Composition operators between the Hardy space Hp and the Bergman spaces Lqa

Introduction

For p>0, α >-1, the weighted Dirichlet space Ⅾ α ｐ is defined to be the space of analytic functions f on D such that .

In case α＝1 and p=2, then Ⅾ 1 2 = H 2 is the classical Hardy space. Furthermore, in case α = p and 1 ≤ p < ∞, then Ⅾ p ｐ = L p a is the usual Bergman space. Also, in case α = p-2 and 1 < p < ∞, Ⅾ p-2 ｐ = Bp is called the Besov space. In particular, Ⅾ 0 2 =Ⅾ is called the Dirichlet space. Also, in case α = p-1 and 0 < p < ∞, Ⅾ p-1 ｐ = Bp、p-1 is called the Besov type space. In particular, is the classical Hardy space.

It is trivial that Bp ⊂ Bp, p-1 (p>1).

For α > 0, the Bloch type space Bα of D is defined to be the space of analytic functions f on D such that For p>0 and α > 0, the space Ｂ p α of D is defined to be the space of analytic functions f on D such that . In case of p = 2 and α = 1, thenＢ ２

1 is the classical Bloch space. Let X be Banach spaces and let T be a linear operator from X into X. Then T is called to be bounded below on X if there exists a positive constant K > 0 such that ∥ Tf ∥≥ K ∥ f ∥ for all f ∈ X. By Schwarz-Pick lemma, the composition operator Cφ is bounded on the Bloch space B. It follows from Littlewood's subordination theorem that Cφ is bounded on all the Bergman spaces L p a. And Cφ is also bounded on all the Hardy space H p . In [14] Nina Zorboska has been characterized the closed range composition operators on the Bergman spaces L p a and Hardy spaces H p . H.Chen and P.Gauthier have been characterized the boundedness from below of composition operators on weighted Bloch space in [4] . On the other hand, Wayne Smith has been characterized the boundedness and compactness of the composition operators between Bergman spaces L p a and Hardy spaces H p in [6]. In this paper we characterize the result that it has not been elucidated yet so far with respect to the boundedness from below of composition operators between Hardy space H p and Bergman spaces L q a, between the Besov type space Bp,p-1 and the Bergman spaces L p a．

Background Material

In [6] Wayne Smith has been characterized the boundedness and compactness of the composition operators between Bergman spaces La p and Hardy spaces H p , between Bergman spaces L p a and Bergman spaces L q a, between Hardy spaces H p and Hardy spaces H q .

In [7] , W.Smith and L. Yang proved the following theorem.

Applying α = 0 and β =-1 in Theorem A implies the following (see [7] ).

Corollary B. Let 0 < q < p and suppose that φ is an analytic self-map of D. Then the following are equivalent.

Corollary B immediately yields the following. Theorem C. Let 0 < q < p. If Cφ: L p a → H q is bounded, then there is no symbol φ such that it is bounded below. Proof. If Cφ: L p a → H q is bounded, then Cφ: L p a → H q is compact. Hence Corollary B implies that there is no symbol φ such that it is bounded below. Theorem A fails for α =-1, that is, we can not apply Theorem A to the case of the composition operator Cφ on the Hardy space H p (see [7] ). In this paper, we do study this case (α =-1) with respect to the bounded below of composition operators from Hardy space H p to Bergman spaces L q a.

The main results

If φ(0) = a and ψ = φa • φ, then Cφ is bounded below on H p if and only if Cψ is bounded below on H p . So we assume from now on that φ(0) = 0 and that Cφ is acting on the subspace of functions that vanish at the origin. In [4] H.Chen and P.Gauthier proved the following result with respect to the composition operators Cφ: Bα → Bβ. Theorem D. ( [4] ) Suppose β ≥ 1 and α ≤ β. Then Cφ: Bα → Bβ is bounded, while Cφ: Bα → Bβ is not bounded below if α < β. In [10] we also proved the following result. Theorem E. ( [10]) Let 0 < p, q < ∞, α,γ > 0. Suppose that Cφ: Ⅾ pα ｐ→ Ⅾ ｑγ ｑ is bounded. If Cφ: Ⅾ pα ｐ→ Ⅾ ｑγ ｑ is bounded below, then there exists a constant K > 0 such that supz∈D |(Cφf) ′ (z)|(1-|z| 2 ) γ ≥ KSp,q,α(f) for all f∈Bα, where If p ≥ q > 1 and Cφ: Bp,p-1→ L q a, then we have the following.

. Proof. Since 1 < q ≤ p, the boundedness of Cφ: Bp, p-1 → L q a follows from H¨older's inequality. Applying α = 1-1/p, γ = 1 and 1 < q ≤ p in Theorem E, it follows from Theorem D that Cφ: Bp, p-1 → L q a is not bounded below. Corollary 2. If 1 < q ≤ 2, then Cφ: H 2 →L q a is bounded, while there is no symbol φ such that it is bounded below. Proof. Since 1 < q ≤ 2, the boundedness of Cφ: H 2 → L q a follows from H¨older's inequality. Applying 1 < q ≤ 2, it follows from Theorem 1 that Cφ: H 2 → L q a is not bounded below.

Let α >-1. For ∀a ∈ D, the following estimate is standard (see, for example, [12] ).

To prove main theorem, we need the well-known following lemma (see [12]). f ∈ L p a, then .

To prove main theorem, we need the following lemma. Lemma 3. Let φ be an analytic self-map of the disk. Then , Proof. Using Schwarz-Pick lemma, we have Using Lemma 3, Lemma F and the estimate (1), we have the following main result that generalizes Corollary 2. Theorem 4. If 0 < q ≤ p, then Cφ: H p → L q a is bounded, while there is no symbol φ such that it is bounded below. Proof. Since 0 < q ≤ p, the boundedness of Cφ: H p → L q a follows from Holder's inequality. Let N = { f ∈ H(D), f(z) ≠ 0 (∀z ∈ D) }. Suppose that Then for a ∈ D, we choose some constant c1 such that For every z ∈ D. Put .

In fact, for , Using the evaluation (1), it holds that there exists a constant K > 0 such that Hence Since , using the boundedness of the composition operator Cφ on L 2 a, there is a positive C > 0 such that .

In the case of p>q, suppose that there exists a symbol φ such that Cφ: H p → L q a is bounded below. Since p/q > 1, using H¨older's inequality and Lemma F, for any a ∈ D, there exists a constant K > 0 such that https://www.mathsjournal.com . Using (2) and (3), we have In the case of p = q, without using Holder's inequality we can also prove this inequality as well.

Here for w ∈ D, we consider the following test function:

Then we see that . In fact, for w ∈ D, the following holds.

. Thus . On the other hand, we have Here, let wn → ∂D. By using Lemma 3, applying w = wn(∈ D) (n =1,2,•••),then Considering (4.3), it contradicts that inequalities (4.1) and (4.2) hold. Hence in the case of 0 < q ≤ p, there don't exist symbol φ such that Cφ: H p → L q a is bounded below.

Acknowledgment

The author wishes to express his sincere gratitude to Professor Mikihiro Hayashi for his many helpful suggestions and advices.

Conclusion

This paper has delved into the intricate realm of composition operators on various function spaces over the open unit disk. Through rigorous analysis and leveraging established theorems, the boundedness and bounded below properties of these operators have been explored, shedding light on fundamental aspects of complex analysis. Notably, the results obtained contribute to a deeper understanding of operator behavior within the context of Hardy and Bergman spaces, Besov spaces, and Bloch type spaces.

The findings presented here enrich the existing body of knowledge in the field and pave the way for further investigations into the interplay between function spaces and composition operators.

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