PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 100(114) (2016), 1–16 DOI: 10.2298/PIM11614001P

LOGARITHMIC BLOCH SPACE AND ITS PREDUAL

Miroslav Pavlović

Abstract. We consider the space 𝔅_{log ^{α}}^{1} , of analytic functions on the unit disk 𝔻, defined by the requirement ∫_{𝔻} |f′(z)|ϕ(|z|) dA(z) < ∞, where ϕ(r) = log^{α} (1/(1 − r)) and show that it is a predual of the “log^{α} -Bloch” space and the dual of the corresponding little Bloch space. We prove that a function f(z) = ∑ _{n=0}^{∞} a_{n} z^{n} with a_{n} ↓ 0 is in 𝔅_{log ^{α}}^{1} iff ∑ _{n=0}^{∞} log^{α} (n + 2)/(n + 1) < ∞ and apply this to obtain a criterion for membership of the Libera transform of a function with positive coefficients in 𝔅_{log ^{α}}^{1} . Some properties of the Cesàro and the Libera operator are considered as well.

1. Introduction and some results

Let H(𝔻) denote the space of all functions analytic in the unit disk 𝔻 of the complex plane. Endowed with the topology of uniform convergence on compact subsets of 𝔻, the class H(𝔻) becomes a complete locally convex space. In this paper we are concerned with the predual of the space 𝔅_{log}^{α} , α ∈ ℝ,

𝔅_{log}^{α} = \{f ∈ H(𝔻) : |f′(z)| = O( (1 − |z|)^{−1} log^{α} \frac{2}{1 − |z|} ) \}.

The norm in 𝔅_{log}^{α} is defined by

∥f∥_{𝔅_{log} ^{α}} = |f(0)| + sup_{z∈𝔻}|f′(z)|(1 − |z|) log^{−α} \frac{1}{1 − |z|} .

The subspace, 𝔟_{log}^{α} , of 𝔅_{log}^{α} is defined by replacing “O” with “o”. It will be proved: (A) The dual of 𝔟_{log}^{α} is isomorphic to 𝔅_{log^{α}}^{1} ,

(1.1) 𝔅_{log^{α}}^{1} = \{f : ∥f∥_{𝔅_{log ^{α}}^{1}} = |f(0)| + ∫_{𝔻} |f′(z)| log^{α} \frac{2}{1 − |z|} dA(z) < ∞\},

2010 Mathematics Subject Classification: 30D55. Key words and phrases: Libera operator, Cesaro operator, Hardy spaces, logarithmic Bloch

type spaces, predual.

Supported by MNTR Serbia, Project ON174017. Communicated by Gradimir Milovanović.

and the dual of 𝔅_{log^{α}}^{1} is isomorphic to 𝔅_{log}^{α} , in both cases with respect to the bilinear form

⟨f, g⟩ = lim_{r↑1} ∑_{n=0}^{∞} f̂(n)ĝ(n)r^{2n} . (1.2)

(In (1.1) dA stands for the normalized Lebesgue measure on 𝔻.) This extends the well-known result on the Bloch space and the little Bloch space 𝔟 := 𝔟_{log}^{0}.

These spaces are Banach spaces, and the space 𝔟_{log}^{α} coincides with the closure in 𝔅_{log}^{α} of the set of all polynomials. The space 𝔅_{log} := 𝔅_{log}^{1} occurs naturally in the study of pointwise multipliers on the usual Bloch space 𝔅 := 𝔅_{log}^{0} (see [3]).

One of interesting properties of 𝔅_{log^{α}}^{1} is described in the following theorem:

Theorem 1.1. Let f(z) = ∑ _{n=0}^{∞} a_{n}z^{n}, where {a_{n}} is a nonincreasing sequence, of real numbers, tending to zero. Let α ⩾ −1. Then f belongs to 𝔅_{log^{α}}^{1} if and only if

(1.3)  S_{α} (f) := ∑_{n=0}^{∞} \frac{a_{n} log^{α} (n + 2)}{n + 1} < ∞.

Moreover, there is a constant C independent of {a_{n}} such that S_{α} (f)/C ⩽ ∥f∥_{𝔅_{log ^{α}}^{1}} ⩽ CS_{α} (f).

Proof. See Section 4. □

In the case α = 0, this assertion is proved in [17]. We can take a_{n} to be the coefficients of the Libera transform of a function with positive coefficients. Namely, if g(z) = ∑ _{n=0}^{∞} ĝ(n)z^{n} and

(1.4)  ∑_{n=0}^{∞} \frac{|ĝ(n)|}{n + 1} < ∞,

then the Libera transform Lg of g is well defined as

Lg(z) = \frac{1}{1 − z} ∫_{z}^{1} f(ζ) dζ = ∑_{n=0}^{∞} z^{n} ∑_{k=n}^{∞} \frac{ĝ(k)}{k + 1} (1.5)

(see, e.g., [12]). If ĝ ⩾ 0, then condition (1.4) is also necessary for the existence of the integral in (1.8): take z = 0 to conclude that (1.8) implies the convergence of the integral

∫_{0}^{1} g(t) dt = ∑_{n=0}^{∞} \frac{ĝ(n)}{n + 1} .

Then, as an application of Theorem 1.1 we get

Theorem 1.2. Let α > −1, let g ∈ H(𝔻), and ĝ ⩾ 0. Then Lg is in 𝔅_{log^{α}}^{1} if

and only if

K_{α} (g) := ∑_{n=0}^{∞} \frac{ĝ(n) log^{α+1} (n + 2)}{n + 1} < ∞.

We have K_{α} (g)/C ⩽ ∥Lg∥_{𝔅_{log ^{α}}^{1}} ⩽ CK_{α} (g).

Proof. See Section 4. □

In the general case, the integral in (1.5) need not exist, but it certainly exists if g ∈ H(\overline{𝔻}), which means that g is analytic in a neighborhood of the closed disk. By using Theorem 1.1 we shall prove that \overline{L} cannot be extended to a bounded operator from 𝔅_{log^{α}}^{1} to H(𝔻), if α < 0. In the case α ⩾ 0, every function g ∈ 𝔅_{log^{α}}^{1} satisfies (1.4), whence L is well defined, and we will show that L maps this space into 𝔅_{log^{α−1}}^{1} , when α > 0. If α = 0 we need a sort of “iterated” logarithmic space.

