Cesàro-like operators acting on spaces of analytic functions

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We study the action of the operators Cμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}_\mu $$\end{document} on distinct spaces of analytic functions in D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {D}}$$\end{document}, such as the Hardy spaces Hp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^p$$\end{document}, the weighted Bergman spaces Aαp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^p_\alpha $$\end{document}, BMOA, and the Bloch space B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document}.


Introduction and main results
Let D = {z ∈ C : |z| < 1} denote the open unit disc in the complex plane C and let Hol(D) be the space of all analytic functions in D endowed with the topology of uniform convergence in compact subsets.
If 0 < r < 1 and f ∈ Hol(D), we set For 0 < p ≤ ∞, the Hardy space H p consists of those f ∈ Hol(D) such that We refer to [13] for the notation and results regarding Hardy spaces. Let d A denote the area measure on D, normalized so that the area of D is 1. Thus d A(z) = 1 π dx dy = 1 π r dr dθ . For 0 < p < ∞ and α > −1 the weighted Bergman space A p α consists of those f ∈ Hol(D) such that where d A α (z) = (α + 1)(1 − |z| 2 ) α d A(z). We refer to [14,25,39] for the notation and results about Bergman spaces.
The space B M O A consists of those functions f ∈ H 1 whose boundary values have bounded mean oscillation on ∂D. We refer to [16] for the theory of B M O A-functions.
Finally, we recall that a function f ∈ Hol(D) is said to be a Bloch function if The space of all Bloch functions is denoted by B. A classical reference for the theory of Bloch functions is [2]. Let us recall that The Cesàro operator C is defined over the space of all complex sequences as follows: k=0 is a sequence of complex numbers then C ((a)) = 1 n + 1 n k=0 a k ∞ n=0 .
Identifying any given function f ∈ Hol(D) with the sequence {a k } ∞ k=0 of its Taylor coefficients, the Cesàro operator C becomes a linear operator from Hol(D) into itself as follows: If f ∈ Hol(D), f (z) = ∞ k=0 a k z k (z ∈ D), then The Cesàro operator is bounded on H p for 0 < p < ∞. For 1 < p < ∞, this follows from a result of Hardy on Fourier series [22] together with the M. Riesz's theorem on the conjugate function [13,Theorem 4.1]. Siskakis [33] used semigroups of composition operators to give an alternative proof of this result and to extend it to p = 1. A direct proof of the boundedness on H 1 was given by Siskakis in [34]. Miao [31] dealt with the case 0 < p < 1. Stempak [36] gave a proof valid for 0 < p ≤ 2 and Andersen [1] provided another proof valid for all p < ∞.
In this paper we associate to every positive finite Borel measure on [0, 1) a certain operator C μ acting on Hol(D) which is a natural generalization of the classical Cesàro operator C.
If μ is a positive finite Borel measure on [0, 1) and n is a non-negative integer, we let μ n denote the moment of order n of μ, that is, 1) t n dμ(t), n = 0, 1, 2, . . . .
It is clear that C μ is a well defined linear operator C μ : Hol(D) → Hol(D). When μ is the Lebesgue measure on [0, 1), the operator C μ reduces to the classical Cesàro operator C.
Our main objective in this work is to characterize those positive finite Borel measures μ on [0, 1) such that the operator C μ is bounded or compact on classical subspaces of Hol(D) such as the Hardy spaces H p , the weighted Bergman spaces A p α , and the spaces B M O A and B.
Measures of Carleson type will play a basic role in the sequel. If I ⊂ ∂D is an interval, |I | will denote the length of I . The Carleson square S(I ) is defined as If s > 0 and μ is a positive Borel measure on D, we shall say that μ is an s-Carleson measure if there exists a positive constant C such that μ (S(I )) ≤ C|I | s , for any intervalI ⊂ ∂D.
A 1-Carleson measure, respectively, a vanishing 1-Carleson measure, will be simply called a Carleson measure, respectively, a vanishing Carleson measure.
Following [38], if μ is a positive Borel measure on D, 0 ≤ α < ∞, and 0 < s < ∞, we say that μ is an α-logarithmic s-Carleson measure if there exists a positive constant C such that If μ (S(I )) log 2 Section 3 will be devoted to present the proofs of Theorem 1 and Theorem 2 as well as some further results concerning the action of the operators C μ on Hardy spaces. Section 4 will deal with the action of the operators C μ on Bergman spaces and, as we have already mentioned, Sect. 5 will be devoted to study the operators C μ acting on B M O A, the Bloch space, and some related spaces. In particular, Sect. 5 will include a proof of Theorem 3 and the substitute of this result concerning compactness.
In Sect. 2 we shall give two alternative representations of the operator C μ , one of them is an integral representation and the other one involves the convolution with a fixed analytic function in D. We shall also introduce a related operator which will be denoted T μ and which will play a basic role in the proofs of some of our results.
Throughout the paper, if μ is a finite positive Borel measure on [0, 1), for n ≥ 0, μ n will denote the moment of order n of μ. Also, we shall be using the convention that C = C( p, α, q, β, . . . ) will denote a positive constant which depends only upon the displayed parameters p, α, q, β . . . (which sometimes will be omitted) but not necessarily the same at different occurrences. Furthermore, for two real-valued functions K 1 , K 2 we write K 1 K 2 , or K 1 K 2 , if there exists a positive constant C independent of the arguments such that K 1 ≤ C K 2 , respectively K 1 ≥ C K 2 . If we have K 1 K 2 and K 1 K 2 simultaneously, then we say that K 1 and K 2 are equivalent and we write K 1 K 2 .
Let us close this section noticing that, since the subspaces X of Hol(D) we shall be dealing with are Banach spaces continuously embedded in Hol(D), to prove that the operator C μ (or T μ , to be defined below) is bounded on X it suffices to show that it maps X into X by appealing to the closed graph theorem.

