This paper aims at studying the boundedness and compactness of weighted composition operator between spaces of analytic functions. We characterize boundedness and compactness of the weighted composition operator uC_ϕ from the Hardy spaces H^p to the Zygmund type spaces and the little Zygmund type spaces 𝒵_α,0 in terms of function theoretic properties of the symbols u and ϕ.

1. Introduction

Let D = {z : |z| < 1} be the open unit disk in the complex plane C and T = {z : |z| = 1} its boundary, and H(D) denotes the set of all analytic functions on D. An analytic self-map φ : D → D induces the composition operator C_φ on H(D), defined by C_φ(f) = f(φ(z)) for f analytic on D. It is a well-known consequence of Littlewood’s subordination principle that the composition operator C_φ is bounded on the classical Hardy H^p (0 < p ≤ ∞) spaces, Bergman A^p (0 < p ≤ ∞) spaces, and Bloch spaces (see, e.g., [1–4]).

Let u be a fixed analytic function on the open unit disk. Define a linear operator uC_φ on the space of analytic functions on D, called a weighted composition operator, by uC_φf = u · (f∘φ), where f is an analytic function on D. We can regard this operator as a generalization of a multiplication operator and a composition operator. In recent years the weighted composition operator has received much attention and appears in various settings in the literature. For example, it is known that isometries of many analytic function spaces are weighted composition operators (see [5], for instance). Their boundedness and compactness have been studied on various Banach spaces of analytic functions, such as Hardy, Bergman, BMOA, Bloch-type, and Zygmund spaces; see, for example, [6–11]. Also, it has been studied from one Banach space of analytic functions to another; one may see [12–23].

The purpose of this paper is to consider the weighted composition operators from the Hardy space H^p (0 < p < ∞) to the Zygmund type spaces 𝒵_α. Our main goal is to characterize boundedness and compactness of the operators uC_φ from H^p to 𝒵_α in terms of function theoretic properties of the symbols u and φ.

Now we give a detailed definition of these spaces. For 0 ≤ r < 1, f(z) ∈ H(D), we set

()

For 0 < p ≤ ∞, the Hardy space H^p consists of those functions f ∈ H(D), for which

()

It is well known that with norm (2) the H^p space is a Banach space if 1 ≤ p ≤ ∞, for 0 < p < 1, H^p space is a nonlocally convex topological vector space, and

is a complete metric for it. For more information about the H^p space, one may see these books, for example, [24, 25].

For α > 0 the α-Bloch space β_α consists of all analytic functions f defined on D such that

()

The space 𝒵_α consists of all analytic functions f defined on D such that

()

When α = 1, it is called the Zygmund space. From a theorem by Zygmund (see [26, vol. I, p. 263] or [24, Theorem 5.3]), we see that f ∈ 𝒵₁ if and only if f is continuous in the close unit disk

and the boundary function f(e^iθ) such that

()

When α > 1, from Proposition 8 of [27], we know that 𝒵_α = β_α−1. Then the space 𝒵_α is called a Zygmund type space if 0 < α ≤ 1. However, all results in this paper are valid for all 𝒵_α spaces (α > 0). An analytic function f ∈ H(D) is said to belong to the little Zymund type space 𝒵_α,0 which consists of all f ∈ 𝒵_α satisfying

. It can be easily proved that 𝒵_α is a Banach space under the norm

()

And the polynomials are norm-dense in closed subspace 𝒵_α,0. For some other information on this space and some operators on it, see, for example, [28–31].

Throughout this paper, constants are denoted by C, C(p), they are positive, and C(p) are only depending on p and may differ from one occurrence to the another.

2. Auxiliary Results

In order to prove the main results of this paper. We need some auxiliary results. The first lemma is well known.

Lemma 1 (see [24], p. 65.)For p > 1, there exists a constant C(p) such that

()

Lemma 2. Suppose that 0 < p < ∞, f ∈ H^p; then

()

for every z ∈ D and all nonnegative integer n = 0,1, 2, ….

Proof. We use induction on n. The case n = 0 holds because it is Exercise 5 in [25, p. 85]. Assume the case n = k holds. Fix 0 < r < 1 and let g(z) = f^(k)(rz). Then g(z) is in H^∞ ⊂ β₁, and . It follows that

()

Let r → 1⁻; we have

()

Then the case n = k + 1 holds. Hence (8) holds.

