- Research Article
- Open access
- Published:
Integral-Type Operators from
Spaces to Zygmund-Type Spaces on the Unit Ball
Journal of Inequalities and Applications volume 2010, Article number: 789285 (2011)
Abstract
Let denote the space of all holomorphic functions on the unit ball
. This paper investigates the following integral-type operator with symbol
,
,
,
, where
is the radial derivative of
. We characterize the boundedness and compactness of the integral-type operators
from general function spaces
to Zygmund-type spaces
, where
is normal function on
.
1. Introduction
Let be the open unit ball of
, let
be its boundary, and let
be the family of all holomorphic functions on
. Let
and
be points in
and
.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ1_HTML.gif)
stand for the radial derivative of . For
, let
, where
is the Möbius transformation of
satisfying
,
, and
. For
,
, we say
provided that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ2_HTML.gif)
The space , introduced by Zhao in [1], is known as the general family of function spaces. For appropriate parameter values
,
, and
,
coincides with several classical function spaces. For instance, let
be the unit disk in
,
if
(see [2]), where
, consists of those analytic functions
in
for which
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ3_HTML.gif)
The space is the classical Bergman space
(see [3]),
is the classical Besov space
, and, in particular,
is just the Hardy space
. The spaces
are
spaces, introduced by Aulaskari et al. [4, 5]. Further,
, the analytic functions of bounded mean oscillation. Note that
is the space of constant functions if
. More information on the spaces
can be found in [6, 7].
Recall that the Bloch-type spaces (or -Bloch space)
, consists of all
for which
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ4_HTML.gif)
The little Bloch-type space consists of all
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ5_HTML.gif)
Under the norm introduced by is a Banach space and
is a closed subspace of
. If
, we write
and
for
and
, respectively.
A positive continuous function on the interval [0,1) is called normal if there are three constants
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ6_HTML.gif)
Let denote the class of all
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ7_HTML.gif)
Write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ8_HTML.gif)
With the norm ,
is a Banach space.
is called the Zygmund space (see [8]). Let
denote the class of all
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ9_HTML.gif)
Let be a normal function on [0,1). It is natural to extend the Zygmund space to a more general form, for an
, we say that
belongs to the space
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ10_HTML.gif)
It is easy to check that becomes a Banach space under the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ11_HTML.gif)
and will be called the Zygmund-type space.
Let denote the class of holomorphic functions
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ12_HTML.gif)
and is called the little Zygmund-type space. When
, from [8, page 261], we say that
if and only if
, and there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ13_HTML.gif)
for all and
, where
is the ball algebra on
.
For , the following integral-type operator (so called extended Cesà ro operator) is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ14_HTML.gif)
where and
. Stević [9] considered the boundedness of
on
-Bloch spaces. Lv and Tang got the boundedness and compactness of
from
to
-Bloch spaces for all
(see [10]). Recently, Li and Stević discussed the boundedness of
from Bloch-type spaces to Zygmund-type spaces in [11]. For more information about Zygmund spaces, see [12, 13].
In this paper, we characterize the boundedness and compactness of the operator from general analytic spaces
to Zygmund-type spaces.
In what follows, we always suppose that ,
,
,
. Throughout this paper, constants are denoted by
; they are positive and may have different values at different places.
2. Some Auxiliary Results
In this section, we quote several auxiliary results which will be used in the proofs of our main results. The following lemma is according to Zhang [14].
Lemma 2.1.
If , then
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ15_HTML.gif)
Lemma 2.2 (see [9]).
For , if
, then for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ16_HTML.gif)
Lemma 2.3 (see [15]).
For every , it holds
.
Lemma 2.4 (see [10]).
Let . Suppose that for each
,
-variable functions
satisfy
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ17_HTML.gif)
Lemma 2.5.
Assume that ,
,
,
, and
is a normal function on
, then
is compact if and only if
is bounded, and for any bounded sequence
in
which converges to zero uniformly on compact subsets of
as
, one has
.
The proof of Lemma 2.5 follows by standard arguments (see, e.g., Lemma 3 in [16]). Hence, we omit the details.
The following lemma is similar to the proof of Lemma 1 in [17]. Hence, we omit it.
Lemma 2.6.
Let be a normal function. A closed set
in
is compact if and only if it is bounded and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ18_HTML.gif)
3. Main Results and Proofs
Now, we are ready to state and prove the main results in this section.
Theorem 3.1.
Let ,
,
, and let
be normal,
and
, then
is bounded if and only if
(i)for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ19_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ20_HTML.gif)
(ii)for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ21_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ22_HTML.gif)
Proof.
-
(i)
First, for
, suppose that
and
. By Lemmas 2.1–2.3, we write
. We have that
(3.5)
Hence, (3.1) and (3.2) imply that is bounded.
Conversely, assume that is bounded. Taking the test function
, we see that
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ24_HTML.gif)
For , set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ25_HTML.gif)
Then, by [14] and
.
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ26_HTML.gif)
From (3.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ27_HTML.gif)
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ28_HTML.gif)
Combing (3.9) and (3.10), we get (3.1).
In order to prove (3.2), let and set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ29_HTML.gif)
It is easy to see that ,
. We know that
; moreover, there is a positive constant
such that
. Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ30_HTML.gif)
From (3.1) and (3.12), we see that (3.2) holds.