Cesàro operator. The dual of H(𝔻) is equal to H(\overline{𝔻}), where “g ∈ H(\overline{𝔻})” means that g is holomorphic in a neighborhood of \overline{𝔻} (depending on g). The duality pairing is given

(1.6)  ⟨f, g⟩ = ∑_{n=0}^{∞} f̂(n)ĝ(n),

where f(z) = ∑ _{n=0}^{∞} f̂(n)z^{n} ∈ H(𝔻) and g(z) = ∑ _{n=0}^{∞} ĝ(n)z^{n} ∈ H(\overline{𝔻}), and the series is absolutely convergent (see, e.g., [8]). The Cesàro operator is defined on H(𝔻) as

(1.7)  Cf(z) = ∑_{n=0}^{∞} z^{n} \frac{1}{n + 1} ∑_{k=0}^{n} a_{k}, f ∈ H(𝔻).

It is easy to verify that the adjoint of C : H(𝔻) ↦ H(𝔻) is equal to \overline{L} : H(\overline{𝔻}) ↦ H(\overline{𝔻}), under the pairing (1.6), and vice versa (see, e.g., [12]).

The operators C and \overline{L} acting on H^{p} spaces were first studied by Siskakis in 1987. In [21] he proved that C is bounded on H^{p} for 1 < p < ∞, and that \overline{L} can be extended to a bounded operator on H^{p}, 1 < p < ∞, and obtained some results on their spectra and norms. A few years later he proved the boundedness of the Cesàro operator on H^{1} ([22]), while Miao proved its boundedness on H^{p} for 0 < p < 1 ([10]). A short proof of the boundedness of C on H^{p}, 0 < p < ∞, as well as a stronger result, can be also found in Nowak [11]. However, H^{∞} is not mapped into itself by C (see [4]). If we write (1.7) as

zCf(z) = ∫_{0}^{z} \frac{f(ζ)}{1 − ζ} dζ,

and hence

(zCf(z))′ = \frac{f(z)}{1 − z} ,

we conclude that C maps H^{∞} into the Bloch space (see [4]).

On the other hand, by using the inequality

|f(z)| = O( log \frac{2}{1 − |z|} ) , f ∈ 𝔅,

and the analogous inequality for f ∈ 𝔟 (replace “O” with “o”), we get: (B) The operator C maps the space 𝔅 into 𝔅_{log}, and 𝔟 into 𝔟_{log}.

One of our aims is to generalize this assertion to some other values of α and then use assertion (A) together with the duality between C and \overline{L} to obtain an alternative proof of some results on the action of L from 𝔅_{α+1}^{1} to 𝔅_{α}^{1} , where

(1.8)  Lf(z) = ∫_{0}^{1} f(t + (1 − t)z) dt.

In particular we have: (C) The operator L is well defined on 𝔅_{log}^{1} and maps it into 𝔅^{1}.

It should be noted that: (a) 𝔅^{1} ⊊ H^{1}; (b) L does not map 𝔅^{1} into H^{1} (see [17]); and (c) L maps 𝔅 into BMOA [12], which improves an earlier result, namely that L maps 𝔅 into 𝔅 ([5, 24]).

The formula (1.8) is obtained from (1.5) by integrating over the straight line joining z and 1. A sufficient (not necessary [18]) condition for the possibility of such integration is (1.4) (g = f).

In proving some of our results, in particular assertions (A) and (B), we use a sequence of polynomials constructed in [6] (see also [7] and [16]) to decompose the space into a sum which resembles a sum of finite-dimensional spaces (see Section 3).

2. Some more results

Some elementary facts concerning the cases when Lf is well defined are collected in the following theorem, where

ℓ_{−1}^{1} = \{g ∈ H(𝔻) : ∥g∥_{ℓ_{−1}^{1}} = ∑_{n=0}^{∞} \frac{|ĝ(n)|}{n + 1} < ∞\}.

Theorem 2.1. Let α ∈ ℝ. Then:

(a) 𝔅_{log}^{α} ⊂ ℓ_{−1}^{1} for all α;

(b) 𝔅_{log^{α}}^{1} ⊂ ℓ_{−1}^{1} if and only if α ⩾ 0;

(c) if α < 0, then \overline{L} cannot be extended to a continuous operator from 𝔅_{log^{α}}^{1}

to H(𝔻).

Proof. See Section 4. □

Remark 2.1. The inclusions in (a) and (b) are continuous. Assertion (c) says much more than simply that 𝔅_{log^{α}}^{1} ⊄ ℓ_{−1}^{1} .

In the context of the action of C and L some new spaces occur: the space 𝔅_{logg} is defined by the requirement

|f′(z)| = O( loglog \frac{4}{1 − |z|} ) ,

the space 𝔟_{logg} defined by replacing “O” with “o”, and the space 𝔅_{logg}^{1} defined by

∫_{𝔻} |f′(z)| loglog \frac{4}{1 − |z|} dA(z) < ∞.

Our next result is

Theorem 2.2. (a) If α > −1, then C maps the space 𝔅_{log}^{α} , resp. 𝔟_{log}^{α} , into 𝔅_{log}^{α+1} , resp. 𝔟_{log}^{α+1} .

(b) C maps the space 𝔅_{log}^{−1} , resp. 𝔟_{log}^{−1} , into 𝔅_{logg} , resp. 𝔟_{logg} .

Proof. See Section 5. □

Remark 2.2. If f ∈ 𝔅_{log}^{α} and α < −1, then, as it can easily be shown, f ∈ A(𝔻), where A(𝔻) is the disk-algebra, i.e., the subset of H^{∞} consisting of those f which have a continuous extension to the closed disk. Moreover, the modulus of continuity of the boundary function f_{∗}(ζ), ζ ∈ ∂𝔻, satisfies the condition

ω(f_{∗}, t) = O( t( log \frac{2}{t})^{α+1} ) , t ↓ 0.