Lemma 3. For 0 < p < ∞, suppose uC_φ : H^p → 𝒵_α,0 is a bounded operator. Then uC_φ : H^p → 𝒵_α is a bounded operator.

This is obvious.

3. Boundedness of uC_φ from H^p (0 < p < ∞) to 𝒵_α and 𝒵_α,0

In this section we characterize bounded weighted composition operators from the Hardy space H^p (0 < p < ∞) to the Zygmund spaces 𝒵_α.

Theorem 4. Let α > 0, 0 < p < ∞, and u be an analytic function on the unit disc D and φ an analytic self-map of D. Then uC_φ is a bounded operator from H^p to the Zygmund spaces 𝒵_α if and only if the following are satisfied:

()

Proof . Suppose uC_φ is bounded from H^p to the Zygmund spaces 𝒵_α. Then we can easily obtain the following results by taking f(z) = 1 and f(z) = z in H^p, respectively:

()

By (14) and the boundedness of the function φ(z), we get

()

Let f(z) = z² in H^p again; in the same way we have

()

Using these facts and the boundedness of the function φ(z) again, we get

()

Fix a ∈ D; we take the test functions

()

for z ∈ D. From Lemma 1 we obtain that f_a ∈ H^p and

with a direct calculation. Since f_a(a) = 0,

, and

, it follows that, for all λ ∈ D with |φ(λ)| > 1/2, we have

()

Let a = φ(λ); it follows that

()

For all λ ∈ D with |φ(λ)| ≤ 1/2, by (17), we have

()

Hence (12) holds.

Next, fix a ∈ D; we take another test functions

()

for z ∈ D. From Lemma 1 we obtain that g_a ∈ H^p and

with a direct calculation. Since g_a(a) = 0,

, and

, it follows that, for all λ ∈ D with |φ(λ)| > 1/2, we obtain that

()

For all λ ∈ D with |φ(λ)| ≤ 1/2, by (15), we have

()

Hence (13) holds.

Finally, fix a ∈ D, and, for all z ∈ D, let

()

From Lemma 1 we obtain that h_a ∈ H^p and

with a direct calculation. Since

, and h_a(a) = −2/(p + 1)(p + 2)(1 − |a|²) ^1/p, it follows that, for all λ ∈ D, we obtain that

()

Then (11) holds.

Conversely, suppose that (11), (12), and (13) hold. For f ∈ H^p, by Lemma 2, we have the following inequality:

()

This shows that uC_φ is bounded. This completes the proof of Theorem 4.

Theorem 5. Let α > 0, 0 < p < ∞, and u be an analytic function on the unit disc D and φ an analytic self-map of D. Then uC_φ : H^p → 𝒵_α,0 is a bounded operator provided that the following are satisfied:

()

Conversely, if uC_φ : H^p → 𝒵_α,0 is a bounded operator, then u ∈ 𝒵_α,0, (11), (12), and (13) hold, and the following are satisfied:

()

Proof. Assume that (28), (29), and (30) hold. Then for any ϵ > 0, there is a constant δ, 0 < δ < 1, such that δ < |z| < 1 implies

()

Then, for any f ∈ H^p, from Lemma 2 we obtain that

()

Hence uC_φf ∈ 𝒵_α,0 for all f ∈ 𝒵_α,0. On the other hand, (25), (28), and (29) imply that (11), (12), and (13) hold; then uC_φ : H^p → 𝒵_α is bounded by Theorem 4. So uC_φ : H^p → 𝒵_α,0 is bounded.

Conversely, assume that uC_φ is bounded from H^p to the little Zygmund type space 𝒵_α,0. Then u = uC_φ1 ∈ 𝒵_α,0. Also uφ = uC_φz ∈ 𝒵_α,0; thus

()

Since |φ| ≤ 1 and u ∈ 𝒵_α,0, we have

. Hence (32) holds.

Similarly, uC_φz² ∈ 𝒵_α,0; then

()

By (32), |φ | ≤ 1, and u ∈ 𝒵_α,0, we get that

; that is, (31) holds.

