 312 (2014). https://doi.org/10.1186/1029-242X-2014-312
- inner functions
- zero set
- Carleson-Newman sequence
- Besov-type space