Weighted composition followed and proceeded by differentiation operators from Q k ( p , q ) spaces to Bloch-type spaces

In this paper, we investigate boundedness and compactness of the weighted composition followed and proceeded by differentiation operators from spaces to Bloch-type spaces and little Bloch-type spaces. Some sufficient and necessary conditions for the boundedness and compactness of these operators are obtained.

MSC: 47B38, 30D45.

Let Δ be an open unit disc in the complex plane, and let be the class of all analytic functions on Δ. The α-Bloch space () is, by definition, the set of all function f in such that

Under the above norm, is a Banach space. When , is the well-known Bloch space. Let denote the subspace of , for f

This space is called a little α-Bloch space.

Assume that μ is a positive continuous function on , having the property that there exist positive numbers s and t, , and , such that

Then μ is called a normal function (see [9]).

Denote (see, e.g., [2,4,10])

It is known that is a Banach space with the norm (see [4]).

Let denote the subspace of , i.e.,

This space is called a little Bloch-type space. When , the induced space becomes the α-Bloch space .

Throughout this paper, we assume that K is a right continuous and nonnegative nondecreasing function. For , , we say that a function belongs to the space (see, [11]), if

where dA denotes the normalized Lebesgue area measure on Δ, is the Green function with logarithmic singularity at a, that is, , where for . When , , the space equals to , which is introduced by Zhao in [13]. Moreover (see [13]), we have that and for , and for . When , is a Banach space with the norm

From [11], we know that , if and only if

Moreover, (see [11], Theorem 2.1]).

Throughout the paper, we assume that

otherwise consists only of constant functions (see [11]).

Let φ be a nonconstant analytic self-map of Δ, and let ϕ be an analytic function in Δ. We define the linear operators

They are called weighted composition followed and proceeded by differentiation operators respectively, where and D are composition and differentiation operators respectively. The boundedness and compactness of on the Hardy spaces were investigated by Hibschweiler and Portnoy in [3] and by Ohno in [8]. In [6], Li and Stević studied the boundedness and compactness of the operator on the α-Bloch spaces. In [7], Li and Stević studied the boundedness and compactness of the composition and differentiation operators between and α-Bloch spaces. In [12], Yang studied the boundedness and compactness of the operator (or ) from to the Bloch-type spaces.

In this paper, we investigate the operators and from spaces to Bloch-type spaces and little Bloch-type spaces. Some sufficient and necessary conditions for the boundedness and compactness of these operators are given. Our results also generalize some known results in [12].

Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other. The notation means that there is a positive constant C such that .

In this paper, we shall prove the following results.

Theorem 2.1Letφbe an analytic self-map of Δ, and letϕbe an analytic function in Δ. Suppose thatμis normal, , , andKis a nonnegative nondecreasing function onsuch that

wheredenote the characteristic function of the setA. Thenis bounded if and only if

Theorem 2.2Letφbe an analytic self-map of Δ, and letϕbe an analytic function in Δ. Suppose thatμis normal, , , andKis a nonnegative nondecreasing function onsuch that (2.1) hold. Thenis compact if and only ifis bounded, and

Theorem 2.3Letφbe an analytic self-map of Δ, and letϕbe an analytic function in Δ. Suppose thatμis normal, , , andKis a nonnegative nondecreasing function onsuch that (2.1) hold. Thenis compact if and only if

From the above three theorems, we get the following

Corollary 2.4Letφbe an analytic self-map of Δ, and letϕbe an analytic function in Δ. Then the following statements hold.

(i) is bounded if and only if

(ii) is compact if and only ifis bounded, and

(iii) is compact if and only if

Theorem 2.5Letφbe an analytic self-map of Δ, and letϕbe an analytic function in Δ. Suppose thatμis normal, , , andKis a nonnegative nondecreasing function onsuch that (2.1) hold. Then the following statements hold.

(i) is bounded if and only if

(ii) is compact if and only ifis bounded, and

(iii) is compact if and only if

From Theorem 2.5, we get the following

Corollary 2.6Letφbe an analytic self-map of Δ, and letϕbe an analytic function in Δ. Then the following statements hold.

(i) is bounded if and only if

(ii) is compact if and only ifis bounded, and

(iii) is compact if and only if

In this section, we will prove our main results. For this purpose, we need some auxiliary results.