This follows from the inequality

ω(f_{∗}, t) ⩽ C ∫_{1−t}^{1} M_{∞} (r, f′) dr,

see [15, Theorem 2.2]. It should be noted that the modulus of continuity of f_{∗} is “proportional” to that of f(z), z ∈ 𝔻, see [23, 19].

Concerning the Libera operator we shall prove, besides Theorem 2.1(c), the following facts.

Theorem 2.3. (a) If α > 0, then L is well defined on 𝔅_{log^{α}}^{1} and maps this space to 𝔅_{log^{α−1}}^{1} .

(b) L is well defined on 𝔅_{logg}^{1} and maps this space into 𝔅_{log^{−1}}^{1} .

(c) L is well defined on 𝔅^{1} and maps it into 𝔅_{α}^{1} for all α < −1.

Proof. See Section 5. □

Theorem 2.4. Let α ∈ ℝ. Then the dual of 𝔟_{log}^{α} , resp. 𝔅_{log^{α}}^{1} , is isomorphic to 𝔅_{log^{α}}^{1} , resp. 𝔅_{log}^{α} under the pairing (1.2). Similarly, the dual of 𝔟_{logg} , resp. 𝔅_{logg}^{1} , is isomorphic to 𝔅_{logg}^{1} , resp. 𝔅_{logg} , under the same pairing.

Proof. See Section 6. □

Remark 2.3. The phrase “the dual of X is isomorphic to Y under the pairing (1.2)” means that if f ∈ X and g ∈ Y, then the limit in (1.2) exists and the functional Φ(f) = ⟨f, g⟩ is bounded on X; and on the other hand, if Φ ∈ X^{∗}, then there exists g ∈ Y such that Φ(f) = ⟨f, g⟩, and moreover, there exists a constant C independent of g such that ∥g∥_{Y}/C ⩽ ∥Φ∥ ⩽ C∥g∥_{Y}.

As an application of Theorems 2.2, 2.3, and 2.4, one can prove the following fact.

Theorem 2.5. Let α > 0. Then the adjoint (with respect to (1.2)) of the operator L : 𝔅_{log^{α}}^{1} ↦ 𝔅_{log^{α−1}}^{1} is equal to C : 𝔅_{log}^{α−1} ↦ 𝔅_{log}^{α} . The adjoint of the operator C : 𝔟_{log}^{α−1} ↦ 𝔟_{log}^{α} is equal to L : 𝔅_{log^{α}}^{1} ↦ 𝔅_{log^{α−1}}^{1} . The analogous assertions hold in the case when α = 0.

3. Decompositions

In [6], a sequence {V_{n}}_{0}^{∞} was constructed in the following way.

Let ω be a C^{∞} -function on ℝ such that

(1) ω(t) = 1 for t ⩽ 1,

(2) ω(t) = 0 for t ⩾ 2,

(3) ω is decreasing and positive on the interval (1, 2). Let φ(t) = ω(t/2) − ω(t), and let V_{0}(z) = 1 + z, and, for n ⩾ 1,

V_{n}(z) = ∑_{k=0}^{∞} φ(k/2^{n−1} )z^{k} = ∑_{k=2^{n−1}}^{2^{n+1} −1} φ(k/2^{n−1} )z^{k}.

The polynomials V_{n} have the following properties:

g(z) = ∑_{n=0}^{∞} V_{n} ∗ g(z), for g ∈ H(𝔻); (3.1)

∥V_{n} ∗ g∥_{p} ⩽ C∥g∥_{p}, for g ∈ H^{p}, p > 0; (3.2)

∥V_{n}∥_{p} ≍ 2^{n(1−1/p)} , for all p > 0, (3.3)

where ∗ denotes the Hadamard product. Here ∥h∥_{p} denotes the norm in the p-Hardy space H^{p},

∥h∥_{p} = sup_{0<r<1} ( \frac{1}{2π} ∫_{0}^{2π} |h(re^{iθ})| dθ)^{1/p} = sup_{0<r<1} M_{p}(r, g).

We need additional properties.

Lemma 3.1. Let P (z) = ∑ _{k=m}^{j} a_{k}z^{k}, m < j. Then

r^{j} ∥P ∥_{p} ⩽ M_{p}(r, P) ⩽ r^{m}∥P ∥_{p}, 0 < r < 1.

When applied to the polynomial P = V_{n} ∗ g′, this gives:

(3.4)  r^{2^{n+1} −1} ∥V_{n} ∗ g′∥_{p} ⩽ M_{p}(r, V_{n} ∗ g′) ⩽ r^{2^{n−1} −1} ∥V_{n} ∗ g′∥_{p} for n ⩾ 1.

Another inequality will be used (see [16, Exercise 7.3.5]):

(3.5)  2^{n−1} ∥V_{n} ∗ g∥_{p}/C ⩽ ∥V_{n} ∗ g′∥_{p} ⩽ C2^{n+1} ∥V_{n} ∗ g∥_{p} for n ⩾ 1,

where C is a constant independent of n and g.

Theorem 3.1. Let α ∈ ℝ, and f ∈ H(𝔻). Then:

(i) f ∈ 𝔅_{log}^{α} if and only if sup_{n⩾0} (n + 1)^{−α} ∥V_{n} ∗ f∥_{∞} < ∞. (ii) f ∈ 𝔟_{log}^{α} if and only if lim_{n→∞} (n + 1)^{−α} ∥V_{n} ∗ f∥_{∞} = 0. (iii) f ∈ 𝔅_{log^{α}}^{1} if and only if ∑ _{n=0}^{∞} (n + 1)^{α} ∥V_{n} ∗ f∥_{1} < ∞.

Moreover, the inequality

C^{−1} ∥f∥_{𝔅_{log} ^{α}} ⩽ sup_{n⩾0}(n + 1)^{−α} ∥V_{n} ∗ g∥_{∞} ⩽ C^{−1} ∥f∥_{𝔅_{log} ^{α}}

holds, where C is independent of f. The analogous inequality holds in the case of (iii) as well.

