A derivative-Hilbert operator acting on Dirichlet spaces

Yun Xu; Shanli Ye; Zhihui Zhou

doi:10.1515/math-2022-0559

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A derivative-Hilbert operator acting on Dirichlet spaces

Yun Xu , Shanli Ye and Zhihui Zhou

Published/Copyright: February 28, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

Let μ be a positive Borel measure on the interval [ 0 , 1 ) . The Hankel matrix H μ = ( μ n , k ) n , k ≥ 0 with entries μ n , k = μ n + k , where μ n = ∫ [ 0 , 1 ) t n d μ ( t ) , induces formally the operator as follows:

DH μ ( f ) ( z ) = ∑ n = 0 ∞ ∑ k = 0 ∞ μ n , k a k ( n + 1 ) z n , z ∈ D ,

where f ( z ) = ∑ n = 0 ∞ a n z n is an analytic function in D . In this article, we characterize those positive Borel measures on [ 0 , 1 ) for which DH μ is bounded (resp. compact) from Dirichlet spaces D α ( 0 < α ≤ 2 ) into D β ( 2 ≤ β < 4 ) .

Keywords: Hilbert operator; Dirichlet space; Carleson measure

MSC 2010: 47B38; 47B35; 30H99

1 Introduction

Let D = { z ∈ C : ∣ z ∣ < 1 } be the open unit disk in the complex plane C , and let H ( D ) denote the class of all analytic functions in D .

For 0 < p < ∞ and f ∈ H ( D ) , the integral means M p ( r , f ) are defined by

M p ( r , f ) = 1 2 π ∫ 0 2 π ∣ f ( r e i θ ) ∣ p d θ 1 p , 0 < r < 1 .

The Hardy space H p ( 0 < p < ∞ ) consists of those functions f ∈ H ( D ) with

‖ f ‖ H p = sup 0 < r < 1 M p ( r , f ) < ∞ ,

and H ∞ is the space of all bounded functions f in H ( D ) . We refer to [1] for the theory of Hardy spaces.

For 0 < p < ∞ , the Bergman space A p consists of all functions f ∈ H ( D ) for which

‖ f ‖ A p p = ∫ D ∣ f ( z ) ∣ p d A ( z ) < ∞ ,

where d A denotes the normalized Lebesgue area measure on D . We refer to [2] for more information about Bergman spaces.

For α ∈ R , the Dirichlet space D α consists of all functions f ( z ) = ∑ n = 0 ∞ a n z n ∈ H ( D ) for which

‖ f ‖ D α 2 = ∑ n = 0 ∞ ( n + 1 ) 1 − α ∣ a n ∣ 2 < ∞ .

We obtain the classical Dirichlet space D = D 0 if α = 0 (see [3]), we obtain the Hardy space H 2 = D 1 if α = 1 (see [1,4]), and we obtain the Bergman space A 2 = D 2 if α = 2 . We mention [3] for complete information on Dirichlet spaces.

Suppose that μ is a positive Borel measure on [0,1), we furtherly define H μ to be the Hankel matrix ( μ n , k ) n , k ≥ 0 with entries μ n , k = μ n + k , where μ n = ∫ [ 0 , 1 ) t n d μ ( t ) . The matrix H μ can be seen as an operator on f ( z ) = ∑ k = 0 ∞ a k z k ∈ H ( D ) by its action on the Taylor coefficients: { a n } n ≥ 0 → ∑ k = 0 ∞ μ n , k a k n ≥ 0 . Furthermore, we can formally induce the Hankel operator H μ as follows:

H μ ( f ) ( z ) = ∑ n = 0 ∞ ∑ k = 0 ∞ μ n , k a k z n , z ∈ D ,

whenever the right-hand side of this equation can be defined as an analytic function in D . If μ is the Lebesgue measure, H μ is the classical Hilbert operator H . This is why H μ is called a generalized Hilbert operator.

The operator H μ has been extensively studied in [5–13]. Galanopoulos and Peláez [13] characterized those measures μ supported on [ 0 , 1 ) such that the generalized Hilbert operator H μ is well defined and is bounded on H 1 . Chatzifountas et al. [5] described those measures μ for which H μ is a bounded operator from H p into H q , where 0 < p , q < ∞ . Diamantopoulos [9] gave many results about the operator induced by Hankel matrices on Dirichlet spaces. Recently, Girela and Merchán [6] have studied the operators H μ acting on certain conformally invariant spaces.

