Integral Operators Induced by Symbols with Non-negative Maclaurin Coefficients Mapping into H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^\infty $$\end{document}

For analytic functions g on the unit disk with non-negative Maclaurin coefficients, we describe the boundedness and compactness of the integral operator Tg(f)(z)=∫0zf(ζ)g′(ζ)dζ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_g(f)(z)=\int _0^zf(\zeta )g'(\zeta )\,d\zeta $$\end{document} from a space X of analytic functions in the unit disk to H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^\infty $$\end{document}, in terms of neat and useful conditions on the Maclaurin coefficients of g. The choices of X that will be considered contain the Hardy and the Hardy–Littlewood spaces, the Dirichlet-type spaces Dp-1p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^p_{p-1}$$\end{document}, as well as the classical Bloch and BMOA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathord {\mathrm{BMOA}}$$\end{document} spaces.

and K H (β) ζ are the reproducing kernels of the Hilbert space H (β). Theoretically, the above relatively simple result offers a characterization of the boundedness in the case of most of the natural spaces one can think of. However, if one tries to apply this in praxis one observes that it looks like a reformulation rather than a solution of the problem. This is due to the fact that treating the function G H (β) g,z in the dual space of X is often laborious if not even frustrating. Because of these reasons, in this study, we restrict ourselves to the case in which the symbol g has nonnegative Maclaurin coefficients, and search for neat and useful conditions in terms of the Maclaurin coefficients of g that can be used to test if T g is either bounded or compact from X to H ∞ . The starting point is the characterization [4,Theorem 2.2] given above, and the choices for X that will be considered in the sequel contain the Hardy and the Hardy-Littlewood spaces, and certain Dirichlet-type spaces, as well as the classical Bloch space and BMOA. Next, the main findings of this study along with necessary definitions are stated.
where d A(z) = dx dy π is the normalized area measure on D. The closely related Hardy-Littlewood space HL p contains those f ∈ H(D) whose Maclaurin coefficients These spaces satisfy the well-known inclusions by [5,6,11]. Each of these inclusions is strict unless p = 2, in which case all the spaces are the same by direct calculations or straightforward applications of Parseval's formula and Green's theorem. Our first main result reveals that T g does not distinguish H p , HL p and D p p−1 when it acts boundedly or compactly from one of these spaces to H ∞ , provided 1 < p < ∞ and the symbol g has non-negative Maclaurin coefficients. Here, as usual, the conjugate index of 1 < p < ∞ is the number p such that 1 p + 1 p = 1. Theorem 1 Let 1 < p < ∞ and g ∈ H ∞ such that g(n) ≥ 0 for all n ∈ N ∪ {0}. Further, let X p ∈ {H p , D Here and from now on T (X , H ∞ ) (resp. T c (X , H ∞ )) denotes the set of g ∈ H(D) such that T g : X → H ∞ is bounded (resp. compact).
Theorem 2 Let g ∈ H ∞ and X p ∈ {H p , D p p−1 , HL p }. Then the following assertions hold: In the proof of Theorem 2, we use identifications of the duals of HL 1 and D p p−1 with 0 < p ≤ 1. Since many dual spaces X can be described, via the H 2 -pairing, as the space of coefficient multipliers from X to the disk algebra [15,Proposition 1.3], a natural characterization of the dual of HL 1 is easy to find by using the relation ( 1 ) ∞ . We do this in Sect. 2 when we prove Lemma 8 which states that (HL 1 ) HL ∞ via the H 2 -pairing with equivalence of norms. The space HL ∞ consists of f ∈ H(D) To find a suitable dual of D The proof of the duality relation (D p p−1 ) B 2 is lengthy, and apart from standard tools, such as Green's theorem and continuous embeddings between different weighted Bergman spaces, it also relies on a use of smooth universal Cesáro basis of polynomials. The last-mentioned creatures are used to show that a certain function, dependent of p, is a coefficient multiplier of B 2 .
The next result is the counterpart of Theorem 1 in the case p = 1. It also proves that the statement in Theorem 2(i) is sharp. Observe that (1.6) is the limit case p = ∞ of (1.4), and that the supremum there is in fact the limit as k → ∞ since the quantity over which the supremum is taken is increasing in k.