For the proof we need the following reformulation of [9, Proposition 4.1].

Lemma 3.2. Let φ be a continuous function on the interval (0, 1] such that φ(x)/x^{γ} (0 < x < 1) is nonincreasing, and φ(x)/x^{β} (0 < x < 1) is nondecreasing, where β and γ are positive constants independent of x. ^{1} Let

F_{1}(r) = (1 − r)^{−1/q} φ(1 − r) sup_{n⩾1}λ_{n}r^{2^{n+1} −1} ,

F_{2}(r) = (1 − r)^{1/q}φ(1 − r) ∑_{n=0}^{∞} λ_{n}r^{2^{n−1} −1} ,

where λ_{n} ⩾ 0, 0 < q ⩽ ∞. If F = F_{1} or F = F_{2}, then

C^{−1} ∥F ∥_{L^{q}(0,1)} ⩽ ∥{φ(2^{−n} )λ_{n}}∥_{ℓ^{q}} ⩽ C∥F ∥_{L^{q}(0,1)} .

Proof of Theorem 3.1. Case (i). Let φ(x) = x log^{−α} (2/x), and q = ∞. That φ is normal follows from the relation

lim_{x↓0} \frac{xφ′_{α} (x)}{φ_{α} (x)} = 1.

Let λ_{n} = 2^{n}∥V_{n} ∗ f∥_{∞} . By (3.1), (3.2), (3.4), and (3.5), we have

C^{−1} |f̂(1)| + C^{−1} sup_{n⩾1}λ_{n}r^{2^{n+1} −1} ⩽ M_{∞} (r, f′) ⩽ C|f̂(1)| + C ∑_{n=1}^{∞} λ_{n}r^{2^{n−1} −1} .

Hence, by Lemma 3.2, we obtain the desired result.

Case (ii). In this case we can proceed in two ways:

1^{∘} Modify the proof of Lemma 3.2 to get the inequalities

C^{−1} ∥F ∥_{C_{0} [0,1]} ⩽ ∥{φ(2^{−n} )λ_{n}}∥_{𝔠_{0}} ⩽ C∥F ∥_{C_{0} [0,1]},

where C_{0}[0, 1] = {u ∈ C[0, 1] : u(1) = 0} and 𝔠_{0} is the set of the sequences tending to zero.

2^{∘} Consider the spaces 𝔟_{log}^{α} ⊂ 𝔅_{log}^{α} and X = {f : ∥V_{n} ∗ f∥ = o((n + 1)^{α})}, which is, by assertion (i) and its proof, a subspace of a space Y isomorphic to 𝔅_{log}^{α} . It is not hard to show that the polynomials are dense in both 𝔟_{log}^{α} and X. This proves (ii).

Case (iii). In this case we use the function φ(x) = x log^{α} (2/x), let q = 1, and then proceed as in the proof of (i). The details are omitted. This concludes the proof of the theorem. □

Remark 3.1. By choosing ϕ(x) = x loglog(\frac{4}{x}), then we can conclude that Theorem 3.1 remains true if log^{α}, resp. (n + 1)^{α}, are replaced with loglog, resp. log(n + 2).

1Following Shields and Williams [20 ], we call such a function normal.

8 PAVLOVIĆ

4. Functions with decreasing coefficients

Proof of Theorem 1.1. Assuming that (1.3) holds, we want to prove that

∥f∥_{𝔅_{log ^{α}}^{1}} ⩽ Ca_{0} + C ∑_{n=1}^{∞} a_{2^{n−1}} (n + 1)^{α} .

According to Theorem 3.1 and its proof, we have

C^{−1} ∥f∥_{𝔅_{log ^{α}}^{1}} ⩽ a_{0} + ∑_{n=1}^{∞} (n + 1)^{α} ∥V_{n} ∗ f∥_{1} ⩽ C∥f∥_{𝔅_{log ^{α}}^{1}} .

Let n ⩾ 1, m = 2^{n−1} , and Q_{k} = ∑ _{j=m}^{k} φ(j/m)e_{j}. Since Q_{4m−1} = V_{n}, we have

V_{n} ∗ f = ∑_{k=m}^{4m−1} φ(k/m)a_{k}e_{k} = ∑_{k=m}^{4m−1} (a_{k} − a_{k+1} )Q_{k} + a_{4m} Q_{4m−1}

= ∑_{k=m}^{4m−1} (a_{k} − a_{k+1} )Q_{k} + a_{4m} V_{n}.

On the other hand, Q_{k} = V_{n} ∗ Δ_{n,k} , where

Δ_{n,k} = ∑_{j=2^{n−1}}^{k} z^{k}, 2^{n−1} ⩽ k ⩽ 2^{n+1} .

By (3.2), with g = Δ_{n,k} , we have

∥Q_{k}∥_{1} ⩽ C∥Δ_{n,k} ∥_{1} ⩽ C log(k + 1 − 2^{n−1} ) ⩽ C(n + 1).

Combining these inequalities we get

∥V_{n} ∗ f∥_{1}(n + 1)^{α} ⩽ C ∑_{k=m}^{4m−1} (a_{k} − a_{k+1} )(n + 1)^{α+1} + Ca_{4m} ∥V_{n}∥_{1}(n + 1)^{α}
⩽ C(n + 1)^{α+1} (a_{m} − a_{4m}) + Ca_{4m} (n + 1)^{α}
= C(n + 1)^{α+1} (a_{2^{n−1}} − a_{2^{n+1}} ) + C(n + 1)^{α} a_{2^{n+1}} .

