Integral-Type Operators from Bloch-Type Spaces to 𝑄_𝐾 Spaces

Abstract

The boundedness and compactness of the integral-type operator 𝐼^{(𝑛)𝜑,𝑔}𝑓(𝑧)=∫^𝑧0𝑓^(𝑛)(𝜑(𝜁))𝑔(𝜁)𝑑𝜁, where 𝑛∈ℕ₀, 𝜑 is a holomorphic self-map of the unit disk 𝔻, and 𝑔 is a holomorphic function on 𝔻, from 𝛼-Bloch spaces to 𝑄_𝐾 spaces are characterized.

1. Introduction

Let 𝔻 be the open unit disk in the complex plane, 𝜕𝔻 be its boundary, 𝐷(𝑤,𝑟) be disk centered at 𝑤 of radius 𝑟, and let 𝐻(𝔻) be the class of all holomorphic functions on 𝔻. Let 𝜂_𝑎(𝑧)=𝑎−𝑧1−𝑎𝑧,𝑎∈𝔻,(1.1) be the involutive Möbius transformation which interchanges points 0 and 𝑎. If 𝑋 is a Banach space, then by 𝐵_𝑋 we will denote the closed unit ball in 𝑋.

The 𝛼-Bloch space, ℬ^𝛼(𝔻)=ℬ^𝛼,𝛼>0, consists of all 𝑓∈𝐻(𝔻) such that sup_𝑧∈𝔻1−|𝑧|²^𝛼||𝑓′(𝑧)||<∞.(1.2) The little 𝛼-Bloch space ℬ^𝛼0(𝔻)=ℬ^𝛼0 consists of all functions 𝑓 holomorphic on 𝔻 such that lim_|𝑧|→1(1−|𝑧|²)^𝛼|𝑓′(𝑧)|=0. The norm on ℬ^𝛼 is defined by ‖𝑓‖_ℬ^𝛼=||𝑓(0)||+sup_𝑧∈𝔻1−|𝑧|²^𝛼||𝑓′(𝑧)||.(1.3) With this norm, ℬ^𝛼 is a Banach space, and the little 𝛼-Bloch space ℬ^𝛼0 is a closed subspace of the 𝛼-Bloch space. Note that ℬ¹=ℬ is the usual Bloch space.

Given a nonnegative Lebesgue measurable function 𝐾 on (0,1] the space 𝑄_𝐾 consists of those 𝑓∈𝐻(𝔻) for which 𝑏^2𝑄_𝐾(𝑓)=sup_𝑎∈𝔻_𝔻||𝑓′(𝑧)||²𝐾1−||𝜂_𝑎(𝑧)||²𝑑𝑚(𝑧)<∞,(1.4) where 𝑑𝑚(𝑧)=(1/𝜋)𝑑𝑥𝑑𝑦=(1/𝜋)𝑟𝑑𝑟𝑑𝜃 is the normalized area measure on 𝔻 [1]. It is known that 𝑏_𝑄𝐾 is a seminorm on 𝑄_𝐾 which is Möbius invariant, that is, 𝑏_𝑄𝐾(𝑓∘𝜂)=𝑏_𝑄𝐾(𝑓),𝜂∈Aut(𝔻),(1.5) where Aut(𝔻) is the group of all automorphisms of the unit disk 𝔻. It is a Banach space with the norm defined by ‖𝑓‖_𝑄𝐾=||𝑓(0)||+𝑏_𝑄𝐾(𝑓).(1.6)

The space 𝑄_𝐾,0 consists of all 𝑓∈𝐻(𝔻) such that lim_|𝑎|→1_𝔻||𝑓′(𝑧)||²𝐾1−||𝜂_𝑎(𝑧)||²𝑑𝑚(𝑧)=0.(1.7) It is known that 𝑄_𝐾,0 is a closed subspace of 𝑄_𝐾. For classical 𝑄 spaces, see [2].

It is clear that each 𝑄_𝐾 contains all constant functions. If 𝑄_𝐾 consists of just constant functions, we say that it is trivial. 𝑄_𝐾 is nontrivial if and only if sup_𝑡∈(0,1)¹⁰𝐾(1−𝑟)(1−𝑡)²1−𝑡𝑟²³𝑟𝑑𝑟<∞.(1.8) Throughout this paper, we assume that condition (1.8) is satisfied, so that the space 𝑄_𝐾 is nontrivial. An important tool in the study of 𝑄_𝐾 spaces is the auxiliary function 𝜆_𝐾 defined by 𝜆_𝐾(𝑠)=sup_0<𝑡≤1𝐾(𝑠𝑡)𝐾(𝑡),0<𝑠<∞,(1.9) where the domain of 𝐾 is extended to (0,∞) by setting 𝐾(𝑡)=𝐾(1) when 𝑡>1. The next two conditions play important role in the study of 𝑄_𝐾 spaces. (a)There is a constant 𝐶>0 such that for all𝑡>0𝐾(2𝑡)≤𝐶𝐾(𝑡).(1.10)(b)The auxiliary function 𝜆_𝐾 satisfies the following condition: ¹⁰𝜆_𝐾(𝑠)𝑠𝑑𝑠<∞.(1.11)

Let Ω(0,∞) denote the class of all nondecreasing continuous functions on (0,∞) satisfying conditions (1.8), (1.10), and (1.11).

A positive Borel measure 𝜇 on 𝔻 is called a 𝐾-Carleson measure [3] if sup_𝐼_𝑆(𝐼)𝐾1−|𝑧|||𝐼||𝑑𝜇(𝑧)<∞,(1.12) where the supermum is taken over all subarcs 𝐼⊂𝜕𝔻, |𝐼| is the length of 𝐼, and 𝑆(𝐼) is the Carleson box defined by 𝑆(𝐼)=𝑧∶1−||𝐼||<|𝑧|<1,𝑧|𝑧|∈𝐼.(1.13)

A positive Borel measure 𝜇 is called a vanishing 𝐾-Carleson measure if lim_||𝐼||→0_𝑆(𝐼)𝐾1−|𝑧|||𝐼||𝑑𝜇(𝑧)=0.(1.14)

We also need the following results of Wulan and Zhu in [3], in which 𝑄_𝐾 spaces are characterized in terms of 𝐾-Carleson measures.