Theorem 3
Let g ∈ H ∞ such that g(n) ≥ 0 for all n ∈ N ∪ {0}, and X 1 ∈ {H 1 , D 1 0 , HL 1 }. Then T g : Moreover, Theorem 3 is relatively straightforward to establish once the tools needed for Theorem 2 are on the table. Both of these theorems are proved in Sect. 4.
Our last result concerns the case when T g acts from the Bloch space or BMOA to H ∞ . Recall that the classical Bloch space B is just the space B 1 defined before Theorem 3. Further, let −1 , and recall that BMOA consists of the functions in the Hardy space H 1 that have bounded mean oscillation on the boundary T. The space BMOA can be equipped with several different norms [8]. We will use the one given by Our last main result says that T g does not distinguish BMOA, B and H ∞ log when it acts boundedly or compactly from one of these spaces to H ∞ , if the symbol g has non-negative Maclaurin coefficients.
Theorem 4 Let X ⊂ H(D) be a Banach space such that BMOA ⊂ X ⊂ H ∞ log and let g ∈ H ∞ with g(n) ≥ 0 for all n ∈ N ∪ {0}. Then the following statements are equivalent: g(n + 1) log(n + 2). (1.9) The proof of Theorem 4, given in Sect. 5, reveals that for each g ∈ H(D). The hypothesis on the coefficients is only used when the right most quantity above is dominated by the operator norm.
Probably the most obvious election for the space X in the statement of Theorem 4 is the classical Bloch space B. However, there are other choices for X which arise naturally in the theory of integral operators, see Sect. 5 for further details.
The hypothesis g ∈ H ∞ in Theorems 1-4 is not a restriction, because it is an obvious necessary condition for T g : X → H ∞ to be bounded.
It is worth mentioning that the smallest space X that we work with regarding bounded and compact operators T g : X → H ∞ is BMOA which is in a sense much larger than H ∞ . It does not seem straightforward to deal with the case X = H ∞ even in the case when the symbol g has non-negative Taylor coefficients. The approach that we use here to prove Theorems 1-4 is based on the abstract solution to characterize T (X , H ∞ ) given in [4]. In the case X = H ∞ , it takes us to calculate sup z∈D G H 2 g,z K < ∞, where K is the space of Cauchy transforms. The space K is endowed the total variation norm which seems pretty untreatable for our auxiliary function G H 2 g,z . To this end, couple of words about the notation used in this paper. The letter C = C(·) will denote an absolute constant whose value depends on the parameters indicated in the parenthesis, and may change from one occurrence to another. If there exists a constant C = C(·) > 0 such that a ≤ Cb, then we write either a b or b a. In particular, if a b and a b, then we denote a b and say that a and b are comparable.