-
(ii)
If
, then, by Lemmas 2.1 and 2.2, we have
, for
, we get
(3.13)
Applying (3.3) and (3.4) in (3.13), for the case , the boundedness of the operator
follows.
Conversely, suppose that is bounded. Given any
, set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ32_HTML.gif)
then by [14].
By the boundedness of , it is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ33_HTML.gif)
By (3.14) and (3.15), in the same way as proving (3.1), we get that (3.3) holds.
Now, given any , set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ34_HTML.gif)
then . Applying Lemma 2.4, we have that
. Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ35_HTML.gif)
From (3.15) and (3.17), we see that (3.4) holds. The proof of this theorem is completed.
Theorem 3.2.
Let ,
,
, and let
be normal,
and
, then the following statements are equivalent:
(A) is compact;
(B) is compact;
(C)
-
(i)
for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ36_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ37_HTML.gif)
-
(ii)
for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ38_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ39_HTML.gif)
Proof.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_IEq193_HTML.gif)
(B) ⇒ (A). This implication is obvious.
(A) ⇒ (C). First, for the case .
Suppose that the operator is compact. Let
be a sequence in
such that
. Denote
, and set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ40_HTML.gif)
It is easy to see that for
and
uniformly on compact subsets of
as
. By Lemma 2.5, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ41_HTML.gif)
By Lemma 2.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ42_HTML.gif)
From (3.23) and (3.24), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ43_HTML.gif)
which means that (3.18) holds.
Similarly, we take the test function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ44_HTML.gif)
Then, for
and
uniformly on compact subsets of
as
. we obtain that (3.20) holds for the case
.
For proving (3.19), we set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ45_HTML.gif)
then , and
converges to 0 uniformly on any compact subsets of
as
. By Lemma 2.5, it yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ46_HTML.gif)
Further, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ47_HTML.gif)
From (3.25), (3.28), and (3.29), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ48_HTML.gif)
which implies that (3.19) holds.
-
(ii)
Second, for the case
, take the test function
(3.31)
Then, by Lemma 2.4 and
uniformly on any compact subset of
. By Lemma 2.5 and condition (
), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ50_HTML.gif)
Hence, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ51_HTML.gif)
From (3.20), (3.32), and (3.33), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ52_HTML.gif)
which implies that (3.21) holds.
(C) ⇒ (B). Suppose that (3.18) and (3.19) hold for . By Lemmas 2.1 and 2.2, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ53_HTML.gif)
Note that (3.18) and (3.19) imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ54_HTML.gif)
Further, they also imply that (3.1) and (3.2) hold. From this and Theorem 3.1, it follows that set is bounded. Using these facts, (3.18), and (3.19), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ55_HTML.gif)
Similarly, we obtain that (3.37) holds for the case by (3.20) and (3.21). Exploiting Lemma 2.6, the compactness of the operator
follows. The proof of this theorem is completed.
Finally, we consider the case .
Theorem 3.3.
Let ,
,
, and let
be normal,
,
, then the following statements are equivalent:
(A) is bounded;
(B) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ56_HTML.gif)
The proof of Theorem 3.3 follows by the proof of Theorem 3.1. So, we omit the details here.
Theorem 3.4.
Let ,
,
, and let
be normal,
and
, then the following statements are equivalent:
(A) is compact;
(B) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ57_HTML.gif)
Proof.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_IEq241_HTML.gif)
(A) ⇒ (B). We assume that is compact. For
, we obtain that
. Exploiting the test function in (3.22), similarly to the proof of Theorem 3.2, we obtain that (3.39) holds. As a consequence, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ58_HTML.gif)
(B) ⇒ (A). Assume that is a sequence in
such that
, and
uniformly on compact of
as
. By Lemma 2.1 and [18, Lemma 4.2],
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ59_HTML.gif)
From (3.39), we have that for every , there is a
, such that, for every
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ60_HTML.gif)
and from (3.39) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ61_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ62_HTML.gif)
Since on compact subsets of
by the Cauchy estimate, it follows that
on compact subsets of
, in particular on
. Taking in (3.44), the supremum over
, letting
, using the above-mentioned facts,
, and since
is an arbitrary positive number, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ63_HTML.gif)
Hence, by Lemma 2.5, the compactness of the operator follows. The proof of this theorem is completed.
Theorem 3.5.
Let ,
,
, and let
be normal,
and
, then the following statements are equivalent:
(A) is compact;
(B) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ64_HTML.gif)
Proof.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_IEq272_HTML.gif)
(A) ⇒ (B). For , we obtain that
. In the same way as in Theorem 3.4, we get that (3.46) holds.
(B) ⇒ (A). By Lemmas 2.1 and 2.2, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F789285/MediaObjects/13660_2010_Article_2247_Equ65_HTML.gif)
This along with Theorem 3.2 implies that is bounded. Taking the supremum over the unit ball in
, letting
in (3.46), using the condition (
), and finally by applying Lemma 2.6, we get the compactness of the operator
. This completes the proof of the theorem.
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Acknowledgments
The author wishes to thank Professor Rauno Aulaskari for his helpful suggestions. This research was supported in part by the Academy of Finland 121281.
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Yang, C. Integral-Type Operators from Spaces to Zygmund-Type Spaces on the Unit Ball.
J Inequal Appl 2010, 789285 (2011). https://doi.org/10.1155/2010/789285
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DOI: https://doi.org/10.1155/2010/789285