Theorem 1.1. Let 𝐾∈Ω(0,∞). Then a positive Borel measure 𝜇 on 𝔻 is a 𝐾-Carleson measure if and only if sup_𝑎∈𝔻_𝔻𝐾1−||𝜂_𝑎(𝑧)||²𝑑𝜇(𝑧)<∞.(1.15) Also, 𝜇 is a vanishing 𝐾-Carleson measure if and only if lim_|𝑎|→1_𝔻𝐾1−||𝜂_𝑎(𝑧)||²𝑑𝜇(𝑧)=0.(1.16)

From Theorem 1.1 and the definition of the spaces 𝑄_𝐾 and 𝑄_𝐾,0, we see that when 𝐾∈Ω(0,∞), then 𝑓∈𝑄_𝐾 if and only if the measure 𝑑𝜇_𝑓=|𝑓′(𝑧)|²𝑑𝑚(𝑧) is a 𝐾-Carleson measure, while 𝑓∈𝑄_𝐾,0 if and only if this measure is a vanishing 𝐾-Carleson measure.

Let 𝜑∈𝑆(𝔻) be the family of all holomorphic self-maps of 𝔻, 𝑔∈𝐻(𝔻), and 𝑛∈ℕ₀. We define an integral-type operator as follows: 𝐼^{(𝑛)𝜑,𝑔}𝑓(𝑧)=^𝑧0𝑓^(𝑛)(𝜑(𝜁))𝑔(𝜁)𝑑𝜁,𝑧∈𝔻.(1.17) Operator (1.17) extends several operators which has been introduced and studied recently (see, e.g., [4–9]). For related operators in 𝑛-dimensional case, see, for example, [10–19]. For some classical operators see, for example, [20, 21] and the related references therein. For other product-type operators, see [22] and the references therein.

Motivated by [23, 24] (see also [25–29]), we characterize when 𝜑 and 𝑔 induce bounded and/or compact operators in (1.17) from 𝛼-Bloch to 𝑄_𝐾 spaces.

Throughout this paper, constants are denoted by 𝐶; they are positive and not necessarily the same at each occurrence. The notation 𝐴≍𝐵 means that there is a positive constant 𝐶 such that 𝐵/𝐶≤𝐴≤𝐶𝐵.

2. Auxiliary Results

Here, we quote several lemmas which will be used in the proofs of the main results in this paper. The following lemma is folklore (see, e.g., [30]).

Lemma 2.1. For any 𝑓∈𝐻(𝔻) and 𝑧∈𝔻, the following inequalities hold ||𝑓(𝑧)||≤𝐶⎧⎪⎪⎨⎪⎪⎩‖𝑓‖^ℬ𝛼‖𝑓‖_ℬ𝛼,if0<𝛼<1,ln𝑒1−|𝑧|²‖𝑓‖_ℬ𝛼,if𝛼=1,1−|𝑧|²^𝛼−1,if𝛼>1,(2.1)||𝑓⁽ⁿ⁾(𝑧)||≤𝐶sup^{𝑤∈𝐷(𝑧,(1−|𝑧|)/2)}1−|𝑤|²^𝛼||𝑓^(𝑤)||1−|𝑧|²^{𝛼+𝑛−1}≤𝐶‖𝑓‖^ℬ𝛼1−|𝑧|²^{𝛼+𝑛−1},if𝑛∈ℕ.(2.2)

The next lemma is obtained in [31, 32].

Lemma 2.2. Let 𝛼>0. Then there are two functions 𝑓₁,𝑓₂∈ℬ^𝛼 such that ||𝑓^1(𝑧)||+||𝑓^2(𝑧)||≥𝐶1−|𝑧|²^𝛼,𝑧∈𝔻.(2.3) Also, if 𝛼≠1, then there are two functions 𝑓₃,𝑓₄∈ℬ^𝛼 and 𝐶>0, such that ||𝑓₃(𝑧)||+||𝑓₄(𝑧)||≥𝐶1−|𝑧|²^𝛼−1,𝑧∈𝔻.(2.4)

The next Schwartz-type lemma [33] is proved in a standard way, so we omit the proof.

Lemma 2.3. Let 𝛼>0,𝐾∈Ω(0,∞),𝜑∈𝑆(𝔻),𝑔∈𝐻(𝔻), and 𝑛∈ℕ₀. Then 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼(𝑜𝑟ℬ^𝛼0)→𝑄_𝐾 is compact if and only if for any bounded sequence (𝑓_𝑚)_𝑚∈ℕ in ℬ^𝛼 converging to zero on compacts of 𝔻, we have lim_𝑚→∞‖𝐼^{(𝑛)𝜑,𝑔}𝑓_𝑚‖_𝑄𝐾=0.

Lemma 2.4. Let 𝛼>0,𝐾∈Ω(0,∞),𝜑∈𝑆(𝔻),𝑔∈𝐻(𝔻), and 𝑛∈ℕ₀. Then 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾(𝑜𝑟𝑄_𝐾,0) is weakly compact if and only if it is compact.

Proof. By a known theorem 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾(or𝑄_𝐾,0) is weakly compact if and only if (𝐼^{(𝑛)𝜑,𝑔})^∗∶𝑄^∗𝐾(or𝑄^∗𝐾,0)→(ℬ^𝛼0)^∗ is weakly compact. Since (ℬ^𝛼0)^∗≅𝐴¹ (the Bergman space) and 𝐴¹ has the Schur property, it follows that it is equivalent to (𝐼^{(𝑛)𝜑,𝑔})^∗∶𝑄^∗𝐾(or𝑄^∗𝐾,0)→(ℬ^𝛼0)^∗, is compact, which is equivalent to 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾(or𝑄_𝐾,0), is compact, as claimed.

Lemma 2.5. Let 𝛼>0,𝐾∈Ω(0,∞),𝜑∈𝑆(𝔻),𝑔∈𝐻(𝔻), and 𝑛∈ℕ₀. Then 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾,0 is compact if and only if 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼→𝑄_𝐾,0 is bounded.

