Vectorial Hankel operators, Carleson embeddings, and notions of BMOA

Let BMOANPL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm{BMOA}_{\mathcal{N}\mathcal{P}} \left(\mathcal{L}\right)}$$\end{document} denote the space of L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{L}}$$\end{document}-valued analytic functions ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\phi}$$\end{document} for which the Hankel operator Γϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Gamma_\phi}$$\end{document} is H2H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H^2\left(\mathcal{H}\right)}$$\end{document}-bounded. Obtaining concrete characterizations of BMOANPL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm{BMOA}_{\mathcal{N}\mathcal{P}} \left(\mathcal{L}\right)}$$\end{document} has proven to be notoriously hard. Let Dα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D^\alpha}$$\end{document} denote fractional differentiation. Motivated originally by control theory, we characterize H2H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H^2\left(\mathcal{H}\right)}$$\end{document}-boundedness of DαΓϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D^\alpha \Gamma_\phi}$$\end{document}, where α>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha > 0}$$\end{document}, in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries: The first is that BMOANPL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm{BMOA}_{\mathcal{N}\mathcal{P}} \left(\mathcal{L}\right)}$$\end{document} is not characterized by said embedding condition. The second is that when we add an adjoint embedding condition, we obtain a sufficient but not necessary condition for boundedness of Γϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Gamma_\phi}$$\end{document}. The third is that there exists a bounded analytic function for which the associated anti-analytic Carleson embedding is unbounded. As a consequence, boundedness of an analytic Carleson embedding does not imply that the anti-analytic ditto is bounded. This answers a question by Nazarov, Pisier, Treil, and Volberg.


Introduction
Throughout this paper we let H denote a separable Hilbert space with inner product ·, · H . Unless we explicitly state otherwise, we assume that H is infinite-dimensional. We denote by L = L (H) the space of bounded linear transformations on H, by S 1 the corresponding trace class, and by S 2 the Hilbert-Schmidt class. X will be used as a generic notation for an element of the set H, L, S 1 , S 2 . We will use Y to denote a general Banach space. By Hol (Y) we denote the space of Y-valued analytic functions on the open unit disc D. For f ∈ Hol(Y), we denote the nth Taylor coefficients at the origin byf (n). We denote by O (Y) the space of functions in Hol (Y) that admit an analytic extension to a larger disc (centered at the origin). If Y = C, then we suppress this in our notation, i.e. Hol = Hol (C), and O = O (C). The same principle will apply to all function spaces discussed below.
For p ∈ [1, ∞] and X ∈ H, S 1 , we let L p (T, X ) denote the standard space of p-Bochner-Lebesgue integrable functions from T to X . Here T denotes the unit circle in C. Similarly, we define L p (T, L) as the natural WOT-analogue of L p (T): A function f : T → L belongs to L p (T, L) if and only if for all x, y ∈ H the function f (·) x, y H is measurable and, moreover, f p L p (T,L) = T f p L dm < ∞. Here m denotes normalized Lebesgue measure on T.

E. RYDHE GAFA
The Hardy space H p (X ) is the space of f ∈ Hol (X ) such that where we have defined the function f r : z → f (rz). An important property of Hardy space functions is that they have boundary values in a natural sense, cf. Proposition 2.1. We denote the boundary values of f ∈ H p (X ) by bf ∈ L p (T, X ). The space H 2 (H) is a Hilbert space, with inner product f, g = ∞ 0 f (n), g(n) H . Of particular importance will be the set of H 2 (H)-normalized functions in O (H), which we denote by O 1 (H).
We now introduce the main topics of this paper. Initially, we consider the scalar setting, rather than the proper vectorial one.