In Ye and Zhou’s works [14,15], they defined the derivative-Hilbert operator DH μ as follows:

DH μ ( f ) ( z ) = ∑ n = 0 ∞ ∑ k = 0 ∞ μ n , k a k ( n + 1 ) z n .

It is closely related to the generalized Hilbert operator, that is,

DH μ ( f ) ( z ) = ( z H μ ( f ) ( z ) ) ′ .

Therefore, we called DH μ to be the derivative-Hilbert operator. In that work, the second and the third authors characterized the measures μ for which DH μ is a bounded (resp. compact) operator from A p into A q for some p and q , and they also characterized the measures μ for which DH μ is a bounded (resp. compact) operator on the Bloch space.

Let us recall the definition of Carleson-type measures that play a very important role in the theory of Banach spaces of analytic functions. We refer to [16,17] for some results about Carleson measures.

If I ⊂ ∂ D in an arc, ∣ I ∣ denotes the length of I , and the Carleson square S ( I ) is defined as follows:

S ( I ) = z = r e i t : e i t ∈ I , 1 − ∣ I ∣ 2 π ≤ r < 1 .

Suppose 0 < p < ∞ and μ is a positive Borel measure on D , then μ is said to be an s -Carleson measure if there exists a positive constant C such that

μ ( S ( I ) ) ≤ C ∣ I ∣ s , for any interval I ⊂ ∂ D .

Here, μ is said to be a vanishing s -Carleson measure if

lim ∣ I ∣ → 0 μ ( S ( I ) ) ∣ I ∣ s = 0 .

If μ is a Borel measure on [0, 1), it can been seen as a Borel measure on D by identifying it as μ ˆ ( A ) = μ ( A ∩ [ 0 , 1 ) ) , for every Borel set A ⊂ D . In this way, for 0 < s < ∞ , we called μ to be an s -Carleson measure if there exists a positive constant C such that

μ ( [ t , 1 ) ) ≤ C ( 1 − t ) s , t ∈ [ 0 , 1 ) .

Also, μ is a vanishing s -Carleson measure on [ 0 , 1 ] if μ satisfies

lim t → 1 − μ ( [ t , 1 ) ) ( 1 − t ) s = 0 .

In this article, we mainly characterize the positive Borel measures μ on [ 0 , 1 ) for which the derivative-Hilbert operator DH μ is bounded (resp. compact) from Dirichlet spaces D α ( 0 < α ≤ 2 ) into D β ( 2 ≤ β < 4 ) .

In this work, C denotes a positive constant that only depends on the displayed parameters but not necessarily the same from one occurrence to the next. In addition, we say that A ≳ B if there exist a constant C (independent of A and B ) such that A ≥ C B , and A ≲ B is the same as A ≳ B .

2 Main results

We shall first give a sufficient condition such that the operator DH μ is well defined on the Dirichlet space D α , for α ≥ − 1 . And we characterize the measure μ such that DH μ is bounded from Dirichlet spaces D α ( 0 < α ≤ 2 ) into D β ( 2 ≤ β < 4 ) .

Theorem 2.1

Suppose that α ≥ − 1 , and let μ be a positive Borel measure on [ 0 , 1 ) . If the moments of μ satisfy that μ n = O n − ( α 2 + ε ) for some ε > 0 , then DH μ is well defined on D α .

Proof

Suppose f ( z ) = ∑ n = 0 ∞ a n z n ∈ D α . By Cauchy-Schwarz inequality, we obtain that

∑ k = 0 ∞ μ n , k a k ≤ ∑ k = 0 ∞ μ n , k ∣ a k ∣ ≲ ∑ k = 0 ∞ ∣ a k ∣ ( n + k + 1 ) α 2 + ε = ∑ k = 0 ∞ ( k + 1 ) α − 1 2 1 ( n + k + 1 ) α 2 + ε ( k + 1 ) 1 − α 2 ∣ a k ∣ ≤ ∑ k = 0 ∞ ( k + 1 ) α − 1 ( n + k + 1 ) α + 2 ε 1 2 ∑ k = 0 ∞ ( k + 1 ) 1 − α ∣ a k ∣ 2 1 2 = ∑ k = 0 ∞ 1 ( k + 1 ) 1 + 2 ε 1 2 ‖ f ‖ D α < ∞ .

This shows that the operator DH μ is well defined on D α . □

Next, we import an auxiliary lemma, which is needed for the main theorem in this article.