Proof. By Lemma 2.4, 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾,0 is compact if and only if it is weakly compact, which, by Gantmacher's theorem ([34]), is equivalent to (𝐼^{(𝑛)𝜑,𝑔})^∗∗((ℬ^𝛼0)^∗∗)⊆𝑄_𝐾,0. Since (ℬ^𝛼0)^∗∗=ℬ^𝛼 and by a standard duality argument (𝐼^{(𝑛)𝜑,𝑔})^∗∗=𝐼^{(𝑛)𝜑,𝑔} on ℬ^𝛼, this can be written as 𝐼^{(𝑛)𝜑,𝑔}(ℬ^𝛼)⊆𝑄_𝐾,0, which by the closed graph theorem is equivalent to 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼→𝑄_𝐾,0 is bounded.

For 𝑎∈𝔻, set Φ_{𝜑,𝑔,𝐾}(𝑎)=_𝔻𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²1−||𝜑(𝑧)||²^{2(1−𝛼−𝑛)}𝑑𝑚(𝑧).(2.5)

Lemma 2.6. Let 𝛼>0,𝐾∈Ω(0,∞),𝜑∈𝑆(𝔻),𝑔∈𝐻(𝔻), and 𝑛∈ℕ₀. If Φ_{𝜑,𝑔,𝐾} is finite at some point 𝑎∈𝔻, then it is continuous on 𝔻.

Proof. We follow the lines of Lemma  2.3 in [24]. From the elementary inequality (1−|𝑎|)1−||𝑎₁||4≤1−||𝜂^𝑎(𝑧)||²1−||𝜂_𝑎1(𝑧)||²≤4(1−|𝑎|)1−||𝑎₁||,𝑎,𝑎¹,𝑧∈𝔻,(2.6) and since 𝐾 is nondecreasing and satisfies (1.10), we easily get 𝐾1−||𝜂_𝑎1(𝑧)||²≤𝐶^{[log2(4/(1−|𝑎|)(1−|𝑎1|))]+1}𝐾1−||𝜂_𝑎(𝑧)||².(2.7) From (2.7) and since Φ_{𝜑,𝑔,𝐾}(𝑎) is finite, it follows that Φ_{𝜑,𝑔,𝐾} is finite at each point 𝑎₁∈𝔻. Let 𝑎∈𝔻 be fixed, and let (𝑎_𝑙)_𝑙∈ℕ⊂𝔻 be a sequence converging to 𝑎.
We have||Φ_{𝜑,𝑔,𝐾}(𝑎)−Φ_{𝜑,𝑔,𝐾}𝑎_𝑙||≤_𝔻||𝑔(𝑧)||²|||𝐾1−||𝜂_𝑎(𝑧)||²−𝐾1−||𝜂^𝑎_𝑙(𝑧)||²|||1−||𝜑(𝑧)||²^{2(𝛼+𝑛−1)}𝑑𝑚(𝑧).(2.8) From (2.6), we have that for 𝑙 such that 1−|𝑎_𝑙|≥(1−|𝑎|)/2, say 𝑙≥𝑙₀, holds 1−||𝜂_𝑎𝑙(𝑧)||²≤8(1−|𝑎|)²1−||𝜂^𝑎(𝑧)||²,(2.9) and consequently for 𝑙≥𝑙₀, it holds |||𝐾1−||𝜂_𝑎(𝑧)||²−𝐾1−||𝜂^𝑎_𝑙(𝑧)||²|||≤1+𝐶^{[log2(8/(1−|𝑎|)2)]+1}𝐾1−||𝜂^𝑎(𝑧)||².(2.10)
From this and since Φ_{𝜑,𝑔,𝐾} is finite at 𝑎, by the Lebesgue dominated convergence theorem, we get that the integral in (2.8) converges to zero as 𝑙→∞ which implies that Φ_{𝜑,𝑔,𝐾}(𝑎_𝑙)→Φ_{𝜑,𝑔,𝐾}(𝑎) as 𝑙→∞, from which the lemma follows.

3. Boundedness and Compactness of 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼(orℬ^𝛼0)→𝑄_𝐾(or𝑄_𝐾,0)

In this section, we characterize the boundedness and compactness of the operators 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼(orℬ^𝛼0)→𝑄_𝐾(or𝑄_𝐾,0). Let 𝑑𝜇_{𝜑,𝑔,𝑛,𝛼}(𝑧)=||𝑔(𝑧)||²1−||𝜑(𝑧)||²^{2(1−𝛼−𝑛)}𝑑𝑚(𝑧).(3.1)

Theorem 3.1. Let 𝛼>0,𝐾∈Ω(0,∞),𝜑∈𝑆(𝔻),𝑔∈𝐻(𝔻), and 𝑛∈ℕ, or 𝑛=0 and 𝛼>1. Then the following statements are equivalent. (a)𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼→𝑄_𝐾 is bounded. (b)𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾 is bounded. (c)𝑀∶=sup_𝑎∈𝔻∫_𝔻𝐾(1−|𝜂_𝑎(𝑧)|²)|𝑔(𝑧)|²(1−|𝜑(𝑧)|²)^{2(1−𝛼−𝑛)}𝑑𝑚(𝑧)<∞. (d)𝑑𝜇_{𝜑,𝑔,𝑛,𝛼}(𝑧) is a 𝐾-Carleson measure. Moreover, if 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼→𝑄_𝐾 is bounded, then the next asymptotic relations hold ‖‖𝐼^{(𝑛)𝜑,𝑔}‖‖_{ℬ𝛼→𝑄𝐾}≍‖‖𝐼^{(𝑛)𝜑,𝑔}‖‖_{ℬ𝛼0→𝑄𝐾}≍𝑀^1/2.(3.2)

