Operators Induced by Radial Measures Acting on the Dirichlet Space

Let 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} be the unit disc in the complex plane. Given a positive finite Borel measure μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} on the radius [0, 1), we let μn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _n$$\end{document} denote the n-th moment of μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} and we deal with the action on 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} of the operator of Hibert-type Hμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}_\mu $$\end{document} and the operator of Cesàro-type 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} which are defined as follows: If f is holomorphic 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}, f(z)=∑n=0∞anzn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)=\sum _{n=0}^\infty a_nz^n$$\end{document} (z∈D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\in {\mathbb {D}})$$\end{document}, then Hμ(f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}_\mu (f)$$\end{document} is formally defined by Hμ(f)(z)=∑n=0∞∑k=0∞μn+kakzn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}_\mu (f)(z) = \sum _{n=0}^\infty \left( \sum _{k=0}^\infty \mu _{n+k}a_k\right) z^n$$\end{document} (z∈D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\in {\mathbb {D}}$$\end{document}) and Cμ(f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}_\mu (f)$$\end{document} is defined by Cμ(f)(z)=∑n=0∞μn∑k=0nakzn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal C_\mu (f)(z) = \sum _{n=0}^\infty \mu _n\left( \sum _{k=0}^na_k\right) z^n$$\end{document} (z∈D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\in {\mathbb {D}}$$\end{document}). These are natural generalizations of the classical Hilbert and Cesàro operators. A good amount of work has been devoted recently to study the action of these operators 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}. In this paper we study the action of the operators Hμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}_\mu $$\end{document} and 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 the Dirichlet space D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document} and, more generally, on the analytic Besov spaces Bp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^p$$\end{document} (1≤p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p<\infty $$\end{document}).


Introduction
The open unit disc in the complex plane C will be denoted by D and Hol(D) will stand for the space of all analytic functions in D. Also, dA will denote the area measure on D, normalized so that the area of D is 1. Thus dA(z) = We refer to [20] for the theory of Hardy spaces. For 0 < p < ∞ and α > −1 the weighted Bergman space A p α consists of those f ∈ Hol(D) such that The unweighted Bergman space A p 0 is simply denoted by A p . We refer to [21,31,48] for the notation and results about Bergman spaces.
The space of Dirichlet type D p α (0 < p < ∞, α > −1) is the space of those f ∈ Hol(D) such that f ∈ A p α . Thus, a function f ∈ Hol(D) belongs to In this paper we shall be mainly concerned with the Dirichlet space D = D 2 0 which consists of those f ∈ Hol(D) whose image Riemann surface has a finite area. We recall that if f ∈ D, f (z) = ∞ n=0 a n z n (z ∈ D), then (1.1) Throughout the paper μ will be a positive finite Borel measure on the radius [0, 1) and, for n = 0, 1, 2, . . . , we shall let μ n denote the moment of order n of μ, that is, μ n = [0,1) t n dμ(t). The matrices H μ and C μ are defined as follows As we shall see in Sects. 2 and 3, these matrices induce operators acting on spaces of analytic functions which are natural generalizations of the classical Hilbert and Cesàro operators. Recently a good amount of work has been devoted to study the action of these operators of Hilbert type and of Cesàro type on distinct subspaces of Hol(D). Carleson-type measures play a basic role in this work.
Let us recall that if μ is a positive finite Borel measure on [0, 1) then: • If s > 0, then μ is said to be an s-Carleson measure if there exists a positive constant C such that • If 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 Let us close this section by saying that, as usual, 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 ≤ CK 2 , respectively K 1 ≥ CK 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 .

Hilbert-Type Operators
The matrix H μ induces formally an operator, which will be also called H μ , on spaces of analytic functions by its action on the Taylor coefficients: a n → ∞ k=0 μ n+k a k , n = 0, 1, 2, . . . .
whenever the right hand side makes sense and defines an analytic function in D.
The finite positive Borel measures μ for which H μ is a bounded operator on distinct spaces of analytic functions in D have been characterized in a number of papers such as [9,14,25,[27][28][29]35,37,38,45]. Obtaining an integral representation of H μ plays a basic role in these works. If μ is as above, we shall write throughout the paper  We improve this result showing that being a 1-logarithmic 1-Carleson measure is enough to insure that H μ is bounded from D into itself and closing the gap between (i) and (ii). Indeed, we shall prove the following result. (ii) If 0 < β < 1, then there exists a positive and finite Borel measure μ on [0, 1) which is a β-logarithmic 1-Carleson measure but such that H μ (D) ⊂ D.
As a corollary of part (i) we obtain the following.

Corollary 2. (a)
Let μ be a positive and finite Borel measure on [0, 1) and suppose that μ is a 1-logarithmic 1-Carleson measure. Then there exists a positive constant C such that Regarding the Bergman space A 2 , Theorem 1.5 of [25] asserts the following.