Hankel operators.
Given φ ∈ Hol and f ∈ O, we define the action of the Hankel operator Γ φ on f by (2) A standard reference on Hankel operators is [Pel03]. We refer to φ as the symbol of Γ φ . We say that Γ φ is bounded if it extends to a bounded operator on H 2 . For Γ φ to be bounded it is necessary for φ to be in H 2 . For φ ∈ H 2 , one shows by computation that Γ φ f = P + φf , where P + denotes the orthogonal projection from L 2 (T) onto H 2 , andf : z → f (z). It is convenient to define the operation of coefficient conjugation, f → f # , f # (z) = f (z). Note that this is an isomorphism on H 2 . A classical result is that H 1 = H 2 ·H 2 : If f, g ∈ H 2 , then f ·g ∈ H 1 , and h H 1 ≤ f H 2 g H 2 . Conversely, if h ∈ H 1 , then there exists f, g ∈ H 2 such that h = f · g and f H 2 g H 2 ≤ C h H 1 , where C > 0 is a constant independent of f and g. Now choose f so that f # g = h. By the calculation Since H 1 may be identified with a subspace of L 1 (T), and L 1 (T) a straightforward application of the Hahn-Banach theorem shows that H 1 * = P + L ∞ (T). The fact that Γ φ is bounded if and only if φ ∈ P + L ∞ (T) is known as Nehari's theorem [Neh57].
1.4 The vectorial setting. Note that (2) makes perfect sense if φ ∈ Hol(L) and f ∈ O(H). We take this as the definition of a vectorial Hankel operator Γ φ . The factorization result H 1 S 1 = H 2 S 2 · H 2 S 2 , due to Sarason [Sar67], implies that Γ φ is H 2 (H)-bounded if and only if φ ∈ H 1 S 1 * , very much like in the scalar setting.
Since L 1 T, S 1 * is not equal to L ∞ (T, L) (L does not have the so-called Radon-Nikodym property, e.g. [DU77]), it is not obvious that H 1 T, S 1 * = P + L ∞ (T, L). However, this follows from a vectorial extension of Nehari's theorem, due to Page [Pag70]: Γ φ is H 2 (H)-bounded if and only if φ ∈ P + L ∞ (T, L).

E. RYDHE GAFA
The space of L-valued analytic functions for which the corresponding Hankel operators are H 2 (H)-bounded is commonly referred to as Nehari-Page BMOA: Definition 1.2. Let φ ∈ Hol (L). We then say that φ ∈ BMOA N P (L) if and only if While BMOA N P (L) can be identified either with P + L ∞ (T, L) or with H 1 S 1 * , these characterizations are of an abstract nature. Finding concrete conditions that characterize BMOA N P (L) has proven to be notoriously difficult. For example, define the class BMOA O (L) as the class of φ ∈ H 1 (L) such that the oscillation condition This fact represents an area of research, where authors consider some aspect of the theory for scalar-valued BMOA (or its harmonic or dyadic analogues), and then discuss to what extent this aspect carries over to the vector-valued case, e.g. [Bla88, BP08,Bou86,GPT04,Mei06,NPTV02,NTV97]. Before we get to the meat of this paper, we define the differentiation operator D : Hol (Y) → Hol (Y) by Df (z) = zf (z) + f (z). With respect to the monomial basis, D acts like a diagonal matrix. This presents an elementary way of taking arbitrary powers of D: For α ∈ R, we set Another convenience of working with D in place of ordinary differentiation is that it does not annihilate constants. In fact we can say more: For each α ∈ R, D α : From a technical point of view, the present paper is mainly concerned with H 2 (H)-boundedness of operators of the type D α Γ φ , with α > 0 and φ ∈ Hol (L). The present paper was originally motivated by the natural appearance of such operators in control theory, e.g. [JRW14]. However, they also have implications to our understanding of BMOA N P (L). Our investigation motivates the definition of a class which we refer to as Carleson BMOA: Since D does not annihilate constants, · BMOAC is a proper norm, and not a semi-norm.