Lemma 3.1Letφbe an analytic self-map of Δ, ϕbe an analytic function in Δ. Suppose, . Thenis compact if and only ifis bounded and for any bounded sequenceinwhich converges to zero uniformly on compact subsets of Δ as, and (or) as.

Lemma 3.1 can be proved by standard way (see [1], Proposition 3.11]).

Lemma 3.2A closed setofis compact if and only if it is bounded and satisfies

Proof First of all, we suppose that is compact and let . By the definition of , we can choose an -net which center at in respectively, and a positive number r () such that , for and . If , for some , so we have

for . This establishes (3.1). □

On the other hand, if is a closed bounded set which satisfies (3.1) and is a sequence in , then by the Montel’s theorem, there is a subsequence which converges uniformly on compact subsets of Δ to some analytic function f, and also converges uniformly to on compact subsets of Δ. According to (3.1), for every , there is an r, , such that for all , , if . It follows that , if . Since converges uniformly to f and converges uniformly to on , it follows that , i.e., , so that is compact.

Lemma 3.3 ([14])

Letand. Then we have

Proof of Theorem 2.1 First, suppose that the conditions in (2.2) hold. Then for any and , by use of the fact and Lemma 3.3, we have

Taking the supremum in (3.2) for , and employing (2.2), we deduce that

is bounded.

Conversely, suppose that is bounded. Then there exists a constant C such that for all . Taking the functions , and , which belong to , we get

and

From (3.3), (3.4), and the boundedness of the function , it follows that

For , let

by direct calculation, we get

From [5], we know that , for each . Moreover, there is a positive constant C such that . Hence, we have

for . Therefore, we obtain

Next, for , let

Then from [5], we see that and . Since

we have

Thus

Inequality (3.5) gives

Therefore, the first inequality in (2.2) follows from (3.9) and (3.10). From (3.7) and (3.8), we obtain

Inequalities (3.3) and (3.11) imply

and

Inequality (3.12) together with (3.13) implies the second inequality of (2.2). The proof of Theorem 2.1 is completed. □

Proof of Theorem 2.2 First, suppose that is bounded and (2.3) hold. Let be a sequence in such that , and converges to 0 uniformly on compact subsets of Δ as . By the assumption, for any , there exists a such that

and

hold for . Since is bounded, it follows from the proof of Theorem 2.1 that

Let . Then we have

From the fact that as on compact subsets of Δ, and Cauchy’s estimate, we conclude that and as on compact subsets of Δ. Letting in (3.14) and using the fact that ε is an arbitrary positive number, we obtain . Applying Lemma 3.1, the result follows.

Conversely, suppose that is compact. Then it is clear that is bounded. Let be a sequence in Δ such that as . For , let

Then and converges to 0 uniformly on compact subsets of Δ as . Since is compact, by Lemma 3.1, we have . On the other hand, from (3.6) we have

which implies that

if one of these two limits exists.

Next, for , set

Then is a sequence in . Notice that ,

And converges to 0 uniformly on compact subsets of Δ as . Since is compact, we have . On the other hand, since

we have

From (3.15) and (3.16), we get

The proof of Theorem 2.2 is completed. □

Proof of Theorem 2.3 First, let . By the proof of Theorem 2.1, we have

Taking the supremum in (3.18) over all such that , we can get

By Lemma 3.2, we see that the operator is compact.

Conversely, suppose that is compact. By taking and using the boundedness of , we get

From this, by taking the test function and using the boundedness of , it follows that

In the following, we distinguish two cases:

First, we assume that . From (3.19) and (3.20), we obtain

and

So the result follows in this case.

Secondly, we assume that . Let be a sequence such that . From the compactness of , we see that is compact. According to Theorem 2.2, we get

and

For any , from (3.19) and (3.22), there exists such that

for , and there exists such that

for . Therefore, when , and , we obtain

On the other hand, if , and , we have

From (3.23) and (3.24), we get the second equality of (2.4). Similarly to the above arguments, by (3.20) and (3.21), we can get the first equality of (2.4). The proof of Theorem 2.3 is completed. □

Similarly to the proofs of Theorems 2.1-2.3, we can get the proofs of Corollary 2.4, Theorem 2.5 and Corollary 2.6. We omit the proofs.

The authors declare that they have no competing interests.

All authors drafted the manuscript, read and approved the final manuscript.

The work is supported by the National Natural Science Foundation of China (Grant No. 11171080), and Foundation of Science and Technology Department of Guizhou Province (Grant No. [2010] 07).

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