Theorem C. Let μ be a positive and finite Borel measure on
We recall that a finite positive Borel measure ν on D is said to be a Dirichlet-Carleson messure if D is continuously embedded in L 2 (dν). Stegenga [43] gave a characterization of these measures involving the logarithmic capacity of a finite union of intervals of ∂D. Shields [39] obtained a simpler characterization when dealing with measures supported on [0, 1). This result of Shields will be used below.
Using Theorem 1 we shall prove the following result.

Theorem 3. (i) Let μ be a positive and finite Borel measure on
In order to prove our results we start using the above mentioned result of Shields [39] to find a weak condition which insures that H μ and I μ are well defined in D and that H μ (f ) = I μ (f ) for all f ∈ D.
then H μ and I μ are well defined in D and, furthermore, Proof. Suppose that μ satisfies (2.6). Shields proved in [39,Theorem 2] that this is equivalent to saying that there exists a positive constant A such that We can express (2.7) simply by saying that μ is a radial Carleson-Dirichlet measure. Also, it is easy to see that (2.6) implies that there exists B > 0 such that This implies that, for all z ∈ D, the integral converges and that So I μ (f ) is a well defined analytic function in D and Vol. 78 (2023) Operators Induced by Radial Measures Acting Page 7 of 24 106 Let us see now that H μ (f ) is also well defined and that H μ (f ) = I μ (f ). Using (2.8), for all n, we have Clearly, this implies that H μ is a well defined analytic function in D. Also, Let us turn now to prove Theorem 1 (2.10) Since μ is a 1-logarithmic 1-Carleson measure,  |a k | (n + k) log(n + k + 1) Now, using a result of Holland and Walsh [30,Theorem 7] and simple estimates we deduce that Also, since, for every n, Set a n = 1 (n+1)[log(n+1)] α (n = 1, 2, . . . ) and g(z) = ∞ n=1 a n z n (z ∈ D). The condition α > 1 2 implies that g ∈ D.
We are going to see that H μ (g) / ∈ D, this will finish the proof.
Proof of Corollary 2. The Dirichlet space is a Hilbert space with the inner product Hence, D is identifiable with its dual with this pairing.
Assume that μ is a finite Borel measure on [0, 1) which is a 1-logarithmic 1-Carleson measure. If f ∈ D, using Theorem 1, we see that H μ (f ) ∈ D and Now, using the definitions, Fubini's theorem, and the reproducing formula for the Bergman space A 2 , we have Using (2.12), we obtain (2.13) |b n |z n (z ∈ D).
Then f 1 , g 1 ∈ D, f 1 D = f D , and g 1 D = g D . Using (2.13) with f 1 and g 1 in the places of f and g, we obtain Taking f = g, (2.3) follows. Part (b) follows taking dμ(t) = log 2 1−t dt in part (a). Proof of Theorem 3. Our proof of Theorem 3 is based on the fact that the pairing is a "duality paring" between the Dirichlet space D and the Bergman space It is a simple exercise to show that < H μ (P ), Q >=< P, H μ (Q) > if P and Q are polynomials. Then it follows that if H μ is a bounded operator from D into itself then its adjoint (via this pairing) is H μ , and then we see that H μ is a bounded operator from A 2 into itself. Using this and Theorem 1 (i) we obtain part (a) of Theorem 3. Similarly, if H μ is a bounded operator from A 2 into itself, then H μ is also a bounded operator from D into itself and then part (b) of Theorem 3 follows using Theorem 1 (ii).