Lemma 2.1

[18, Theorem 318] Let K ( x , y ) be a real function of two variables and has the following properties:

K ( x , y ) is non-negative and homogeneous of degree − 1 ;
∫ 0 ∞ K ( x , 1 ) x − 1 2 d x = ∫ 0 ∞ K ( 1 , y ) y − 1 2 d y = C ;
K ( x , 1 ) x − 1 2 is a strictly decreasing function of x , and K ( 1 , y ) y − 1 2 of y ; or, more generally;
K ( x , 1 ) x − 1 2 decreases from x = 1 onwards, while the interval ( 0 , 1 ) can be divided into two parts, ( 0 , ξ ) and ( ξ , 1 ) , of which one may be null, in the first of which it decreases and in the second of which it increases; and K ( 1 , y ) y − 1 2 has similar properties; and K ( x , x ) = 0 .

Then for every sequence { a n } n ≥ 0 such that ∑ n = 0 ∞ ∣ a n ∣ 2 < ∞ , we obtain

∑ n = 1 ∞ ∑ k = 1 ∞ K ( n , k ) ∣ a k ∣ 2 ≤ C 2 ∑ n = 1 ∞ ∣ a n ∣ 2 .

In short, if f ( z ) = ∑ n = 0 ∞ a n z n ∈ H 2 , we have

∑ n = 1 ∞ ∑ k = 1 ∞ K ( n , k ) ∣ a k ∣ 2 ≤ C 2 ‖ f ‖ H 2 2 .

Theorem 2.2

Suppose that 0 < α ≤ 2 , 2 ≤ β < 4 , and let μ be a positive Borel measure on [ 0 , 1 ) , which satisfies the condition in Theorem 2.1. Then the following conditions are equivalent:

μ is a 2 − β − α 2 -Carleson measure.
μ n = O 1 n 2 − β − α 2 .
DH μ is a bounded operator from D α into D β .

Before giving the proof, let us recall some classical conclusions about the Beta function. The Beta function B ( s , t ) can be defined as follows:

B ( s , t ) = ∫ 0 ∞ x s − 1 ( 1 + x ) s + t d x ,

for each s , t with Re ( s ) > 0 , Re ( t ) > 0 . The value B ( s , t ) can be expressed in terms of the Gamma function as follows:

B ( s , t ) = Γ ( s ) Γ ( t ) Γ ( s + t ) .

Now we continue to complete the proof of the Theorem 2.2.

Proof

(i) ⇔ (ii). The result can be found in [5,10].

(ii) ⇒ (iii). First, we define two operators. For f ( z ) = ∑ n = 0 ∞ a n z n ∈ D α , we define V α ( f ) by the formula

V α ( f ) ( z ) = ∑ n = 0 ∞ ( n + 1 ) 1 − α 2 a n z n ,

and for g ( z ) = ∑ n = 0 ∞ b n z n ∈ H 2 , we define T β ( g ) by the formula:

T β ( g ) ( z ) = ∑ n = 0 ∞ ( n + 1 ) β − 1 2 b n z n .

It is easy to check that V α is a bounded operator from D α into H 2 , and T β is a bounded operator from H 2 into D β .

Now suppose that 0 < α ≤ 2 and 2 ≤ β < 4 . We consider a new operator S μ defined as follows: If h ( z ) = ∑ n = 0 ∞ c n z n ∈ H 2 , we define S μ ( h ) by

S μ ( h ) ( z ) = ∑ n = 0 ∞ ∑ k = 0 ∞ ( n + 1 ) 3 − β 2 ( k + 1 ) α − 1 2 μ n , k c k z n .

A direct calculation shows that

‖ S μ ( h ) ‖ H 2 2 = ∑ n = 0 ∞ ∑ k = 0 ∞ ( n + 1 ) 3 − β 2 ( k + 1 ) α − 1 2 μ n , k c k 2 ≤ ∑ n = 0 ∞ ∑ k = 0 ∞ ( n + 1 ) 3 − β 2 ( k + 1 ) α − 1 2 μ n , k ∣ c k ∣ 2 ≲ ∑ n = 0 ∞ ∑ k = 0 ∞ ( n + 1 ) 3 − β 2 ( k + 1 ) α − 1 2 ∣ c k ∣ ( n + k + 2 ) 2 − β − α 2 2 = ∑ n = 1 ∞ ∑ k = 1 ∞ n 3 − β 2 k α − 1 2 ∣ c k − 1 ∣ ( n + k ) 2 − β − α 2 2 .