Alternative representations of C and a related operator
A simple calculation with power series gives the following integral representation of the operators C μ .

Proposition 1
If μ is a positive finite Borel measure on [0, 1) and f ∈ Hol(D) then Next we shall give another expression for C μ ( f ) involving the convolution of analytic functions. If f and g are two analytic functions in the unit disc, f (z) = ∞ n=0 a n z n , g(z) = ∞ n=0 b n z n , z ∈ D, the convolution f g of f and g is defined by f g(z) = ∞ n=0 a n b n z n , z ∈ D.
Lemma 1 Let μ be a positive finite Borel measure on [0, 1) and set If f ∈ Hol(D) and The proof is elementary and will be omitted.
The following result regarding the radial measures μ we are considering will be used in our work.

Lemma 2
Let μ be a finite positive Borel measure on the interval [0, 1) and, for n ≥ 0, let μ n denote the moment of order n of μ.

(i) μ is a Carleson measure if and only if
Proof (i) is Proposition 8 of [8] and (ii) follows with a similar argument. Lemma 2. 7 of [19] gives one implication of (iii) and the other one follows from the from the simple inequality Finally, (iv) can be proved with an argument similar the the one used to prove (iii). Now we define a new operator operator T μ associated to μ which will be important in our work because it will become the adjoint of C μ in distinct instances.
If μ is a finite positive Borel measure on [0, 1) and f ∈ Hol(D), f (z) = ∞ n=0 a n z n (z ∈ D) we set whenever the right hand side makes sense and defines an analytic function in D.
Clearly, the operator T μ is not defined over the whole space Hol(D). We have the following result. (a) If P is a polynomial then T μ (P) is well defined and it also a polynomial. Lemma 2). This and Hardy's inequality [13, p. 48] shows that if It is well known that, for 1 < p < ∞, the dual of H p is identifiable with H q , and f (z) = ∞ n=0 a n z n ∈ A (i) If 1 < p < ∞, f ∈ H p , and g is a polynomial then Proposition 3, together with the fact that the polynomials are dense in all the spaces H p ( p < ∞) and A p α ( p < ∞, α > −1), readily implies the following result.

Proposition 4
Suppose that 1 < p < ∞ and let μ be a positive finite Borel measure on [0, 1). Let q be the conjugate exponent of p, that is, 1 p + 1 q = 1.

(i) If C μ is a bounded operator from H p into itself, then there exists a positive constant C such that
for every polynomial P. Consequently, T μ extends to a bounded linear operator from H q into itself. This extension, which will be also denoted by T μ , is the adjoint for every polynomial P. Consequently, T μ extends to a bounded linear operator from A q α into itself. This extension, which will be also denoted by T μ , is the adjoint of C μ .

The operators C acting on Hardy spaces
In this section we shall study the action of the operators C μ on Hardy spaces.
We shall use complex interpolation to prove some of our results. Let us refer to [39,Chapter 2] for the terminology and basic results concerning complex interpolation.