Main result and corollaries
Theorem 1.4. Let H be a separable Hilbert space, L its space of bounded linear transformations. Let α > 0 and suppose that φ : D → L is analytic. Moreover, Theorem 1.4 generalizes a result by Janson and Peetre [JP88] who obtained essentially the above characterization in the case where H = C. We point out that, in the case where φ is L-valued, we are forced to avoid the Schur multiplier techniques used in [JP88]. This is made evident by the discussion in [DP97, Section 4].
Operators of the type DΓ φ received a lot of attention in connection to the so called Halmos problem [Hal70, Problem 6]: If a Hilbert space operator is similar to a Hilbert space contraction, then it is also polynomially bounded (by von Neumann's inequality). Is the converse true?
Following the works of many authors [AP96,Bou86,Fog64,Pau84,Pel82,Sz-N59], Pisier [Pis97] answered this question in the negative. Subsequently, different proofs of the same result have been given in several papers [DP97,Kis97]. All of these proofs exploit boundedness properties of operators of the type DΓ φ . The following two propositions are essentially from Davidson and Paulsen [DP97]: Proposition 1.5. Let α > 0, and H be a separable, infinite-dimensional Hilbert space, L its space of bounded linear transformations. Then there exists an analytic Moreover, φ may be chosen to be rank one-valued.  . Proposition 1.6 is essentially stated for α = 1 in [DP97, Corollary 4.2], but this does not explicitly mention the boundedness of φ, even though it follows from the original proof. A dyadic analogue of this result has been proved by Mei [Mei06]. For the convenience of the reader, we present proofs of the above propositions in Sect. 5.
Combining the results by Davidson and Paulsen with Theorem 1.4, we are able to derive several interesting results.
Given φ ∈ Hol (L), we define the function φ # : z → φ (z) * . This is the function obtained by taking the Hilbert space conjugate of each Taylor coefficient of φ. Note that Γ φ D = DΓ φ # * . By Proposition 1.5 and Theorem 1.4, it follows that On the other hand, BMOA N P (L) is obviously closed under coefficient conjugation. We obtain the following corollary: Consider now the relation which is obtained by duality, and the Leibniz rule for D. The operator Γ φ is bounded on H 2 (H), whenever any of the other terms in (6) is bounded, since then Dφ is a Bloch function (cf. Lemma 3.2 below). In the light of Theorem 1.4, it is then clear from (6) that We point out that the above inclusion also follows implicitly from the proof of [NPTV02, Theorem 0.8]. However, we obtain also that the inclusion is strict. To see that this is so, suppose that it is not. This would only be possible if BMOA N P (L) was contained in BMOA C (L). By another application of Theorem 1.4, this would contradict Proposition 1.6. We summarize the above discussion: Corollary 1.10. Let H be a separable Hilbert space, L its space of bounded linear transformations. If φ : D → L is an analytic function such that Moreover, If H is infinite dimensional, then the converse statement does not hold.
, where μ is a certain operator valued measure. It is natural to think of this as an embedding of anti-analytic functions, rather than analytic ones. For this reason, we call (5) the anti-analytic Carleson embedding, to be distinguished from the analytic one, which is given by the straightforward modification (8) below. In the scalar case it is obvious that these two conditions are equivalent. In the general case, this is no longer obvious. In fact, whether or not the two conditions define the same class of functions was posed as an open question by Nazarov, Treil, and Volberg in [NTV97]. They later restated the question in a joint paper with Pisier [NPTV02]. We answer this question in the negative: Corollary 1.11. Let H be a separable infinite-dimensional Hilbert space, L its space of bounded linear transformations. Then there exists a bounded analytic func- while The proof is as follows: Since D is an isomorphism from H 2 (H) to the standard weighted Bergman space A 2 1 (H), it follows from the Leibniz rule that (8) is satisfied whenever φ is bounded. On the other hand, by Proposition 1.6 and Theorem 1.4, there exists φ ∈ H ∞ (L) that satisfies (5 ).
Using standard arguments involving duality, A similar statement holds for boundedness of D α Γ φ # . This yields an alternative formulation of Theorem 1.4: We return for a moment to the scalar case. By the square function characterization of H 1 , due to Fefferman and Stein [FS72], it follows that A generalization to general α > 0, which also yields equality of function spaces in (9), has been obtained by Cohn and Verbitsky [CV00]. By Corollary 1.12, the dual inclusion becomes Combined with Corollary 1.10, this implies the well-known result that BMOA C = H 1 * . For this argument to work, it suffices to use Theorem 1.4 with (say) α = 1, a special case which is substantially simpler to prove.
The paper is structured as follows: In Sect. 2 we fix notation, and review some preliminary material. Of particular importance are some Bergman type spaces of analytic functions. In Sect. 3 we prove Theorem 1.4. In Sect. 4 we discuss and compare the special cases of H-valued, and H * -valued symbols, and point out some significant differences between these. In Sect. 5 we provide proofs of the Propositions 1.5 and 1.6.