Dualities
In this section we will discuss the duality relations employed to prove the main results of the paper. Apart from the well-known relation (H p ) H p , 1 < p < ∞, we will need to know the dual spaces, with respect to appropriate pairings, of D p p−1 and HL p for 0 < p ≤ 1 and 1 < p < ∞, respectively.
The following lemma describes the dual of the Dirichlet-type space D p p−1 when 1 < p < ∞, and it will be needed in the proof of Theorem 1. We believe that the result itself must be known at least by experts working on the field, but since we do not know an exact reference, we include a proof here.
Proof Let us first show that each g ∈ D p p −1 induces a bounded linear functional on D p p−1 . Green's theorem implies from which Hölder's inequality yields where the first step is an easy consequence of the inequality − log t ≤ We note that this duality relation of the weighted Bergman spaces is essentially contained in [12,Theorem 2.1] as a special case, but with respect to a slightly different pairing. The method there would certainly work also in our setting and therefore offers an alternative way to deduce this duality. Getting back to the proof of the lemma, we observe that, Further, since G ∈ A p p −1 , there exists a unique g ∈ D p p −1 such that g = G and g(0) = L(1). Consequently, there exists a unique g ∈ D p p −1 such that where the last identity follows from (2.1). Moreover, g and the assertion is proved.
To prove Theorems 2 and 3, we need to know the dual of D p p−1 with 0 < p ≤ 1. In order to do that, some more notation is needed.
It is well known that for each 0 < α < ∞.
We will also need background on certain smooth polynomials defined in terms of Hadamard products. Recall that the Hadamard product of f ∈ H(D) and g ∈ H(D) is formally defined as A direct calculation shows that and consider the polynomials With this notation we can state the next auxiliary result that follows by [17,p. 111-113].
Theorem A shows that the polynomials {W n } n∈N can be seen as a universal Césaro basis for H p for any 0 < p < ∞. A particular case of the previous construction is useful for our purposes. By following [9,Section 2], let : R → R be a C ∞ -function such that is decreasing and positive on (1,2), These polynomials have the following properties with regard to smooth partial sums, see [9,p. 175-177] or [16,p. 143-144] for details: By denoting In the case p = 1, we have and hence K 1 ∈ B 2 by (2.2). To obtain the same conclusion for each 0 < p < 1, we first observe that J (z) = ∞ k=0 (k + 1) G(k)z k+1 = d dz (zG(z)), and thus J ∈ B 2 . Therefore it suffices to show that λ p (z) = ∞ n=0 w n, p w n+1, p z n+1 is a coefficient multiplier of B 2 for each 0 < p ≤ 1.
To see this, for each β ∈ N, denote D β f (z) = ∞ n=0 (n + 1) β f (n)z n for all f ∈ H(D), and for simplicity write D f instead of D 1 f . We claim that the proof of which is postponed for a moment. Direct calculations show that and hence (2.7) yields It follows that f * λ p ∈ B 2 for all f ∈ B 2 and 0 < p ≤ 1. Thus K p ∈ B 2 for each To complete the proof, it remains to establish (2.7). To do this, we will use the families of polynomials defined by (2.4) and (2.5). It follows from (2.6) that where (Dλ p ) r (z) = ∞ n=1 n w n−1, p w n, p r n z n . Next, for each n ∈ N \ {1} and r ∈ 1 2 , 1 , consider Since for each radial weight ν, there exists a constant C = C(ν) > 0 such that it follows by a direct calculation that for some constant C = C(ω) > 0. Therefore, This together with Theorem A, (2.9) and (2.6) implies V n * (Dλ) r H 1 = W n 1 * V n H 1 A n ,2 V n H 1 2 n r 2 n−1 V n H 1 2 n r 2 n−1 , r ∈ 1 2 , 1 , n ∈ N \ {1}, which combined with (2.8) gives This implies (2.7), and finishes the proof.
In the proof of Theorem 1, we need to know the dual space of the Banach space HL p , with 1 < p < ∞, with respect to the H 2 -pairing. It is given in the next lemma, the proof of which is standard.
HL p via the H 2 -pairing with equivalence of norms.
Recall that the space HL ∞ consists of f ∈ H(D) such that The last lemma of the section describes (HL 1 ) . It will be used in the proof of Theorem 2. The proof of this lemma is also standard and is therefore omitted.

Hardy, Hardy-Littlewood and Dirichlet-Type Spaces with 1 < p < ∞
The main aim of this section is to prove Theorem 1. To do that some notation and auxiliary results are needed. For each g ∈ H(D), with Maclaurin series expansion g(z) = ∞ k=0 g(k)z k , consider the dyadic polynomials defined by 0 g(z) = g(0) and n g(z) = 2 n+1 −1 k=2 n g(k)z k for all n ∈ N and z ∈ D. Then, obviously, g = ∞ n=0 n g. Further, write 0 = 1 and n (z) = 2 n+1 −1 k=2 n z k for all n ∈ N and z ∈ D. Then [4,Lemma 2.7] shows that With these preparations we can state the first auxiliary result.