Proof. By Theorem 1.1, it is clear that (c) and (d) are equivalent.
(c) ⇒ (a). Let 𝑓∈𝐵_ℬ^𝛼. First note that 𝐼^{(𝑛)𝜑,𝑔}𝑓(0)=0 for each 𝑓∈𝐻(𝔹) and 𝑛∈ℕ₀. From this and by Lemma 2.1, we have‖‖𝐼^{(𝑛)𝜑,𝑔}𝑓‖‖^2𝑄_𝐾=sup_𝑎∈𝔻_𝔻|||𝐼_{(𝑛)𝜑,𝑔}=sup_𝑎∈𝔻𝑓′(𝑧)|||²𝐾1−||𝜂_𝑎(𝑧)||²𝑑𝑚(𝑧)_𝔻||𝑓^(𝑛)(𝜑(𝑧))||²||𝑔(𝑧)||²𝐾1−||𝜂_𝑎(𝑧)||²≤𝐶‖𝑓‖_2ℬ𝛼𝑑𝑚(𝑧)sup_𝑎∈𝔻_𝔻𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²1−||𝜑(𝑧)||²^{2(1−𝛼−𝑛)}𝑑𝑚(𝑧),(3.3) from which the boundedness of 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼→𝑄_𝐾 follows, and moreover ‖‖𝐼^{(𝑛)𝜑,𝑔}‖‖_{ℬ𝛼→𝑄𝐾}≤𝐶𝑀^1/2.(3.4)
(a)⇒(b). This implication is obvious.
(b)⇒(c). By Lemma 2.2, if 𝑛∈ℕ, there are two functions 𝑓₁,𝑓₂∈ℬ^𝛼 such that (2.3) holds, and if 𝑛=0 and 𝛼>1 such that (2.4) holds. Letℎ₁(𝑧)=𝑓₁(𝑧)−^𝑛−1_𝑘=1𝑓^1(𝑘)(0)𝑘!𝑧^𝑘,ℎ₂(𝑧)=𝑓₂(𝑧)−^𝑛−1_𝑘=1𝑓^2(𝑘)(0)𝑘!𝑧^𝑘.(3.5) It is known (see [30]) that for each 𝑓∈𝐻(𝔻) and 𝑛∈ℕ, we have 1−|𝑧|²^{𝛼+𝑛−1}||𝑓^(𝑛)(𝑧)||+^𝑛−1_𝑘=1||𝑓^(𝑘)(0)||≍1−|𝑧|²^𝛼||𝑓^(𝑧)||.(3.6)
From this, Lemma 2.2, and since ℎ^1(𝑘)(0)=ℎ^2(𝑘)(0)=0, 𝑘=0,1,…,𝑛−1, we have that there is a 𝛿>0 such that𝐶1−|𝑧|²^{−(𝛼+𝑛−1)}≤||ℎ^1(𝑛)(𝑧)||+||ℎ^2(𝑛)(𝑧)||,for|𝑧|>𝛿.(3.7) Now note that for any 𝑓∈ℬ^𝛼, the functions 𝑓_𝑟(𝑧)=𝑓(𝑟𝑧), 𝑟∈(0,1) belong to ℬ^𝛼, and moreover, sup_0<𝑟<1‖𝑓_𝑟‖_ℬ^𝛼≤‖𝑓‖_ℬ^𝛼.
Applying (3.7), using an elementary inequality, the boundedness of 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾, and the last inequality, we obtain_{|𝑟𝜑(𝑧)|>𝛿}≤𝐶𝑟^2𝑛_𝔻𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²1−𝑟||𝜑(𝑧)||²^{2(1−𝛼−𝑛)}𝑑𝑚(𝑧)=𝐶_𝔻𝑟^2𝑛𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²||ℎ^1(𝑛)(𝑟𝜑(𝑧))||²+||ℎ^2(𝑛)(𝑟𝜑(𝑧))||²𝑑𝑚(𝑧)+𝐶_𝔻𝐾1−||𝜂_𝑎(𝑧)||²|||𝐼^{(𝑛)𝜑,𝑔}ℎ₁_𝑟^(𝑧)|||²𝑑𝑚(𝑧)≤‖‖𝐼𝐾1−||𝜂_𝑎(𝑧)||²|||𝐼^{(𝑛)𝜑,𝑔}ℎ₂_𝑟^(𝑧)|||²𝑑𝑚(𝑧)_{(𝑛)𝜑,𝑔}‖‖^2ℬ_{𝛼0→𝑄𝐾}‖‖ℎ¹‖‖^2ℬ_𝛼+‖‖ℎ₂‖‖^2ℬ_𝛼.(3.8) Letting 𝑟→1 in (3.8) and using the monotone convergence theorem, we get _{||||𝜑(𝑧)>𝛿}𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²1−||𝜑(𝑧)||²^{2(1−𝛼−𝑛)}𝑑𝑚(𝑧)≤𝐶‖‖𝐼^{(𝑛)𝜑,𝑔}‖‖^2ℬ_{𝛼0→𝑄𝐾}.(3.9) On the other hand, for 𝑓₀(𝑧)=𝑧^𝑛/𝑛!∈ℬ^𝛼0, we get 𝐼^{(𝑛)𝜑,𝑔}𝑓₀∈𝑄_𝐾 which implies sup_𝛼∈𝔻_{||||𝜑(𝑧)≤𝛿}𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²1−||𝜑(𝑧)||²^{2(1−𝛼−𝑛)}𝑑𝑚(𝑧)≤‖‖𝐼^{(𝑛)𝜑,𝑔}‖‖^2ℬ_{𝛼0→𝑄𝐾}‖‖𝑓₀‖‖^2ℬ_𝛼1−𝛿²^{2(𝛼+𝑛−1)}.(3.10) From (3.9) and (3.10), (c) follows. Moreover we get 𝑀^1/2≤𝐶‖𝐼^{(𝑛)𝜑,𝑔}‖_{ℬ𝛼0→𝑄𝐾}. From this, (3.4) and since ‖𝐼^{(𝑛)𝜑,𝑔}‖_{ℬ𝛼0→𝑄𝐾}≤‖𝐼^{(𝑛)𝜑,𝑔}‖_{ℬ𝛼→𝑄𝐾} the asymptotic relations in (3.2) follow, finishing the proof of the theorem.