Proof of Theorem 1
We shall split it in several cases. Proof of the implication (i) ⇒ (ii) when p = 1. Assume that μ is a Carleson measure and take f ∈ H 1 . Set Using the integral representation on C μ , we see that, for 0 < r < 1, Using this, the Hardy-Littlewood maximal theorem [13, Theorem 1.9], the fact that integral means M 1 (s, g) increase with s, and the Cauchy-Schwarz inequality, we obtain Making the change of variables t = rs in the last integral and setting f r (z) = f (r z) (z ∈ D), it follows that Using a result of Hardy and Littlewood [23] (see also [34]) we see that Then it follows that This Assume that μ is a Carleson measure and take f ∈ H 2 , f (z) = ∞ n=0 a n z n (z ∈ D). Using [8, Proposition 1] we see that |μ n | 1 n+1 . Using this, the definition of C μ ( f ), and the fact that the Cesàro operator is bounded on H 2 , it follows that Proof of the implication (i) ⇒ (ii) for 1 < p < 2. Since (i) ⇒ (ii) when p = 1 and p = 2, the fact that (i) ⇒ (ii) when 1 < p < 2 follows using (3) and Theorem 2. 4 of [39].
To prove the remaining case, that is, the implication (i) ⇒ (ii) for 2 < p < ∞ we shall use ideas of Andersen [1]. Actually, our next argument works for 1 < p < ∞.
Using this and (5) it follows that Then the argument in p. 622 of [1] yields that and For 0 < a < 1, set a n (2/ p)a n z n , z ∈ D.
We have that f a ∈ H p and f a H p = 1, 0 < a < 1.
Since C μ is bounded on H p , we have Now Using the fact that 1 ≤ p ≤ 2, [13, Theorem 6.2], (7), and the fact that the sequence {μ n } is decreasing, we obtain for every positive integer N and every a ∈ (0, 1). Taking a = 1 − 1 N and using the fact that C μ is bounded on H p , we obtain This and (8) imply that μ N 1 N . Using again Lemma 2, this yields that μ is a Carleson measure. Proof of the implication (ii) ⇒ (i) for 2 ≤ p < ∞. Suppose that 2 < p < ∞ and that C μ is a bounded operator on H p . Let q be the conjugate exponent of p, that is, 1 p + 1 q = 1. Bearing in mind Proposition 2 and Proposition 3, we see that the operator T μ , initially defined over polynomials, extends to a bounded operator on H q .
We have that for all a ∈ (0, 1), f a ∈ H q and f a H q = 1. Since T μ is bounded on H q , it follows that Also, for every a, f a,N → f a , as N → ∞ in H q and uniformly on compact subsets of D. Now, T μ f a,N (z) = (1 − a 2 ) 1/q N n=0 N k=n μ k a k (2/q)a k z n (z ∈ D) and then, using that 1 < q < 2 and [13, Theorem 6.2], we have that Letting N tend to ∞, we obtain Taking a = 1 − 1 N and letting [N /2] denote the largest integer less than or equal to N /2, we obtain Using (10), it follows that μ N 1 N and then Lemma 2 implies that μ is a Carleson measure.