2)
for all f ∈ H(D) and j ∈ N.
Proof For each j ∈ N and z ∈ D, let us consider the C ∞ -function Then Further, and hence (3.4) Therefore, by using (3.3) and (3.4), we can find a C ∞ -function 2 j ,z and an absolute constant C > 0 such that supp 2 j ,z ⊂ 1 2 , 4 , 2 j ,z (s) = 2 j ,z (s) for all s ∈ [1, 2] and Using now Theorem A, we find a constant C = C(q) > 0 such that This finishes the proof.
The next result gives a sufficient condition for T g : H p → H ∞ to be bounded and establishes the operator norm estimate (1.5) announced in the introduction.
Proof We begin with the case X p = D Therefore, by combining (3.5), (3.6), Proposition 9 and (3.1), we deduce Thus the assertion is proved for X p = D p p−1 . Next we deal with the case X p = HL p . By Lemma 7 and [4,Theorem 1.1], T g : HL p → H ∞ is bounded if and only if sup z∈D G H 2 g,z HL p < ∞, and moreover, and thus T g : HL p → H ∞ is bounded and Bearing in mind (1.1) and (1.2), the remaining case X p = H p follows from [4,Theorem 1.1], the well-known identification (H p ) H p via the H 2 -pairing and the two cases already proven.
Despite the inclusions in (1.1) and (1.2) are strict unless p = 2, if one restricts to the class of power series with non-negative decreasing coefficients, then the following statements hold by [10], [18] and [30,Chapter XII, Lemma 6.6].
n=0 form a sequence of non-negative numbers decreasing to zero. In particular, for every such f .
We are now ready to prove Theorem 1.

Proof of Theorem 1 Assume first that
(3.9) Lemmas 5 and 7 together with the well-known identification of (H p ) as H p via the H 2 -pairing imply (X p ) X p . Therefore [ (3.10) ζ k , for each x ∈ (0, 1), the Maclaurin coefficients form a sequence of non-negative and decreasing numbers. Therefore (3.10), Lemma B and (3.9) imply (n + 1) g(n + 1) The dominated convergence theorem now implies (3.11). If 1 < p ≤ 2, then Proposition 9 and an argument similar to that used in the proof of (3.7) allows us to find a constant C = C( p) > 0 such that This together with (1.1) implies Consequently, (3.11) holds for each 1 < p < ∞.
Thus (v) holds and ∞ n=0 g(n + 1) log(n + 2) It arises naturally when the Hardy-Stein-Spencer formula is applied to the dilatation f r in order to establish the identity see [20,Theorem 4.2] for details. Because the Laplacian of | f | p contains the factor | f | 2 , which can be interpreted as the Jacobian of the non-univalent change of variable w = f (z), this equivalent norm is useful, for example, in the study of composition operators [23]. But the associated weight comes to the picture also in some other instances which are more closely related to the topic of the present paper. To explain this, we say that a radial weight ω belongs to the class D if there exists a constant C = C(ω) ≥ 1 such that ω(r ) ≤ C ω( 1+r 2 ) for all 0 ≤ r < 1. Moreover, if there exist K = K (ω) > 1 and C = C(ω) > 1 such that ω(r ) ≥ C ω 1 − 1−r K for all 0 ≤ r < 1, then we write ω ∈ q D. The intersection D ∩ q D is denoted by D. For ω ∈ D, the space C 1 (ω ) consists of f ∈ H(D) such that the measure | f | 2 ω d A is a 1-Carleson measure for A p ω [19,Theorem 6.1]. As usual, we say that a positive Borel measure μ on D is a p-Carleson measure for X if X is continuously embedded into L p μ . The space C 1 (ω ) arises in the study of integration operators acting on weighted Bergman spaces. Indeed, it is known that, for each 0 < p < ∞, the operator T g is bounded from A p ω into itself if and only if g ∈ C 1 (ω ) [19,Theorem 6.4]. For ω ∈ D, the space C 1 (ω ) is nothing else but the Bloch space by the proof of [19,Theorem 6.1(C)], but it may be a proper subspace of B by [19,Theorem 6.1(D)], yet it always contains BMOA. Therefore we may choose X = C 1 (ω ) in Theorem 4. It is worth observing that while BMOA and B are conformally invariant, there exists ω ∈ D \ D such that C 1 (ω ) is not [20,Proposition 5.6]. Recall that a Banach space X ⊂ H(D), equipped with a semi-norm ρ X , is conformally invariant if there exists a constant C = C(X ) > 0 such that ρ X ( f • ϕ) ≤ Cρ( f ) X for all f ∈ X and for all automorphisms ϕ of D.
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