Theorem 3.2. Let 𝛼>0,𝐾∈Ω(0,∞),𝜑∈𝑆(𝔻),𝑔∈𝐻(𝔻), and 𝑛∈ℕ, or 𝑛=0 and 𝛼>1. Let 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼→𝑄_𝐾 be bounded. Then the following statements are equivalent. (a)𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼→ 𝑄_𝐾 is compact. (b)𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾 is compact. (c)𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾 is weakly compact. (d)sup_𝑎∈𝔻∫_𝔻𝐾(1−|𝜂_𝑎(𝑧)|²)|𝑔(𝑧)|²𝑑𝑚(𝑧)<∞, andlim_𝑟→1sup_𝑎∈𝔻_{|𝜑(𝑧)|>𝑟}𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²1−||𝜑(𝑧)||²^{2(1−𝛼−𝑛)}𝑑𝑚(𝑧)=0.(3.11)

Proof. By Lemma 2.4, we have that (b) is equivalent to (c).
(d) ⇒ (a). Let (𝑓_𝑙)_𝑙∈ℕ be a bounded sequence in ℬ^𝛼, say by 𝐿, converging to zero uniformly on compacts of 𝔻. Then 𝑓^𝑙(𝑛) also converges to zero uniformly on compacts of 𝔻. From (3.11) we have that for every 𝜀>0 there is an 𝑟₁∈(0,1) such that for 𝑟∈(𝑟₁,1)sup_𝑎∈𝔻_{|𝜑(𝑧)|>𝑟}𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²1−||𝜑(𝑧)||²^{2(1−𝛼−𝑛)}𝑑𝑚(𝑧)<𝜀.(3.12) Therefore, by Lemma 2.1 and (3.12), we have that for 𝑟∈(𝑟₁,1)‖‖𝐼^{(𝑛)𝜑,𝑔}𝑓_𝑙‖‖^2𝑄_𝐾=_{|𝜑(𝑧)|≤𝑟}+_{|𝜑(𝑧)|>𝑟}||𝑓^𝑙(𝑛)<sup_{||||𝜑(𝑧)≤𝑟}(𝜑(𝑧))||²𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²𝑑𝑚(𝑧)||𝑓^𝑙(𝑛)(𝜑(𝑧))||²_𝔻𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²𝑑𝑚(𝑧)+𝐶𝐿²𝜀.(3.13) Letting 𝑙→∞ in (3.13), using the first condition in (d) and sup_{|𝑤|≤𝑟}|𝑓^𝑙(𝑛)(𝑤)|→0 as 𝑙→∞, it follows that lim_𝑙→∞‖𝐼^{(𝑛)𝜑,𝑔}𝑓_𝑙‖_𝑄𝐾=0. Thus, by Lemma 2.3, 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼→ 𝑄_𝐾 is compact.
(a) ⇒ (b). The implication is trivial since ℬ^𝛼0⊂ℬ^𝛼.
(b) ⇒ (d). By choosing 𝑓(𝑧)=𝑧^𝑛/𝑛!∈ℬ^𝛼0, 𝑛∈ℕ₀, we have that the first condition in (d) holds. Let 𝑓_𝑙(𝑧)=𝑧^𝑙/𝑙, 𝑙∈ℕ. It is easy to see that (𝑓_𝑙)_𝑙∈ℕ is a bounded sequence in ℬ^𝛼0 converging to zero uniformly on compacts of 𝔻. Hence, by Lemma 2.3, it follows that ‖𝐼^{(𝑛)𝜑,𝑔}(𝑓_𝑙)‖_𝑄𝐾→0 as 𝑙→∞. Thus, for every 𝜀>0, there is an 𝑙₀∈ℕ, 𝑙₀>𝑛 such that for𝑙≥𝑙₀^𝑛−1_𝑗=1(𝑙−𝑗)²sup_𝑎∈𝔻_𝔻||𝜑(𝑧)||^{2(𝑙−𝑛)}𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²𝑑𝑚(𝑧)<𝜀.(3.14) From (3.14) we have that for each 𝑟∈(0,1) and 𝑙≥𝑙₀𝑟^{2(𝑙−𝑛)}^𝑛−1_𝑗=1(𝑙−𝑗)²sup_𝑎∈𝔻_{|𝜑(𝑧)|>𝑟}𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²𝑑𝑚(𝑧)<𝜀.(3.15) Hence, for 𝑟∈[(∏^{𝑛−1𝑗=1}(𝑙₀−𝑗))^{−1/(𝑙0−𝑛)},1), we have that sup_𝑎∈𝔻_{||||𝜑(𝑧)>𝑟}𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²𝑑𝑚(𝑧)<𝜀.(3.16)