Proof of Theorem 2
Proof Let us start with the implication (ii) ⇒ (i). We shall consider the cases 1 ≤ p ≤ 2 and 2 < p < ∞ separately.
Suppose first that 1 ≤ p ≤ 2 and C μ is compact from H p into itself. As in the proof of Theorem 1, for 0 < a < 1, set We have that f a H p = 1 for all a and, also, f a → 0, as a → 1, uniformly on compact subsets of D. Hence, C μ ( f a ) H p → 0, as a → 1. But in the course of the proof of the implication (ii) ⇒ (i) of Theorem 1, we obtained that μ N N C μ ( f a ) H p for a = 1 − 1 N (see (9)). Then it follows that μ N = o 1 N and this implies that μ is a vanishing Carleson measure.
Suppose now that 2 < p < ∞ and C μ is compact from H p into itself. By Theorem 1, μ is a Carleson measure and then it follows that the operator T μ is well defined on H q ( 1 p + 1 q = 1) and it is the adjoint of C μ . For 0 < a < 1, set f a (z) = 1−a 2 (1−az) 2 1/q , (z ∈ D). We have that f a H q = 1 for all a and, also, f a → 0, as a → 1, uniformly on compact subsets of D. By Schauder's theorem [10, p. 174], T μ is a compact operator from H q into itself and, hence, T μ ( f a ) H q → 0. In the course of the proof of the implication (ii) ⇒ (i) of Theorem 1, we obtained that μ N N T μ ( f a ) H q for a = 1 − 1 N (see (11)). Then it follows that μ N = o 1 N and, hence, μ is a vanishing Carleson measure.
Let us start with the case p = 2. So assume that μ is a vanishing Carleson measure and let { f n } be a sequence of functions in H 2 with f n H 2 ≤ 1, for all n, and such that f n → 0, uniformly on compact subsets of D.
Since μ is a vanishing Carleson measure μ k = o 1 k , as k → ∞. Say Then {ε k } → 0. Say that, for every n, Since the Cesàro operator C is bounded on H 2 , there exists M > 0 such that Take ε > 0 and next take a natural number N such that We have Now, since f n → 0, uniformly on compact subsets of D, it follows that Then it follows that that there exist n 0 ∈ N such that C μ ( f n ) 2 H 2 < ε for all n ≥ n 0 . So, we have proved that C μ ( f n ) 2 H 2 → 0. The compactness of C μ on H 2 follows. Let us move to the case p = 1. Assume that μ is a vanishing Carleson measure and let { f n } be a sequence of functions in H 1 with f n H 1 ≤ 1, for all n, and such that f n → 0, uniformly on compact subsets of D. Set As in the proof of the implication (i) ⇒ (ii) in Theorem 1 when p = 1 we see that, for 0 < r < 1 and n ∈ N, and, hence, Since μ is a vanishing Carleson measure μ ([t k−1 , t k ]) = o(2 −k ) and, hence, we have On the other hand, looking at the proof of Theorem 1, we see that there exists C > 0 such that Take ε > 0 and then take N ∈ N so that ε k ≤ ε 2C K , for all k ≥ N , where K is the constant in the Hardy-Littlewood maximal estimate Using (13) we see that Using (14), we obtain Since f n → 0, uniformly on compact subsets of D, it is clear that I (n) → 0, as n → ∞. Then it follows that there exists n 0 ∈ N such that C μ ( f n ) H 1 < ε whenever n ≥ n 0 . Thus,we have shown that C μ ( f n ) H 1 → 0, as n → ∞ and the compactness of C μ on H 1 follows.
To deal with the cases 1 < p < 2 and 2 < p < ∞, we use again complex interpolation.
Suppose first that 1 < p < 2 and μ is a vanishing Carleson measure. Recall that We have also that if 2 < s < ∞ then for a certain α ∈ (0, 1), namely, α = 1 2 − 1 s / 1 − 1 s . Since H 2 is reflexive, and C μ is compact from H 2 into H 2 and from H 1 into H 1 , Theorem 10 of [11] gives that and C μ is compact from H p into H p .
Suppose now that 2 < p < ∞ and μ is a vanishing Carleson measure. Let q be conjugate exponent of p. Take q 1 with 1 < q 1 < q < 2. We have that T μ is compact from H 2 into itself and continuous from H q 1 into H q 1 . Also, H q = (H 2 , H q 1 ) θ for a certain θ ∈ (0, 1). Then, Theorem 10 of [11] gives that and T μ is compact from H q into H q and, hence, C μ is compact from H p into itself.