Preliminaries and further notation
We use the standard notation Z, R, and C for the respective rings of integers, real numbers, and complex numbers. By N we denote the set of strictly positive elements of Z, while N ∪ {0} is denoted by N 0 . Given two parametrized sets of nonnegative numbers (A i ) i∈I and (B i ) i∈I , we use the notation A i B i , i ∈ I to indicate the existence of a positive constant C such that ∀i ∈ I, A i ≤ CB i . We then say that A i is bounded by B i , and refer to C as a bound. Sometimes we allow ourselves to not mention the index set I and instead let it be implicit from the context. If A i B i and B i A i , then we write A i ≈ B i . We then say that A i and B i are comparable.
The Hilbert space adjoint of A ∈ L is denoted by A * . We sometimes identify x ∈ H with the rank one operator C c → cx ∈ H. Note that x * is then the linear functional H y → y, x H ∈ C.
The dual of a Banach space Y will be denoted by Y * . With Hilbert spaces in mind, we equip Y * with an anti linear structure, rather than the standard linear ditto. Thus, the duality pairing y, y * Y , of y ∈ Y and y * ∈ Y * , becomes anti linear in y * .
We define the tensor product of two elements x, y ∈ H as the rank one operator x ⊗ y : z → z, y H x. The tensor product is anti linear in its second argument. The projective tensor product H⊗H, is the closed linear span of {x ⊗ y} x,y∈H , with respect to the norm The space H⊗H can be isometrically identified with S 1 . The dual of S 1 is isometrically identified with L via the pairing An important property of Hardy spaces H p (X ) is that, given certain properties of X , H p (X ) may be isometrically identified as a subspace of L p (T, X ). The precise statement is as follows: In particular, we may identify the Taylor coefficients of f with the Fourier coefficients of bf .
In the scalar case, the above result is proved in any serious introduction to Hardy spaces. We mention [Gar07]. We refer to [Nik02] for the case X = H, and [RR85] for the case X = L. The statement for X = S 1 holds because S 1 has the so-called analytic Radon-Nikodym property, see [BD82,HP89]. We define the formal duality pairing between f ∈ Hol (Y) and g ∈ Hol (Y * ) as The pairing is well defined if f ∈ O (Y) or g ∈ O (Y * ), and generalizes the inner product on H 2 (H). Note that D α f, g = f, D α g , and, in the case where Y = H, f, Γ φ g = f ⊗g, φ . We will make use of two related notions of weighted Bergman spaces. For β > −1, we define two finite measures on D: dA β (z) = 1 + β π 1 − |z| 2 β dA (z) and dA β,log (z) = 1 + β π log 1 |z| 2 β dA (z) .
Here dA denotes area measure on C. For p ∈ [1, ∞), we denote by L p β (Y) the space of strongly measurable functions f : D → Y such that We then define the standard weighted Bergman space A p β (Y) = L p β (Y)∩Hol (Y). We similarly define the logarithmically weighted spaces L p β,log (Y) and A p β,log (Y), with dA β,log in place of dA β . An enlightening reference for standard weighted Bergman spaces with Y = C is [HKZ00]. We remark that many of the results presented below for Y-valued functions follow by the same proofs as in the scalar case.
The above two notions of Bergman spaces are to a large extent interchangeable: Proposition 2.2. Let p ∈ [1, ∞) , β > −1, and Y be an arbitrary Banach space.
We then have that . The corresponding bounds depend on p and α.

GAFA VECTORIAL HANKEL OPERATORS, CARLESON EMBEDDINGS 437
One of the above bounds is obtained using the pointwise estimate 1 − |z| 2 ≤ log 1 |z| 2 , z ∈ D, and the other by using subharmonicity of the function z → f (z) p Y . We refer the interested reader to the easily modified proof of [Gar07, Lemma VI.3.2] for details.
A multiplier is an operator λ : Hol (Y) f → λf ∈ Hol (Y) given by for some scalar sequence (λ n ) ∞ n=0 . With some abuse of the terminology in [BKV99], we say that a multiplier is small if |λ n | 1 1+n . Using ideas from the proof of [BA02, Theorem 3.2], one can prove the following result, which we refer to as the small multiplier property for Bergman spaces. and Here Γ : C\ {−1, −2, . . .} → C is the standard Γ-function. By (10) and (11) we are allowed to define P β,log φf and P β φf as elements of Hol (H), whenever φ ∈ Hol (L) and f ∈ O (H). In this sense, they are analogues of (2). Using Parseval's identity we obtain In the literature the Bloch space is typically defined by finiteness of the quantity We leave it as an exercise to show that these definitions are equivalent.
The Bloch space has the simple property that as can be seen by interchanging the order of suprema.
This follows mostly as in [HKZ00]. The major difference is that B (Y * ) is the Bergman projection of a certain class of measures, rather than L ∞ (D, Y * ), see [BA03].