On the other hand, from Lemma 3 and Theorem 4, we obtain that (11), (12), and (13) hold.

4. Compactness of uC_φ

In order to prove the compactness of uC_φ from H^p to the Zygmund spaces 𝒵_α, we require the following lemmas.

Lemma 6. Let α > 0, 0 < p < ∞, and u be an analytic function on the unit disc D and φ an analytic self-map of D. Suppose that uC_φ is a bounded operator from H^p to 𝒵_α. Then uC_φ is compact if and only if, for any bounded sequence {f_n} in H^p which converges to 0 uniformly on compact subsets of D, one has as n → ∞.

The proof is similar to that of Proposition 3.11 in [32]. The details are omitted.

Theorem 7. Let α > 0,0 < p < ∞, u be an analytic function on the unit disc D and φ an analytic self-map of D. Then uC_φ is a compact operator from H^p to 𝒵_α if and only if uC_φ is a bounded operator and the following are satisfied:

()

Proof. Suppose that uC_φ is compact from H^p to the Zygmund type space 𝒵_α. Let {z_n} be a sequence in D such that |φ(z_n)| → 1 as n → ∞. If such a sequence does not exist, then (37) are automatically satisfied. Without loss of generality we may suppose that |φ(z_n)| > 1/2 for all n. We take the test functions

()

By a direct calculation, we may easily prove that {f_n} converges to 0 uniformly on compact subsets of D and

. Then {f_n} is a bounded sequence in H^p which converges to 0 uniformly on compact subsets of D. Then

by Lemma 6. Note that

()

It follows that

()

Then

()

Next, let

()

By a direct calculation we obtain that g_n⇉0 (n → ∞) on compact subsets of D and

. Consequently, {g_n} is a bounded sequence in H^p which converges to 0 uniformly on compact subsets of D. Then

by Lemma 6. Note that g_n(φ(z_n)) ≡ 0,

and

; it follows that

()

Then lim _n→∞(1−|z_n|²) ^α(2u^′(z_n)φ^′(z_n) + φ^′′(z_n)u(z_n))/(1−|φ(z_n)|²) ^1/p+1 = 0.

Finally, let

()

By a direct calculation we obtain that h_n⇉0 (n → ∞) on compact subsets of D and

. Consequently, {h_n} is a bounded sequence in H^p which converges to 0 uniformly on compact subsets of D. Then

by Lemma 6. Note that h_n(φ(z_n)) = 2/(p + 1)(p + 2)(1−|φ(z_n)|²) ^1/p,

, and

; it follows that

()

Then lim _n→∞((1 − |z_n|²) ^α|u^′′(z_n)|/(1−|φ(z_n)|²) ^1/p) = 0. The proof of the necessary is completed.

Conversely, Suppose that (37) hold. Since uC_φ is a bounded operator, from Theorem 4, we have

()

Let {f_n} be a bounded sequence in H^p with

and f_n → 0 uniformly on compact subsets of D. We only prove

by Lemma 6. By the assumption, for any ϵ > 0, there is a constant δ, 0 < δ < 1, such that δ<|φ(z)| < 1 implies

()

Let K = {w ∈ D:|w | ≤ δ}. Note that K is a compact subset of D. Then from Lemma 2 it follows that

()

As n → ∞,

()

Hence uC_φ is compact. This completes the proof of Theorem 7.

In order to prove the compactness of uC_φ on the little Zygmund spaces 𝒵_α,0, we require the following lemma.

Lemma 8. Let U ⊂ 𝒵_α,0. Then U is compact if and only if it is closed, bounded and satisfies

()

The proof is similar to that of Lemma 1 in [1], but we omit it.

Theorem 9. Let α > 0, 0 < p < ∞, u be an analytic function on the unit disc D and φ an analytic self-map of D. Then uC_φ is compact from H^p to the little Zygmund type spaces 𝒵_α,0 if and only if (28), (29), and (30) hold.