Here we have used the relation ∥V_{n}∥_{1} ⩽ C (see (3.3))! Thus

(n + 1)^{α} ∥V_{n} ∗ f∥_{1} ⩽ C(n + 1)^{α+1} (a_{2^{n−1}} − a_{2^{n}} )
+ C(n + 1)^{α+1} (a_{2^{n}} − a_{2^{n+1}} ) + C(n + 1)^{α} a_{2^{n+1}} ,

and therefore it remains to estimate the sums

S_{1} = ∑_{n=1}^{∞} (n + 1)^{α+1} (a_{2^{n−1}} − a_{2^{n}} ) and S_{2} = ∑_{n=1}^{∞} (n + 1)^{α+1} (a_{2^{n}} − a_{2^{n+1}} ).

If α > −1, then

(n + 1)^{α+1} ⩽ C ∑_{k=1}^{n} (k + 1)^{α} ,

and hence

S_{1} ⩽ C ∑_{n=1}^{∞} (a_{2^{n−1}} − a_{2^{n}} ) ∑_{k=1}^{n} (k + 1)^{α} = C ∑_{k=1}^{∞} (k + 1)^{α} ∑_{n=k}^{∞} (a_{2^{n−1}} − a_{2^{n}} )

= C ∑_{k=1}^{∞} (k + 1)_{2^{k−1}}^{α} .

In the case of S_{2} we get

S_{2} ⩽ C ∑_{k=1}^{∞} (k + 1)^{α} a_{2^{k}} ,

which completes the proof of “if” part of the theorem in the case α > −1. If α = −1, then

∥V_{n} ∗ f∥_{1}(n + 1)^{−1} ⩽ C(a_{2^{n−1}} − a_{2^{n+1}} ) + C(n + 1)^{−1} a_{2^{n+1}} ,

from which we get the desired result in the case α = −1.

To prove “only if” part we use Hardy’s inequality in the form

πM_{1}(r, g) ⩾ ∑_{n=0}^{∞} \frac{|ĝ(n)|}{n + 1}r^{n}.

It follows that

∫_{𝔻} |f′(z)| log^{α} \frac{2}{1 − |z|} dA(z) = 2 ∫_{0}^{1} M_{1}(r, f′) log^{α} \frac{2}{1 − r} r dr

⩾ \frac{2}{π} ∑_{n=1}^{∞} a_{n} \frac{n}{n + 1} ∫_{0}^{1} log^{α} \frac{2}{1 − r} r^{n} dr

Now the desired result follows from the inequality

∫_{0}^{1} \frac{φ(1 − r)}{1 − r} r^{n} dr ⩾ cφ( \frac{1}{n + 1} ) (c = const. > 0),

valid for any normal function φ (see [9, Lemma 4.1]). □

Before proving Theorem 1.2, some remarks are in order. Let f(z) = ∑ _{n=0}^{∞} a_{n}z^{n}, a_{n} ⩾ 0. In order that Lf be well defined by (1.8) it is necessary and sufficient that

(4.1)  ∑_{n=0}^{∞} \frac{a_{n}}{n + 1} < ∞.

We already have mentioned in Introduction that this condition implies the existence of the integral in (1.8). In fact, this integral converges uniformly on compact subsets of 𝔻, which means that the limit

lim_{x↑1}∫_{0}^{x} f(t + (1 − t)z) dt

exists and is uniform in |z| < ρ, for every ρ < 1. This guarantees that Lf is analytic. On the other hand, if the integral in (1.8) exists, then we take z = 0 to conclude that (4.1) holds.

Proof of Theorem 1.2. The Taylor coefficients of Lf are

b_{n} = ∑_{k=n}^{∞} \frac{a_{n}}{n + 1} .

The sequence {b_{n}} is nonincreasing so we can apply Theorem 1.1 to conclude that Lf ∈ 𝔅_{log^{α}}^{1} if and only if

∑_{n=0}^{∞} \frac{log^{α} (n + 2)}{n + 1} ∑_{k=n}^{∞} \frac{a_{k}}{k + 1} = ∑_{k=0}^{∞} \frac{a_{k}}{k + 1} ∑_{n=0}^{k} \frac{log^{α} (n + 2)}{n + 1} < ∞.

Now the desired result follows from the estimate

C^{−1} log^{α+1} (k + 2) ⩽ ∑_{n=0}^{k} \frac{log^{α} (n + 2)}{n + 1} ⩽ C log^{α+1} (k + 2),

which holds because α > −1. □

Remark 4.1. The above proof shows that Lf belongs to 𝔅_{log^{−1}}^{1} if and only if

∑_{n=0}^{∞} \frac{a_{n}}{loglog(n + 4)} < ∞.

Now we pass to the proof of Theorem 2.1.

Proof of Theorem 1.1(c). Since 𝔅_{log}^{α} ⊂ 𝔅_{log}_{β} for β < α, we may assume that −1 < α < 0. Let

f(z) = ∑_{n=0}^{∞} a_{n}z^{n}, a_{n} = log^{−ε−α} (n + 2),

where ε > 1. Condition (4.1) holds because ε > 1. For every r ∈ (0, 1) the function f_{r}(z) = f(rz) belongs to H(\overline{𝔻}) and, by Theorem 1.1 and its proof, the set {f_{r} : 0 < r < 1} is bounded in 𝔅_{log^{α}}^{1} . On the other hand,

\overline{L}(f_{r})(0) = ∑_{k=0}^{∞} \frac{r^{k}}{(k + 1) log^{α+ε} (k + 2)} .

Now choose ε = 1 − α > 1 (because α < 0) to get

\overline{L}(f_{r})(0) = ∑_{k=0}^{∞} \frac{r^{k}}{(k + 1) log(k + 2)} −→ ∞ (r ↑ 1).

This contradicts the fact that if a set X ⊂ 𝔅_{log^{α}}^{1} is bounded and \overline{L} is bounded on 𝔅_{log^{α}}^{1} , then the set \{\overline{L}f(0) : f ∈ X\} is bounded because the functional h ↦ h(0) is continuous on H(𝔻). This completes the proof. □

Proof of Theorem 2.1(a). Let g ∈ 𝔅_{log}^{α} . Then

M_{2}(r, g′) ⩽ C(1 − r)^{−1} log \frac{2}{1 − r} .