Let 𝑓∈𝐵_ℬ^𝛼0, and let 𝑓_𝑡(𝑧)=𝑓(𝑡𝑧),0<𝑡<1. Then sup_0<𝑡<1‖𝑓_𝑡‖_ℬ^𝛼≤‖𝑓‖_ℬ^𝛼, 𝑓_𝑡∈ℬ^𝛼0, 𝑡∈(0,1), and 𝑓_𝑡→𝑓 uniformly on compact subsets of 𝔻 as 𝑡→1. The compactness of 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾 implieslim_𝑡→1‖‖𝐼^{(𝑛)𝜑,𝑔}𝑓_𝑡−𝐼^{(𝑛)𝜑,𝑔}𝑓‖‖_𝑄𝐾=0.(3.17) Hence, for every 𝜀>0, there is a 𝑡∈(0,1) such that sup_𝑎∈𝔻_𝔻||𝑓^𝑡(𝑛)(𝜑(𝑧))−𝑓^(𝑛)(𝜑(𝑧))||²𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²𝑑𝑚(𝑧)<𝜀.(3.18)
From this and (3.16), we have that for such 𝑡 and each 𝑟∈[(∏^{𝑛−1𝑗=1}(𝑙₀−𝑗))^{−1/(𝑙0−𝑛)},1)sup_𝑎∈𝔻_{||||𝜑(𝑧)>𝑟}||𝑓^(𝑛)(𝜑(𝑧))||²𝐾1−||𝜂_𝑎(𝑧)||²≤2sup_𝑎∈𝔻||𝑔(𝑧)||²𝑑𝑚(𝑧)_{|𝜑(𝑧)|>𝑟}||𝑓^𝑡(𝑛)(𝜑(𝑧))−𝑓^(𝑛)(𝜑(𝑧))||²𝐾1−||𝜂_𝑎(𝑧)||²+2sup_𝑎∈𝔻||𝑔(𝑧)||²𝑑𝑚(𝑧)_{|𝜑(𝑧)|>𝑟}||𝑓^𝑡(𝑛)(𝜑(𝑧))||²𝐾1−||𝜂_𝑎(𝑧)||²<2𝜀1+‖‖𝑓_𝑡(𝑛)||𝑔(𝑧)||²𝑑𝑚(𝑧)‖‖^2∞.(3.19) From (3.19) we conclude that for every 𝑓∈𝐵_ℬ^𝛼0, there is a 𝛿₀∈(0,1) and 𝛿₀=𝛿₀(𝑓,𝜀) such that for 𝑟∈(𝛿₀,1)sup_𝑎∈𝔻_{||||𝜑(𝑧)>𝑟}||𝑓^(𝑛)(𝜑(𝑧))||²𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²𝑑𝑚(𝑧)<𝜀.(3.20)
Since 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾 is compact, we have that for every 𝜀>0 there is a finite collection of functions 𝑓₁,𝑓₂,…,𝑓_𝑘∈𝐵_ℬ^𝛼0 such that, for each 𝑓∈𝐵_ℬ^𝛼0, there is a 𝑗∈{1,…,𝑘}, such thatsup_𝑎∈𝔻_𝔻||𝑓^(𝑛)(𝜑(𝑧))−𝑓^𝑗(𝑛)(𝜑(𝑧))||²𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²𝑑𝑚(𝑧)<𝜀.(3.21) On the other hand, from (3.20), it follows that if 𝛿̂∶=max_{1≤𝑗≤𝑘}𝛿_𝑗(𝑓_𝑗,𝜀), then for 𝑟∈(𝛿̂,1) and all 𝑗∈{1,…,𝑘}, we have sup_𝑎∈𝔻_{|𝜑(𝑧)|>𝑟}||𝑓^𝑗(𝑛)(𝜑(𝑧))||²𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²𝑑𝑚(𝑧)<𝜀.(3.22) From (3.21) and (3.22), we have that for 𝑟∈(𝛿̂,1) and every 𝑓∈𝐵_ℬ^𝛼0sup_𝑎∈𝔻_{|𝜑(𝑧)|>𝑟}||𝑓^(𝑛)(𝜑(𝑧))||²𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²𝑑𝑚(𝑧)<4𝜀.(3.23) If we apply (3.23) to the delays of the functions in (3.5) which are normalized so that they belong to 𝐵_ℬ^𝛼 and then use the monotone convergence theorem, we easily get that for𝑟>max{𝛿,𝛿̂} where 𝛿 is chosen as in (3.7) sup_𝑎∈𝔻_{||||𝜑(𝑧)>𝑟}𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²1−||𝜑(𝑧)||²^{2(1−𝛼−𝑛)}𝑑𝑚(𝑧)<𝐶𝜀,(3.24) from which (3.11) follows, as desired.