Let

K ( x , y ) = x 3 − β 2 y α − 1 2 1 ( x + y ) 2 − β − α 2 , x > 0 , y > 0 .

Then we obtain that

∫ 0 ∞ K ( x , 1 ) x − 1 2 d x = ∫ 0 ∞ x 1 − β 2 ( x + 1 ) 2 − β − α 2 d x = B 2 − β 2 , α 2 , ∫ 0 ∞ K ( 1 , y ) y − 1 2 d y = ∫ 0 ∞ y α 2 − 1 ( y + 1 ) 2 − β − α 2 d y = B α 2 , 2 − β 2 .

And it is clear that the functions K ( x , 1 ) x − 1 2 and K ( 1 , y ) y − 1 2 are strictly decreasing. By applying Lemma 2.1, we have

∑ n = 1 ∞ ∑ k = 1 ∞ n 3 − β 2 k a − 1 2 ∣ c k − 1 ∣ ( n + k ) 2 − β − α 2 2 ≲ B ( 2 − β 2 , α 2 ) 2 ‖ h ‖ H 2 2 .

This implies that the operator S μ is bounded on H 2 .

For each f ∈ D α , it is easy to check that

T β ∘ S μ ∘ V α ( f ) ( z ) = ∑ n = 0 ∞ ( n + 1 ) β − 1 2 ∑ k = 0 ∞ ( n + 1 ) 3 − β 2 ( k + 1 ) α − 1 2 ( k + 1 ) 1 − α 2 μ n , k a k z n = ∑ n = 0 ∞ ∑ k = 0 ∞ μ n , k a k ( n + 1 ) z n = DH μ ( f ) ( z ) .

Hence, DH μ is bounded from D α into D β .

(iii) ⇒ (i). For 0 < t < 1 , let f t ( z ) = ( 1 − t 2 ) 1 − α 2 ∑ n = 0 ∞ t n z n . We have that

‖ f t ‖ D α 2 = ( 1 − t 2 ) 2 − α ∑ n = 0 ∞ ( n + 1 ) 1 − α t 2 n ≈ 1 .

Therefore,

‖ DH μ ( f t ) ‖ D β 2 = ∑ n = 0 ∞ ( n + 1 ) 1 − β ∑ k = 0 ∞ ( n + 1 ) μ n , k ( 1 − t 2 ) 1 − α 2 t k 2 ≳ ( 1 − t 2 ) 2 − α ∑ n = 0 ∞ ( n + 1 ) 3 − β ∑ k = 0 ∞ t k ∫ t 1 ς n + k d μ ( ς ) 2 ≳ ( 1 − t 2 ) 2 − α ∑ n = 0 ∞ ( n + 1 ) 3 − β ∑ k = 0 n t n + 2 k μ ( [ t , 1 ) ) 2 .

Since DH μ is bounded from D α into D β , we obtain that

‖ DH μ ‖ D β 2 ‖ f t ‖ D β 2 ≥ ‖ DH μ ( f t ) ‖ D β 2 ≳ ( 1 − t 2 ) 2 − a ∑ n = 0 ∞ ( n + 1 ) 3 − β ∑ k = 0 n t n + 2 k μ ( [ t , 1 ) ) 2 ≳ ( 1 − t 2 ) 2 − a ∑ n = 0 ∞ ( n + 1 ) 5 − β t 6 n μ ( [ t , 1 ) ) 2 ≈ μ ( [ t , 1 ) ) 2 ( 1 − t 2 ) 4 + α − β .

This implies that

μ ( [ t , 1 ) ) ≲ ( 1 − t 2 ) 2 − β − α 2 ,

which is equivalent to saying that μ is a ( 2 − β − α 2 ) -Carleson measure.□

In particular, if we take α = β = 2 in Theorem 2.2, we can obtain the following corollary, which the second and the third authors have proved in [14].

Corollary 2.1

The operator DH μ is bounded on A 2 if and only if the measure μ is a 2-Carleson measure.

Lemma 2.2

Let 0 < α ≤ 2 , 2 ≤ β < 4 , and DH μ is a bounded operator from D α into D β . Then DH μ is a compact operator if and only if DH μ ( f n ) → 0 in H 2 , for any bounded sequence { f n } in D α , which converges to 0 uniformly on every compact subset of D .

Proof

The proof is similar to that of in [19, Proposition 3.11], and we omit the details.□

Theorem 2.3

Suppose that 0 < α ≤ 2 , 2 ≤ β < 4 , and let μ be a positive Borel measure on [ 0 , 1 ) , which satisfies the condition in Theorem 2.1. Then the following conditions are equivalent:

μ is a vanishing ( 2 − β − α 2 ) -Carleson measure.
μ n = o 1 n 2 − β − α 2 .
DH μ is a compact operator from D α into D β .