It follows that

2^{n}( ∑_{k=2^{n}}^{2^{n+1} −1} |ĝ(k)|^{2}) ^{1/2} r^{2^{n+1}} ⩽ C(1 − r)^{−1} log^{α} \frac{2}{1 − r} .

Taking r = 1 − 2^{−n} , n ⩾ 1, we get

( ∑_{k=2^{n}}^{2^{n+1} −1} |ĝ(k)|^{2}) ^{1/2} ⩽ C log^{α} (n + 1).

Hence,

2^{−n} ∑_{k=2^{n}}^{2^{n+1} −1} |ĝ(k)| ⩽ ( 2^{−n} ∑_{k=2^{n}}^{2^{n+1} −1} |ĝ(k)|^{2}) ^{1/2} ⩽ 2^{−n/2} log^{α} (n + 1).

This gives the result. □

Proof of Theorem 2.1(b). In this case we use Hardy’s inequality as in the proof of Theorem 1.1 to get

∥g∥_{𝔅_{log ^{α}}^{1}} ⩾ c ∑_{n=0}^{∞} \frac{|ĝ(n)| log^{α} (n + 2)}{n + 1} .

This proves the result because α ⩾ 0. □

5. Proofs of Theorem 2.2 and 2.3

Define the operator R : H(𝔻) ↦ H(𝔻) by

Rf(z) = ∑_{n=0}^{∞} (n + 1)f̂(n)z^{n} = \frac{d}{dz}(zf(z)).

By using Theorem 3.1 and the relation

(5.1)  C^{−1} 2^{n}∥V_{n} ∗ f∥_{p} ⩽ ∥V_{n} ∗ Rf∥_{p} ⩽ C2^{n}∥V_{n} ∗ f∥_{p} (n ⩾ 0),

one proves that the norm in 𝔅_{log}^{α} is equivalent to

sup_{z∈𝔻}(1 − |z|) log^{−α} \frac{2}{1 − |z|} |Rf(z)|.

Proof of Theorem 2.2(a). Let α > −1 and f ∈ 𝔅_{log}^{α} . Then, by integration,

|f(z)| ⩽ log^{α+1} \frac{1}{1 − |z|} .

Since

RCf(z) = \frac{f(z)}{1 − z} ,

we see that

|RCf(z)| ⩽ C(1 − |z|)^{−1} log^{α+1} \frac{1}{1 − |z|} .

The result follows. □

Proof of Theorem 2.2(b). The function φ(x) = x loglog(4/x) is normal because lim_{x→0} xφ′(x)/φ(x) = 1. Hence, arguing as in the proof of Theorem 3.1 we conclude that f ∈ 𝔅_{logg} if and only if sup_{n⩾0} ∥V_{n} ∗ f∥_{∞} / log(n + 2) < ∞. Then using (5.1) we find that g ∈ 𝔅_{logg} if and only if

|Rg(z)| ⩽ C(1 − |z|)^{−1} loglog \frac{4}{1 − |z|} .

The rest of the proof is the same as in the case of (a). □

Remark 5.1. In the case of the small spaces the proofs are similar and therefore omitted.

For the proof of Theorem 2.3 we need the following lemma [18]:

Lemma 5.1. If f ∈ ℓ_{−1}^{1} , then Lf is well defined by (1.8) and the inequality

(5.2)  rM_{1}(r, (Lf)′) ⩽ 2(1 − r)^{−1} ∫_{r}^{1} M_{1}(s, f′) ds, 0 < r < 1,

holds.

Before passing to the proof observe that 𝔅_{log^{α}}^{1} ⊂ 𝔅^{1} and 𝔅_{logg}^{1} ⊂ 𝔅^{1}, and, since 𝔅_{1} ⊂ H^{1}, we see that in all cases of Theorem 2.3 the operator L is well defined.

Proof of Theorem 2.3(a). We have, by (5.2),

∫_{𝔻} |(Lf)′(z)| log^{α−1} \frac{2}{1 − |z|} dA(z) = 2 ∫_{0}^{1} M_{1}(r, (Lf)′) log^{α−1} \frac{2}{1 − r} r dr

⩽ 4 ∫_{0}^{1}(1 − r)^{−1} log^{α−1} \frac{2}{1 − r} dr ∫_{r}^{1} M_{1}(s, f′) ds

= 4 ∫_{0}^{1} M_{1}(s, f′) ds ∫_{0}^{s}(1 − r)^{−1} log^{α−1} \frac{2}{1 − r} dr

⩽ C ∫_{0}^{1} M_{1}(s, f′) log^{α} \frac{2}{1 − s} ds.

A standard application of the maximum modulus principle shows that the inequality remains valid if we replace ds with s ds. This gives the result. □

The proofs of Theorem 2.3, (b) and (c), are similar and we omit them.

6. Proof of Theorem 2.4

We consider a more general situation. Let X ⊂ H(𝔻) (with continuous inclusion) be a Banach space such that the functions f_{w}(z) = f(wz), |w| ⩽ 1, belong to X whenever f ∈ X, and sup_{|w|⩽1} ∥f_{w}∥_{X} ⩽ ∥f∥_{X}. Such a space is said to be homogeneous (see [2]). A homogeneous space satisfies the condition

(6.1)  ∥V_{n} ∗ f∥_{X} ⩽ C∥f∥_{X} , f ∈ X,

where C is independent of n and f.

If in addition

(6.2)  lim_{r↑1}∥f − f_{r}∥_{X} = 0, f ∈ X,

then the dual of X can be identified with the space, X′, of those g ∈ H(𝔻) for which limit (1.2) exists for all f ∈ X (see [1, 2]). Also, the dual of a homogeneous space X satisfying (6.2) can be realized as the space of coefficient multipliers, (X, A(𝔻)), from X to A(𝔻); in this case we have (X, A(𝔻)) = (X, H^{∞} ) =: X^{∗} (see [2]). The norm in X^{∗} is introduced as

∥g∥_{X^{∗}} = sup∥f ∗ g∥_{∞} : f ∈ X, ∥f∥_{X} ⩽ 1},

and, if X is homogeneous and satisfies (6.2), it is equal to

∥g∥_{X′} = sup{|⟨f, g⟩| : f ∈ X, ∥f∥_{X} ⩽ 1}.