The operators C acting on H ∞
For the constant function 1 we have If μ is positive finite Borel measure on [0, 1) then So, it follows that This easily implies the following result.
Theorem 4 Let μ be positive finite Borel measure on [0, 1). Then the following conditions are equivalent.
Danikas and Siskakis [12] proved that We extend this result obtaining a characterization of those positive finite Borel measure μ on [0, 1) for which C μ (H ∞ ) ⊂ B.
Theorem 5 Let μ be positive finite Borel measure on [0, 1). Then the following conditions are equivalent Proof Let us start with the implication (i) ⇒ (ii). So, assume that C μ (H ∞ ) ⊂ B. Then C μ (1) ∈ B, but, as we have seen above and then, using the fact that the sequence {μ n } is a decreasing sequence of nonnegative numbers and Lemma B, we see that μ n = O 1 n which is equivalent to saying that μ is a Carleson measure.
Let us turn now to prove the other implication. So, assume that μ is a Carleson measure and take f ∈ H ∞ . Using the integral representation of C μ we see that Hence, using that f ∈ H ∞ ⊂ B, we obtain Take z ∈ D and set r = |z|. Set also Integrating by parts and using the fact that μ is a Carleson measure, we obtain This and (15) in [9] that Λ 2 1/2 ⊂ B M O A and this result was generalized by Bourdon, Shapiro and Sledd who proved in [4] that This was shown to be sharp in a very strong sense in [3].
Lemma B Let f ∈ Hol(D), f (z) = ∞ n=0 a n z n (z ∈ D). Suppose that 1 < p < ∞ and that the sequence {a n } is a decreasing sequence of nonnegative numbers. If 1 < p < ∞ and X is a subspace of Hol(D) with Λ We shall also use some results on pointwise multipliers and coefficient multipliers of Bergman spaces and Hardy spaces.
Let us start recalling that for g ∈ Hol(D), the multiplication operator M g is defined by If X and Y are two spaces of analytic functions in D (which will always be assumed to be Banach or F-spaces continuously embedded in Hol(D)) and g ∈ Hol(D) then g is said to be a pointwise multiplier from X to Y if M g (X ) ⊂ Y . The space of all multipliers from X to Y will be denoted by M(X , Y ). Using the closed graph theorem we see that if g ∈ M(X , Y ) then M g is a bounded operator from X into Y . The following result is a particular case of Theorem C of [37].
Theorem C Suppose that 1 < p < ∞ and α > −1. Then If X and Y are two spaces of analytic functions in D, a function F ∈ Hol(D) is said to be a coefficient multiplier (or a convolution multiplier) from X to Y if The following result is due to Duren and Shields, it is a particular case of [15,Theorem 4].
Theorem D Suppose that 1 < p < ∞ and F ∈ Hol(D). Let m be a positive integer such that (m + 1) −1 ≤ p p+1 < m −1 . Then F is a coefficient multiplier from H p/( p+1) to H p if and only if the (m + 1)-th derivative F (m+1) of F satisfies We can now proceed to prove Theorem 6. Proof of the implication (i) ⇒ (ii) in Theorem 6. Assume that μ is a Carleson measure and set Since μ is a Carleson measure μ n = O 1 n . This, the simple fact that {μ n } is a deceasing sequence of nonnegative numbers, and Lemma B imply that F ∈ Λ p 1/ p and, hence Using [13,Theorem 5.5], we see that this implies and then Theorem D gives that F is a coefficient multiplier from H p/( p+1) into H p . Trivially, this implies that F is also a coefficient multiplier from . A simple computation shows that 1 1−z ∈ A 1 α . Then, using Theorem C we deduce that g ∈ A p/( p+1) α . This and (16) imply that F g ∈ A p α . By Lemma 1 this is equivalent to saying that C μ ( f ) ∈ A p α . Proof of the implication (ii) ⇒ (i) in Theorem 6. Suppose that C μ is a bounded operator on A p α . Let q be the exponent conjugate to p, that is, 1 p + 1 q = 1. Let T μ be the adjoint of C μ , it is a bounded operator on A q α . For 0 < b < 1, set Using [39, Lemma 3.10], we see that Also, Bearing in mind Proposition 2 and Proposition 3, we see that Since the coefficients a k,b are nonnegative, it follows that the sequence of the Taylor coefficients of T μ ( f b,N ) is a decreasing sequence of nonnegative numbers, then (see, e. g., [20,Proposition 1]) (17), and simple estimations, we deduce that Hence, μ is a Carleson measure.

The operators C acting on BMOA and on the Bloch space
Let λ be defined by λ(z) = log 1 1−z (z ∈ D). Then λ ∈ B M O A. In fact, it is true that λ ∈ Λ p 1/ p for all p > 1. Danikas and Siskakis [12] Proof Let us start showing that (i) ⇒ (ii). So assume that μ is a 1-logarithmic 1-Carleson measure and take f ∈ X . We recall that μ being a 1-logarithmic 1-Carleson measure is equivalent to Take f ∈ X , f (z) = ∞ n=0 a n z n (z ∈ D). Since X ⊂ B, we have that f ∈ B. Then, using a result of Kayumov and Wirths (see [27,Corollary 4] The estimates (18) and (19) yield Hence Suppose now that C μ (X ) ⊂ Y . As above, set λ(z) = log 1 1−z = ∞ n=1 z n n (z ∈ D).
We have the following result concerning compactness. Proof Clearly, it suffices to prove that (i) and (ii) are equivalent. Let us prove first that (i) implies (ii). So, assume that μ is a vanishing 1-logarithmic 1-Carleson measure and Λ 2 1/2 ⊂ X , Y ⊂ B. Take { f j } ⊂ X with f j X ≤ 1, for all j, and f j → 0, as j → ∞, uniformly on compact subsets of D. Since X is continuously embedded in B, { f j } ⊂ B and there exists K 1 > 0 such that f B ≤ K 1 , for all j.
Say f j (z) = ∞ k=0 a ( j) k z k (z ∈ D). Using the result of Kayumov and Wirths that we have mentioned above, we see that there exists K 2 > 0 such that n k=0 a ( j) k ≤ K 2 f j B log(n + 1) ≤ K 1 K 2 log(n + 1), for all n and j.
A simple calculation gives that for 0 < a < 1 and z ∈ D, Then it follows that, for 0 < a < 1, This and (20) imply that μ is a vanishing 1-logarithmic 1-Carleson measure.
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