Proof

(i) ⇔ (ii). The result can be found in [5,10].

(ii) ⇒ (iii). Take f ( z ) = ∑ n = 0 ∞ a n z n ∈ D α and h ( z ) = ∑ n = 0 ∞ c n z n ∈ H 2 . Let

S μ , m ( h ) ( z ) = ∑ n = 0 m ∑ k = 0 ∞ ( n + 1 ) 3 − β 2 ( k + 1 ) α − 1 2 μ n , k c k z n .

Notice that S μ , m is a finite rank operator, then S μ , m is compact on H 2 . Since μ n satisfies μ n = o ( n − ( 2 − β − α 2 ) ) , we obtain that for any ε > 0 , and there exists an N > 0 such that ∣ μ m ∣ < ε n − ( 2 − β − α 2 ) when m > N . Then we note

( S μ − S μ , m ) ( h ) ( z ) = ∑ n = m + 1 ∞ ∑ k = 0 ∞ ( n + 1 ) 3 − β 2 ( k + 1 ) α − 1 2 μ n , k c k z n ,

( T β ∘ S μ ∘ V a − T β ∘ S μ , m ∘ V a ) ( f ) ( z ) = ∑ n = m + 1 ∞ ∑ k = 0 ∞ ( n + 1 ) μ n , k a k z n = T β ∘ ( S μ − S μ , m ) ∘ V a ( f ) ( z ) = ( DH μ − DH μ , m ) ( f ) ( z ) .

Therefore,

‖ ( S μ − S μ , m ) ( h ) ‖ H 2 2 = ∑ n = m + 1 ∞ ∑ k = 0 ∞ ( n + 1 ) 3 − β 2 ( k + 1 ) α − 1 2 μ n , k c k 2 .

Then for m > N , we have

‖ ( S μ − S μ , m ) ( h ) ‖ H 2 2 ≲ ε 2 ∑ n = m + 1 ∞ ∑ k = 0 ∞ ( n + 1 ) 3 − β 2 ( k + 1 ) α − 1 2 ∣ c k ∣ ( n + k + 2 ) 2 − β − α 2 2 .

By Lemma 2.1 and the proof of Theorem 2.2, we obtain

‖ ( S μ − S μ , m ) ( h ) ‖ H 2 2 ≲ ε 2 ‖ h ‖ H 2 2 .

Thus,

‖ S μ − S μ , m ‖ H 2 → H 2 ≲ ε .

It is clear that

‖ DH μ − DH μ , m ‖ D α → D β ≲ ε .

Hence, DH μ is compact from D α into D β .

(iii) ⇒ (i). For 0 < t < 1 , let f t ( z ) = ( 1 − t 2 ) 1 − α 2 ∑ n = 0 ∞ t n z n , we have

‖ f t ‖ D α 2 = ( 1 − t 2 ) 2 − α ∑ n = 0 ∞ ( n + 1 ) 1 − α t 2 n ≈ 1 ,

and lim t → 1 f t ( z ) = 0 for any z ∈ D . Since all Hilbert spaces are reflexive, we obtain that f t is convergent weakly to 0 in D α as t → 1 . By the assumption that DH μ is compact from D α into D β , we have

lim t → 1 ‖ DH μ ( f t ) ‖ D β = 0 .

Similar to the proof of Theorem 2.2, we obtain that

μ ( [ t , 1 ) ) ≲ ( 1 − t ) 2 − β − α 2 ‖ DH μ ( f t ) ‖ D β .

Therefore,

lim t → 1 μ ( [ t , 1 ) ) ( 1 − t ) 2 − β − α 2 = 0 .

Thus, μ is a vanishing ( 2 − β − α 2 ) -Carleson measure.□

Acknowledgments

The authors are grateful to the referees for instructive comments and numerous stylistic corrections.

Funding information: The research was supported by the National Natural Science Foundation of China (Grant No. 11671357) and the Zhejiang Provincial Natural Science Foundation (Grant No. LY23A010003).
Conflict of interest: The authors state no conflict of interest.

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Received: 2022-07-06

Revised: 2023-01-23

Accepted: 2023-01-24

Published Online: 2023-02-28

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Keywords for this article

Hilbert operator; Dirichlet space; Carleson measure

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