Theorem 3.3. Let 𝛼>0,𝐾∈Ω(0,∞),𝜑∈𝑆(𝔻), 𝑔∈𝐻(𝔻) and 𝑛∈ℕ, or 𝑛=0 and 𝛼>1. Then the next statements are equivalent. (a)𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼→𝑄_𝐾,0 is bounded. (b)𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼→𝑄_𝐾,0 is compact. (c)𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾,0 is compact. (d)𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾,0 is weakly compact. (e)lim_|𝑎|→1∫_𝔻𝐾(1−|𝜂_𝑎(𝑧)|²)|𝑔(𝑧)|²(1−|𝜑(𝑧)|²)^{2(1−𝛼−𝑛)}𝑑𝑚(𝑧)=0. (f)𝑑𝜇_{𝜑,𝑔,𝑛,𝛼}(𝑧) is a vanishing 𝐾-Carleson measure.

Proof. By Theorem 1.1, (e) and (f) are equivalent; by Lemma 2.4, (c) is equivalent to (d), while, by Lemma 2.5, (a) is equivalent to (c). Also (b) obviously implies (a).
(a) ⇒ (e) Let ℎ₁ and ℎ₂ be as in (3.5). Then from (3.7) and an elementary inequality, we get_{||𝜑(z)||>𝛿}≤𝐶𝐾_𝔻1−||𝜂_𝑎(𝑧)||²1−||𝜑(𝑧)||²^{2(1−𝛼−𝑛)}||𝑔(𝑧)||²𝑑𝑚(𝑧)+𝐶_𝔻𝐾1−||𝜂_𝑎(𝑧)||²|||𝐼^{(𝑛)𝜑,𝑔}ℎ₁^(𝑧)|||²𝑑𝑚(𝑧)𝐾1−||𝜂_𝑎(𝑧)||²|||𝐼^{(𝑛)𝜑,𝑔}ℎ₂^(𝑧)|||²𝑑𝑚(𝑧).(3.25) For 𝑓₀(𝑧)=𝑧^𝑛/𝑛!∈ℬ^𝛼, we get 𝐼^{(𝑛)𝜑,𝑔}𝑓₀∈𝑄_𝐾,0. From this and since 𝐼^{(𝑛)𝜑,𝑔}(ℎ_𝑗)∈𝑄_𝐾,0,𝑗=1,2, by letting |𝑎|→1, we get that (e) holds.
(e) ⇒ (b). We have that for every 𝜀>0 there is a 𝛿∈(0,1) so that for |𝑎|>𝛿Φ_{𝜑,𝑔,𝐾}(𝑎)<𝜀.(3.26) On the other hand, by Lemma 2.6, Φ_{𝜑,𝑔,𝐾} is continuous on |𝑎|≤𝛿, so uniformly bounded therein, which along with (3.26) gives the boundedness of Φ_{𝜑,𝑔,𝐾} on 𝔻. Hence, by Theorem 3.1, 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼→𝑄_𝐾 is bounded. By Lemma 2.1, we have lim_|𝑎|→1sup_{‖𝑓‖ℬ𝛼≤1}_𝔻|||𝐼_{(𝑛)𝜑,𝑔}≤𝐶sup_{‖𝑓‖ℬ𝛼≤1}𝑓^(𝑧)|||²𝐾1−||𝜂_𝑎(𝑧)||²𝑑𝑚(𝑧)‖𝑓‖^2ℬ_𝛼lim_|𝑎|→1Φ_{𝜑,𝑔,𝐾}(𝑎)=𝐶lim_|𝑎|→1Φ_𝜑,𝑔,K(𝑎)=0,(3.27) so 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼→𝑄_𝐾,0 is bounded.
Now assume that (𝑓_𝑙)_𝑙∈ℕ is a bounded sequence in ℬ^𝛼, say by 𝐿, converging to zero uniformly on compacta of 𝔻 as 𝑙→∞. To show that the operator 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼→𝑄_𝐾,0 is compact, it is enough to prove that there is a subsequence (𝑓_𝑙𝑘)_𝑘∈ℕ of (𝑓_𝑙)_𝑙∈ℕ such that 𝐼^{(𝑛)𝜑,𝑔}𝑓_𝑙𝑘 converges in 𝑄_𝐾,0 as 𝑘→∞. By Lemma 2.1 and Montel's theorem, it follows that there is a subsequence, which we may denote again by (𝑓_𝑙)_𝑙∈ℕ converging uniformly on compacta of 𝔻 to an 𝑓∈ℬ^𝛼, such that ‖𝑓‖_ℬ^𝛼≤𝐿. Since 𝐼^{(𝑛)𝜑,𝑔}(ℬ^𝛼)⊆𝑄_𝐾,0, then clearly 𝐼^{(𝑛)𝜑,𝑔}𝑓∈𝑄_𝐾,0. We show thatlim_𝑙→∞‖‖𝐼^{(𝑛)𝜑,𝑔}𝑓_𝑙−𝐼^{(𝑛)𝜑,𝑔}𝑓‖‖_𝑄𝐾=0.(3.28) From (3.26), Lemma 2.1, and some simple calculation, we obtain sup_{𝛿<|𝑎|<1}_𝔻|||𝐼_{(𝑛)𝜑,𝑔}𝑓_𝑙(𝑧)−𝐼^{(𝑛)𝜑,𝑔}𝑓(𝑧)^|||²𝐾1−||𝜂_𝑎(𝑧)||²𝑑𝑚(𝑧)<4𝐶𝐿²𝜀.(3.29)
For 𝑎∈𝔻 and 𝑡∈(0,1), letΨ_𝑡(𝑎)=_{𝔻⧵𝑡𝔻}𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²1−||𝜑(𝑧)||²^{2(1−𝛼−𝑛)}𝑑𝑚(𝑧).(3.30) Lemma 2.6 essentially shows that Ψ_𝑡 is continuous on 𝔻. Hence, for each 𝑎∈𝔻, there is a 𝑡(𝑎)∈(𝑟,1) such that Ψ_𝑡(𝑎)(𝑎)<𝜀/2. Moreover, there is a neighborhood 𝒪(𝑎) of 𝑎 such that, for every 𝑏∈𝒪(𝑎), Ψ_𝑡(𝑎)(𝑏)<𝜀. From this and since the set |𝑎|≤𝛿 is compact, it follows that there is a 𝑡₀∈(0,1) such that Ψ_𝑡0(𝑎)<𝜀 when |𝑎|≤𝛿. This along with Lemma 2.1 implies that sup_{|𝑎|≤𝛿}_{𝔻⧵𝑡0𝔻}|||𝐼_{(𝑛)𝜑,𝑔}𝑓_𝑙(𝑧)−𝐼^{(𝑛)𝜑,𝑔}𝑓(𝑧)^|||²≤𝐶‖‖𝑓_𝑙𝐾1−||𝜂_𝑎(𝑧)||²𝑑𝑚(𝑧)−𝑓‖‖^2ℬ_𝛼sup_{|𝑎|≤𝛿}Ψ_𝑡0(𝑎)<4𝐶𝐿²𝜀.(3.31)
By the Weierstrass theorem 𝑓^𝑙(𝑛)→𝑓^(𝑛) uniformly on compacta as 𝑙→∞, from which along with (2.2) and since 𝜑(𝑡₀𝔻) is compact, for 𝑟=sup_{𝑤∈𝜑(𝑡0𝔻)}|𝑤|, it follows thatsup_{|𝑎|≤𝛿}_𝑡0𝔻|||𝐼_{(𝑛)𝜑,𝑔}𝑓_𝑙(𝑧)−𝐼^{(𝑛)𝜑,𝑔}𝑓(𝑧)^|||²≤𝐶sup_{|𝑧|≤𝑟}𝐾1−||𝜂_𝑎(𝑧)||²𝑑𝑚(𝑧)|||𝑓_𝑙−𝑓^(𝑛)(𝑧)|||²sup_{|𝑎|≤𝛿}Φ_{𝜑,𝑔,𝐾}(𝑎)⟶0,as𝑙⟶∞.(3.32)
From (3.29)–(3.32) and since 𝐼^{(𝑛)𝜑,𝑔}𝑓(0)=0 for each 𝑓∈𝐻(𝔻), we easily get (3.28), from which (b) follows, finishing the proof of this theorem.