Cesàro-Type Operators
For μ a finite positive Borel measure on [0, 1) as above, the matrix C μ induces a linear operator, also called C μ , from Hol(D) into itself as follows: If f ∈ Hol(D), f (z) = ∞ n=0 a n z n (z ∈ D), Let us remark that the operator C μ has the following integral representation: When μ is the Lebesgue measure on [0, 1), the operator C μ reduces to the classical Cesàro operator C.
The operators C μ were introduced in [23] where, among other results, it was proved that the following conditions are equivalent: (iv) 1 < p < ∞, α > −1, and C μ is bounded from A p α into itself. Blasco [12] has generalized the definition of the operators C μ by dealing with complex Borel measures on [0, 1) and he has extended results of [23] to this more general setting.
A further generalization has been given in [24] by working with the operators C μ associated to arbitrary complex Borel measures on D, not necessarily supported on a radius. The complex Borel measures on D for which the operator C μ is bounded or Hilbert-Schmidt on H 2 or on A 2 α (α > −1) are characterized in the mentioned paper [24].
We devote this section to study the operators C μ on the Dirichlet space, a question which has not been considered in the just mentioned papers. Our main results are contained in the following two theorems. (i) If μ is a 1-logarithmic 1-Carleson measure, then C μ is a bounded operator from the Dirichlet space D into itself. (ii) If C μ is a bounded operator from D into itself then μ is a 1/2-logarithmic 1-Carleson measure.
n=0 a n z n (z ∈ D). Using (3.2) and Theorem 7 of [30], we obtain Proof of Theorem 5 (ii). Suppose that C μ is a bounded operator from D into itself. For N ∈ N, set Then, Since C μ is bounded on D, bearing in mind that the sequence of moments {μ n } is decreasing, we have We have that A n 1 (n + 1)[log(n + 2)] 1−α .
Since α < 1 2 , we have that g ∈ D. Also Danikas and Siskakis [15] proved that C(H ∞ ) ⊂ H ∞ and that C(H ∞ ) ⊂ BM OA. This was improved by Essén and Xiao who proved in [22] that C(H ∞ ) ⊂ Q p for 0 < p < ∞. This result has been sharpened in [10].
We recall that BM OA is the space of those functions f ∈ H 1 whose boundary values have bounded mean oscillation. Alternatively, a function f ∈ Hol(D) belongs to BM OA if and only if where Aut(D) denotes the set of all Möbius transformations from D onto itself. We refer to [26] for the theory of BM OA-functions.
For 0 < s < ∞ the space Q s consists of those f ∈ Hol(D) such that The spaces Q s were introduced in [6] and [7]. We refer to [46] for the theory of Q s spaces. Let us recall that D Q s1 Q s2 Q 1 = BM OA, 0 < s 1 < s 2 < 1.
The paper [3] is an excellent reference for the theory of Bloch functions. Let us recall that BM OA B. Blasco [12] has proved that Here, for p ≥ 1, Λ p 1/p is the space of those functions f ∈ Hol(D) having a nontangential limit at almost every point of ∂D and so that ω p (·, f), the integral modulus of continuity of order p of the boundary values f (e iθ ) of f , satisfies ω p (δ, f ) = O(δ 1/p ), as δ → 0. Classical results of Hardy and Littlewood (see [13] and [20,Chapter 5]) show that Λ p 1/p ⊂ H p and that In particular, Λ 1 1 is the space of those f ∈ Hol(D) such that f ∈ H 1 . The spaces Λ p 1/p increase with p and they are all contained in BM OA [13]. Since Λ 2 1/2 ⊂ Q s for all s > 0 (see [5, p. 427 Indeed, set a n = 1 (n+1) log(n+1) (n ≥ 1) and f (z) = ∞ n=1 a n z n (z ∈ D). Then f ∈ D and, setting A n = n k=1 a k , we have, for 0 < r < 1, Hence, C(f ) ∈ B. The next natural step is trying to characterize the measures μ such that C μ (D) ⊂ B. We have the following result. (i) If μ is a 1 2 -logarithmic 1-Carleson measure, then C μ is a bounded operator from D into X.
Proof. Suppose that μ is a 1 2 -logarithmic 1-Carleson measure. Then μ n 1 n[log(n + 1)] 1/2 . (3.5) Take f ∈ D, f (z) = ∞ n=0 a n z n (z ∈ D). We have This and (3.5) imply that |A n | f D n a fact which easily yields that C μ (f ) ∈ Λ 2 1/2 . This finishes the proof of (i). Let us turn to prove (ii). Assume that 0 < β < 1 2 and that C μ is a bounded operator from D into X.
Since X ⊂ B, C μ is a bounded operator from D into B.

Extensions to Besov Spaces
The Dirichlet space is one among the analytic Besov spaces B p . For 1 < p < ∞, the analytic Besov space B p is the space D p p−2 . Thus B 2 = D. The minimal Besov space B 1 requires a special definition. It is the space of all f ∈ Hol(D) such that f ∈ A 1 . It is a Banach space with the norm · B 1 defined by The Besov spaces B p form a nested scale of conformally invariant spaces and they are all contained in BM OA: We mention [4,11,18,30,47,48] for information on Besov spaces. Let us remark that, letting dλ be the Möbius invariant measure on D defined by dλ(z) = dA(z) (1−|z| 2 ) 2 , we have: (a) The Bergman projection P is a continuous linear operator from L ∞ (D) onto the Bloch space B, (b) For 1 < p < ∞, the Bergman projection P is a continuous linear operator from L p (dλ) onto B p (see [48,Chapter 5]).
Our aim in this section is trying to extend to the spaces B p some of the results obtained in the preceding ones for the Dirichlet space.
For the space B 1 we have the following result.
Thus, (iii) follows. Let us prove next the equivalence (i) ⇔ (iv). Suppose (i). Take f ∈ B 1 . Bearing in mind (3.1) and using Fubini's theorem, we see that We now estimate each of the three terms in the last formula separately. For the first one we have For the second one, we use the fact that B 1 ⊂ Λ 1 1 to obtain For the last integral, we use that B 1 ⊂ H ∞ and Lemma 3.10 of [48] to see that