Proof. Assume that (28), (29), and (30) hold. By Theorem 5, we know that uC_φ is bounded from H^p to the little Zygmund type spaces 𝒵_α,0. Suppose that f ∈ H^p with ∥f∥_p ≤ 1. From Lemmas 1 and 2 we obtain that

()

thus

()

and it follows that

()

Hence uC_φ : H^p → 𝒵_α,0 is compact by Lemma 8.

Conversely, suppose that uC_φ : H^p → 𝒵_α,0 is compact.

Firstly, it is obvious uC_φ is bounded, from Theorem 5 we have u ∈ 𝒵_α,0, and (31), (32) hold. On the other hand, we have

()

for some M > 0 by Lemma 6.

Next, note that the proof of Theorem 4 and the fact that the functions given in (18) are in H^p and have norms bounded independently of a; we obtain that

()

for |φ(z)| > 1/2. However, if |φ(z)| ≤ 1/2, by (31), we easily have

()

Similarly, note that the functions given in (22) and (25) are in H^p and have norms bounded independently of a, we obtain that

()

for |φ(z)| > 1/2. However, if |φ(z)| ≤ 1/2, from u ∈ 𝒵_α,0 and (32), we easily have

()

This completes the proof of Theorem 9.

Remark 10. From Theorems 5 and 9, we conjecture that uC_φ : H^p → 𝒵_α,0 is compact if and only if uC_φ : H^p → 𝒵_α,0 is bounded.

Taking u(z) = 1 from Theorems 4, 7, and 9, we obtain the following results about the characterization of the boundedness and compactness of the composition operator C_φ : H^p → 𝒵_α(or 𝒵_α,0).

Corollary 11. Let α > 0, 0 < p < ∞, and φ be an analytic self-map of D. Then C_φ : H^p → 𝒵_α is a bounded operator if and only if the following are satisfied:

()

Corollary 12. Let α > 0, 0 < p < ∞, and φ be an analytic self-map of D. Then C_φ : H^p → 𝒵_α is a compact operator if and only if C_φ is bounded and the following are satisfied:

()

Corollary 13. Let α > 0, 0 < p < ∞, and φ be an analytic self-map of D. Then C_φ : H^p → 𝒵_α,0 is a compact operator if and only if

()

In the formulation of corollary, we use the notation M_u on H(D) defined by M_uf = uf for f ∈ H(D). Taking φ(z) = z from Theorems 4, 5, 7, and 9, we obtain the following results about the characterization of the boundedness and compactness of pointwise multiplier M_u : H^p → 𝒵_α(or 𝒵_α,0).

Corollary 14. Let α > 0, 0 < p < ∞. Then the pointwise multiplier M_u : H^p → 𝒵_α is a bounded operator if and only if

(i)
u = 0 if α < 2 + 1/p;
(ii)
u ∈ H^∞ if α = 2 + 1/p;
(iii)
sup _z∈D(1−|z|²) ^α−2−1/p | u(z)| < ∞ if α > 2 + 1/p.

Corollary 15. Let α > 0, 0 < p < ∞. Then the pointwise multiplier M_u : H^p → 𝒵_α,0 is a bounded operator if and only if M_u : H^p → 𝒵_α,0 is a compact operator if and only if M_u : H^p → 𝒵_α is a compact operator if and only if

(i)
u = 0 if α ≤ 2 + 1/p,
(ii)
if α > 2 + 1/p.

Acknowledgments

The research was supported by Special Fund of Colleges and Universities in Fujian Province (no. JK2012010) and Natural Science Foundation of Fujian Province, China (no. 2009J01004).

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Weighted Composition Operators from Hardy to Zygmund Type Spaces

Abstract

1. Introduction

2. Auxiliary Results

3. Boundedness of uC_φ from H^p (0 < p < ∞) to 𝒵_α and 𝒵_α,0

4. Compactness of uC_φ

Acknowledgments

References

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Weighted Composition Operators from Hardy to Zygmund Type Spaces

Abstract

1. Introduction

2. Auxiliary Results

3. Boundedness of uCφ from Hp (0 < p < ∞) to 𝒵α and 𝒵α,0

4. Compactness of uCφ

Acknowledgments

References

Citing Literature

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Information

3. Boundedness of uC_φ from H^p (0 < p < ∞) to 𝒵_α and 𝒵_α,0

4. Compactness of uC_φ