VECTORIAL HANKEL OPERATORS, CARLESON EMBEDDINGS 439
Theorem 1.4 is the statement that φ 1,α ≈ φ 6,α . We will prove that the quantities φ k,α , 1 ≤ k ≤ 6 are pairwise comparable. The outline of the proof is as follows. We show in detail that φ 1,α φ 2,α . The reverse estimate, as well as the estimates φ 2,α ≈ φ 3,α , and φ 3,α ≈ φ 4,α are similar, although the last one is substantially simpler than the preceding ones. The statement that φ 4,α ≈ φ 5,α is just a special case of Proposition 2.2. Furthermore, it is trivial that φ 5,α ≤ φ 6,α . The reverse of this last estimate follows in a routine manner from the following remarkable result by Aleman and Perfekt [AP12]: Lemma 3.1. There exists a constant C > 0 such that whenever ψ ∈ Hol (L) it holds that To prove that φ 1,α φ 2,α we will need some lemmata. The first result gives us some preliminary control of φ.
Proof. We consider only the case k = 1. The other cases are similar. By (12) it suffices to prove that Given w ∈ D, x, y ∈ H, let f (z) = ∞ n=0 w n 1 + n − n n 1 + n α z n y, A calculation shows that 1 + n − n n 1+n α is bounded in n, and so f H 2 g H 2 x H y H 1−|w| 2 . The definition of φ α,1 now yields (13).
Remark 3.3. Another proof of Lemma 3.2 is to use (12) together with the (already known) scalar version of Theorem 1.4. Our approach is chosen so that our results do not depend on the scalar case.
The qualitative content of the next lemma is known, and due to Peller [Pel82]. See also [Pel03, Chapter 6.9]. However, the original proof gives a much worse quantitative dependence on l. The proof we present is a bit lengthy, and is postponed to the next subsection.
We are now ready for the main part of the argument. Given f ∈ O (H), and φ ∈ Hol (L), we use the formulas (2) and (11) to write A well known fact about standard weighted Bergman spaces is that there exists many bounded projections from L p γ onto A p γ , eg. [HKZ00, Theorem 1.10]. This inspires us to replace P α+β−1,log with P 2β−1,log . By the triangle inequality We carry out a few manipulations with the Taylor coefficients of ψ and f , use the power series expansion at the origin of the function z → (1 − z) β−α , and apply Minkowski's inequality to obtain (1 + n) α (1 + k) l (1 + n + k) α+lψ (n + k) where f l is defined byf l (k) = k 1+k lf (k). By Stirling's formula, the binomial coefficients β−α l decay like 1 l 1+β−α , and since the map f → f l is obviously H 2 (H)contractive for each l, we use Lemma 3.4 to conclude that since β > 1 + α. Lemma 3.2 then implies that This proves that φ 1,α φ 2,α . It is straightforward to use the same type of argument to show that φ 2,α φ 1,α . In order to prove that φ 2,α ≈ φ 3,α , we note that We repeat the above argument in order to replace P β,log with P 1,log . This time instead of β > 1 + α, we use that β > 2. A third application of the argument allows us to replace P 1,log with P 1 , yielding φ 3,α ≈ φ 4,α .
As was pointed out earlier, φ 4,α ≈ φ 5,α is just a special case of Proposition 2.2, while the estimate φ 5,α ≤ φ 6,α is trivial. For the reverse inequality, if we identify H as a subspace of rank one operators in S 2 , it is obvious that By a simple argument holds whenever f, g ∈ O S 2 . By Lemma 3.1, φ 6,α φ 5,α . This completes the proof of Theorem 1.4.
A calculation shows that Going to the case where ψ ∈ B (L) , f, g ∈ O (H), we obtain that .
Since ψ ∈ B (L), we have that .
Following the ideas in [BKV99], we use Stirling's formula to see thatD α D −α acts like the identity plus a small multiplier. By Propositions 2.2 and 2.3, we can now complete the proof of Lemma 3.4 by showing that First we perform a simple decomposition of f and g into low and high frequencies. Assume that f and g are of degree at most l. By the triangle inequality we have and so by Cauchy-Schwarz's inequality. Thus the low frequencies exhibit the desired behaviour.
We now consider the high frequencies. Assume that (D α f ) ⊗ (D l g) has a zero of order l at the origin. We can then use Lemma 3.5, followed by Cauchy-Schwarz's inequality, and Parseval's identity to obtain that where in the last step, we have used Stirling's formula. Assuming Lemma 3.5, this completes the proof of Lemma 3.4.
Lemma 3.5. Let α > 0, N ∈ N 0 , and assume that h ∈ Hol (Y) has a zero of order N at the origin. Then Proof. We will use an idea of Flett [Fl72]. Term by term integration of the power series of h shows that Therefore We now replace the variable s with s δ , where δ = δ (x) will soon be chosen. (1 − x) l−1 x α−1 2 + N 1 + N + x α+l dx.
Replacing the variable x with (N +1)x N +2−x we obtain and the proof of Lemma 3.5 is complete.
Remark 3.6. The bound in Lemma 3.5 is sharp, as is seen by testing on the function h (z) = z N . In particular we have that where ∞ l=1 Cl l γ < ∞ whenever γ > 1, then in the proof of Theorem 1.4 one could immediately prove that φ 1,α ≈ φ 3,α , instead of using two iterations of the same argument.