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,
and
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 [1] 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 [7].
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 [8] such that
where g denotes the Green function given by
, . It is easy to check that is a Möbius invariant Besov-type space. From [8], when , , for more information on spaces, we refer to [9] and [10]. When , , the space of analytic functions in the Hardy space whose boundary functions have bounded mean oscillation. When , , the Bloch space (see [11]).
The following result was proved by Peláez in [[12], Theorem 1].
Theorem A Suppose that and θ is an inner function. Then the following conditions are equivalent:
-
(1)
;
-
(2)
θ is -improving;
-
(3)
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:
-
(1)
;
-
(2)
θ is -improving;
-
(3)
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 [13])
Corollary 1 Let , and . Suppose θ is an inner function. Then the following conditions are equivalent:
-
(1)
;
-
(2)
θ is -improving;
-
(3)
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
where .
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 [16], Shapiro and Shields [17], Taylor and Williams [18]. Recently, Pau and Peláez obtained a theorem on zero sets of Dirichlet spaces in [[19], Theorem 1]. In the next theorem, we characterize the zero sets of spaces which generalizes the Pau and Peláez’s result in [19]. 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:
-
(1)
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 [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 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 .