There is another way to express ⟨f, r⟩, when f ∈ X, X satisfies (6.2), and g ∈ X′; namely, in this case,the function f∗g belongs to A(𝔻), and we have ⟨f, g⟩ = (f∗g)(1) (see [13, 2]).

We fix a sequence λ = {λ_{n}}_{0}^{∞} of positive real numbers such that

(6.3)  0 < inf_{n⩾0} \frac{λ_{n+1}}{λ_{n}} , sup_{n⩾0} \frac{λ_{n+1}}{λ_{n}} < ∞.

It is clear that the spaces H^{p} (0 < p ⩽ ∞), A(𝔻), 𝔅_{log}^{α} , 𝔟_{log}^{α} , and 𝔅_{log^{α}}^{1} are homogeneous. Among them only H^{∞} and 𝔅_{log}^{α} do not satisfy condition (6.2).

Consider the following three spaces of sequences {f_{n}}_{0}^{∞} , f_{n} ∈ H(𝔻):

(a) 𝔠_{0}(λ, X) = {{f_{n}} : lim_{n→∞} ∥f_{n}∥_{X} /λ_{n} = 0};
(b) ℓ^{∞} (λ, X) = {{f_{n}} : sup_{n⩾0} λ_{n}∥V_{n} ∗ f∥_{X} < ∞};
(c) ℓ^{1}(λ, X) = {{f_{n}} : ∑ _{n=0}^{∞} ∥f_{n}∥_{X} /λ_{n} < ∞}.

We also define the spaces v_{0}(λ, X), V^{∞} (λ, X), and V^{1}(λ, X) (as subsets of H(𝔻)) by replacing f_{n} with V_{n} ∗ f in (a), (b), and (c), respectively. The proof of the following lemma is rather easy, and is therefore left to the reader.

Lemma 6.1. If X is a homogeneous space, then so are v(λ, X), V^{∞} (λ, X), and V^{1}(λ, X). The spaces v_{0}(λ, X) and V^{1}(λ, X) satisfy (6.2). The space v_{0}(λ, X) is equal to the closure in V^{∞} (λ, X) of the sets of all polynomials.

Theorem 2.4 will be deduced from Theorem 3.1 and the following.

Proposition 6.1. If X is a homogeneous space satisfying (6.2), then the dual of v_{0}(λ, X), resp. V^{1}(λ, X), is isomorphic to V^{1}(λ, X′), resp. V^{∞} (λ, X′), with respect to (1.2).

In proving we use ideas from [13, 14, 7]. For the proof we need the following lemma.

Lemma 6.2. The operator T ({f_{n}}) = ∑ _{n=0}^{∞} V_{n} ∗ f_{n} acts as a bounded operator from Y to Z, where Y is one of the spaces 𝔠_{0}(λ, X), ℓ^{∞} (λ, X), and ℓ^{1}(λ, X), while Z is v_{0}(λ, X), V^{∞} (λ, X), and V^{1}(λ, X), respectively.

14 PAVLOVIĆ

Proof. We have V_{n} ∗ V_{j} = 0 for |j − n| ≧ 2 and hence

V_{n} ∗ T ({f_{j} }) = ∑_{j=n−1}^{n+1} V_{n} ∗ V_{j} ∗ f_{j} , n ⩾ 0,

where, by definition, w_{j} = f_{j} = 0 for j < 0. It follows that

∥V_{n} ∗ T ({f_{j} }∥_{X} ⩽ C ∑_{j=n−1}^{n+1} ∥f_{j} ∥_{X} ,

where we have used (6.1). Now the proof is easily completed by using (6.3). □

Lemma 6.3. Let g ∈ (v_{0}(λ, X))′, resp. g ∈ (V^{1}(λ, X))′, and define the operator S on 𝔠_{0}(λ, X), resp. ℓ^{1}(λ, X), by

S({f_{n}}) = T ({f_{n}}) ∗ g = ∑_{k=0}^{∞} f_{k} ∗ V_{k} ∗ g.

Then S maps 𝔠_{0}(λ, X), resp. ℓ^{1}(λ, X), into H^{∞} and ∥S∥ ⩽ C∥g∥_{(v_{0} (λ,X))′} , resp. ∥S∥ ⩽ C∥g∥_{(V^{1} (λ,X))′} .

Proof. By the preceding lemma, we have

∥S({f_{n}})∥_{∞} ⩽ ∥T ({f_{n}})∥_{v_{0} (λ,X)} ∥g∥_{(v_{0} (λ,X))∗} ⩽ C∥{f_{n}}∥_{𝔠_{0} (λ,X)} ∥g∥_{(v_{0} (λ,X))∗} . This proves the result in one case. In the other case the proof is the same. □

Proof of Proposition 6.1. Define the polynomials P_{n} (n ⩾ 0) by P_{n} = V_{n−1} + V_{n} + V_{n+1} .

Hence

V_{n} = ∑_{j=0}^{∞} V_{j} ∗ V_{n} = (V_{n−1} + V_{n} + V_{n+1} ) ∗ V_{n} = P_{n} ∗ V_{n}.

Let f ∈ v_{0}(λ, X) and g ∈ V^{1}(λ, X′). It is easily verified that, when 0 < r < 1,

(f ∗ g)(z) = ∑_{n=0}^{∞} (f ∗ V_{n} ∗ g)(z) = ∑_{n=0}^{∞} (P_{n} ∗ f ∗ V_{n} ∗ g)(z), z ∈ 𝔻.

the series being absolutely convergent. Since

∥P_{n} ∗ f_{r} ∗ V_{n} ∗ g∥_{∞} ⩽ ∥P_{n} ∗ f∥_{X} ∥V_{n} ∗ g∥_{X^{∗}} ,

we have

∥f ∗ g∥_{∞} ⩽ ∑_{n=0}^{∞} ∥P_{n} ∗ f∥_{X} ∥V_{n} ∗ g∥_{X^{∗}} ⩽ C ∑_{n=0}^{∞} ∥P_{n} ∗ f∥_{X} ∥V_{n} ∗ g∥_{X^{∗}}

= C ∑_{n=0}^{∞} (∥P_{n} ∗ f∥_{X} /λ_{n}) (λ_{n}∥V_{n} ∗ g∥_{X^{∗}} ) ⩽ C∥f∥_{v_{0} (λ,X)} ∥g∥_{V^{1} (λ,X^{∗})}

This proves the inclusion V^{1}(λ, X′) ⊂ (v_{0}(λ, X))′.

To prove the converse, let g ∈ (v_{0}(λ, X))^{∗}. Let S denote the operator defined in Lemma 6.3. By Lemma 6.3, S acts as a bounded operator from 𝔠_{0}(λ, X) into H^{∞} and we have ∥S∥ ⩽ C∥g∥_{v_{0} (λ,X)^{∗}} . Now it suffices to prove that

∥S∥ ⩾ (1/2)∥{g_{n}}∥_{ℓ^{1} (λ,X^{∗})} = (1/2)∥g∥_{V^{1} (λ,X)} .

For each n ⩾ 0 choose f_{n} ∈ X so that ∥f_{n}∥_{X} = 1 and ⟨f_{n}, g_{n}⟩ is a real number such that ⟨f_{n}, g_{n}⟩ ⩾ (1/2)∥g_{n}∥_{X^{∗}} . If {a_{n}} is a finite sequence of nonnegative real numbers, then

S({a_{n}f_{n}}) = ∑_{n=0}^{∞} a_{n}⟨f_{n}, g_{n}⟩ ⩾ (1/2) ∑_{n=0}^{∞} a_{n}∥g_{n}∥_{X^{∗}} = (1/2) ∑_{n=0}^{∞} (a_{n}/λ_{n})λ_{n}∥g_{n}∥_{X^{∗}} .

Hence, by taking the supremum over all {a_{n}} such that 0⟨a_{n}⟩λ_{n}, we get S{λ_{n}f_{n}} ⩾ (1/2) ∑ _{n=0}^{∞} λ_{n}∥g_{n}∥_{X^{∗}} . Since ∥{a_{n}f_{n}}∥_{𝔠_{0} (λ,X)} ⩽ 1, where a_{n} = λ_{n} for 0 ⩽ n ⩽ N (N ∈ ℕ) and a_{n} = 0 for n > N we see that ∥S∥ ⩾ (1/2)∥g∥_{V^{1} (λ,X^{∗})} , as desired. This completes the proof that v_{0}(λ, X)′ = V^{1}(λ, X′). In a similar way one proves that V^{1}(λ, X)′ = V^{∞} (λ, X′), which is all what has to be proved. □

Proof of Theorem 2.4. First we prove that (𝔟_{log}^{α} )′ = 𝔅_{log^{α}}^{1} . By Theorem 3.1, we have 𝔟_{log}^{α} = v_{0}(λ, A(𝔻)), where λ_{n} = (n + 1)^{α}. Hence, by Proposition 6.1, the dual of 𝔟_{log}^{α} is isomorphic to V^{1}(λ, A(𝔻)′). In order to estimate ∥V_{n} ∗ g∥_{A()′} first observe that H^{1} ⊂ A(𝔻)′and moreover ∥V_{n} ∗g∥_{A(𝔻)′} ⩽ ∥V_{n} ∗g∥_{1}. On the other hand, let Φ be a bounded linear functional on A(𝔻), let Φ_{0} be the Hahn/Banach extension of Φ to hC(𝔻), and choose g ∈ A(𝔻)^{a} so that Φ(f) = ⟨f, g⟩ for all f ∈ A(𝔻). By the Riesz representation theorem, we have

Φ_{0}(f) = \frac{1}{2π} ∫_{0}^{2π} f(e^{−iθ} ) dμ(e^{iθ})

= lim_{r→1^{−}} \frac{1}{2π} ∫_{0}^{2π} f(re^{−iθ} )g(re^{iθ}) dθ = lim_{r→1^{−}} ∑_{n=0}^{∞} f̂(n)ĝ(n)r^{2n} ,

and ∥μ∥ = ∥Φ∥ = ∥Φ_{0}∥. In particular, taking f(w) = (1 − zw)^{−1} , where z ∈ 𝔻 is fixed, we get \frac{1}{2π} ∫_{0}^{2π} (1 − e^{−iθ} z)^{−1} dμ(e^{iθ}) = g(z). Hence

R^{1}g(z) = \frac{1}{2π} ∫_{0}^{2π} (1 − e^{−iθ} z)^{−2} dμ(e^{iθ}),

and hence, by integration, M_{1}(r, R^{1}f) ⩽ ∥μ∥(1 − r^{2})^{−1} = ∥g∥_{A(𝔻)′} (1 − r^{2})^{−1} . Now we proceed as in the proof of Theorem 3.1 to conclude that ∥V_{n} ∗ g∥_{1} ⩽ C∥V_{n} ∗ g∥_{A(𝔻)′}. It follows that g ∈ (𝔟_{log}^{α} )′ if and only if g ∈ V^{1}(λ, H^{1}), i.e., by Theorem 3.1, g ∈ 𝔅_{log^{α}}^{1} .

In proving that (𝔅_{log^{α}}^{1} )′ is isomorphic to 𝔅_{log}^{α} , we use the inclusions H^{∞} ⊂ (H^{1})′ ⊂ 𝔅, and then proceed as above. □

Remark 6.1. The above proof of Theorem 2.4 certainly is not the simplest one. However, it can be applied to prove some general duality and multipliers theorems (see [13, 14, 7]. For instance, the dual of 𝔟_{logg} is isomorphic to 𝔅_{logg}^{1} , and the dual of 𝔅_{logg}^{1} is isomorphic to 𝔅_{logg} .

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Faculty of Mathematics, University of Belgrade (Received 24 12 2015) Belgrade, Serbia

 pavlovic@matf.bg.ac.rs