Theorem 3.4. Let 𝛼>0,𝐾∈Ω(0,∞),𝜑∈𝑆(𝔻),𝑔∈𝐻(𝔻), and 𝑛∈ℕ, or 𝑛=0 and 𝛼>1. Then the following statements are equivalent.(a)𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾,0 is bounded, (b)sup_𝑎∈𝔻∫_𝔻|𝑔(𝑧)|²𝐾(1−|𝜂_𝑎(𝑧)|²)(1−|𝜑(𝑧)|²)^{2(1−𝛼−𝑛)}𝑑𝑚(𝑧)<∞, andlim_|𝑎|→1_𝔻||𝑔(𝑧)||²𝐾1−||𝜂_𝑎(𝑧)||²𝑑𝑚(𝑧)=0.(3.33)

Proof. Suppose (b) holds and 𝑓∈ℬ^𝛼0. Then by Theorem 3.1, 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾 is bounded. We show 𝐼^{(𝑛)𝜑,𝑔}𝑓∈𝑄_𝐾,0, for every 𝑓∈ℬ^𝛼0. Since 𝑓∈ℬ^𝛼0, we have that, for every 𝜀>0, there is an 𝑟∈(0,1) such that (see, e.g., the idea in [35, Lemma 2.4]) ||𝑓^(𝑛)(𝜑(𝑧))||²1−||𝜑(𝑧)||²^{2(𝛼+𝑛−1)}<𝜀for||𝜑(𝑧)||>𝑟.(3.34) Thus, sup_𝑎∈𝔻_{||||𝜑(𝑧)>𝑟}|||𝐼_{(𝑛)𝜑,𝑔}𝑓(𝑧)^|||²<𝜀sup_𝑎∈𝔻𝐾1−||𝜂_𝑎(𝑧)||²𝑑𝑚(𝑧)_𝔻𝐾1−||𝜂_𝑎(𝑧)||²1−||𝜑(𝑧)||²^{2(1−𝛼−𝑛)}||𝑔(𝑧)||²𝑑𝑚(𝑧).(3.35) We also have lim_|𝑎|→1_{|𝜑(𝑧)|≤𝑟}|||𝐼_{(𝑛)𝜑,𝑔}𝑓(𝑧)^|||²≤𝐶‖𝑓‖_2ℬ𝛼𝐾1−||𝜂_𝑎(𝑧)||²𝑑𝑚(𝑧)1−𝑟²^{2(𝛼+𝑛−1)}lim_|𝑎|→1_{|𝜑(𝑧)|≤𝑟}≤𝐶‖𝑓‖_2ℬ𝛼𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²𝑑𝑚(𝑧)1−𝑟²^{2(𝛼+𝑛−1)}lim_|𝑎|→1_𝔻𝐾1−||𝜂_𝑎(𝑧)||²||𝑔(𝑧)||²𝑑𝑚(𝑧)=0.(3.36) Combining (3.35) and (3.36), we get 𝐼^{(𝑛)𝜑,𝑔}𝑓∈𝑄_𝐾,0. Hence, 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾,0 is bounded.
Conversely, if 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾,0 is bounded, then 𝐼^{(𝑛)𝜑,𝑔}∶ℬ^𝛼0→𝑄_𝐾 is bounded too. Thus, by Theorem 3.1, we get the first condition in (b). For 𝑓₀(𝑧)=𝑧^𝑛/𝑛!∈ℬ^𝛼0, we get 𝐼^{(𝑛)𝜑,𝑔}𝑓₀∈𝑄_𝐾,0, which is equivalent to (3.33), finishing the proof of the theorem.

If 𝑛=0, we simply denote the operator 𝐼^(0)𝜑,𝑔 by 𝐼_𝜑,𝑔.

Theorem 3.5. Let 𝛼∈(0,1),𝐾∈Ω(0,∞),𝜑∈𝑆(𝔻), and 𝑔∈𝐻(𝔻). Then the following statements are equivalent. (a)𝐼_𝜑,𝑔∶ℬ^𝛼→𝑄_𝐾 is bounded. (b)𝐼_𝜑,𝑔∶ℬ^𝛼0→𝑄_𝐾 is bounded. (c)𝑀₁∶=sup_𝑎∈𝔻∫_𝔻𝐾(1−|𝜂_𝑎(𝑧)|²)|𝑔(𝑧)|²𝑑𝑚(𝑧)<∞. (d)𝑑𝜇₁(𝑧)=|𝑔(𝑧)|²𝑑𝑚(𝑧) is a 𝐾-Carleson measure. (e)𝐼_𝜑,𝑔∶ℬ^𝛼→ 𝑄_𝐾 is compact. (f)𝐼_𝜑,𝑔∶ℬ^𝛼0→𝑄_𝐾 is compact. (g)𝐼_𝜑,𝑔∶ℬ^𝛼0→𝑄_𝐾 is weakly compact. Moreover, if 𝐼_𝜑,𝑔∶ℬ^𝛼→𝑄_𝐾 is bounded, then the next asymptotic relations hold ‖‖𝐼_𝜑,𝑔‖‖_{ℬ𝛼→𝑄𝐾}≍‖‖𝐼_𝜑,𝑔‖‖_{ℬ𝛼0→𝑄𝐾}≍𝑀^11/2.(3.37)

Proof. The proof of the equivalence of statements (a)–(d) of this theorem is similar to the proof of Theorem 3.1; moreover, the implication (b) ⇒ (c) is much simpler since it follows by using the test function 𝑓₀(𝑧)≡1. That (c) is equivalent to (e)–(g) is proved similarly as in Theorem 3.2, by using the well-known fact that if a bounded sequence (𝑓_𝑙)_𝑙∈ℕ in ℬ^𝛼, 𝛼∈(0,1) converges to zero uniformly on compacts of 𝔻, then it converges to zero uniformly on the whole 𝔻. The details are omitted.

The proof of the next theorem is similar to the proofs of Theorems 3.3 and 3.4 and will be omitted.

Theorem 3.6. Let 𝛼∈(0,1),𝐾∈Ω(0,∞),𝜑∈𝑆(𝔻), and 𝑔∈𝐻(𝔻). Then the following statements are equivalent. (a)𝐼_𝜑,𝑔∶ℬ^𝛼0→𝑄_𝐾,0 is bounded. (b)𝐼_𝜑,𝑔∶ℬ^𝛼→𝑄_𝐾,0 is bounded. (c)𝐼_𝜑,𝑔∶ℬ^𝛼→𝑄_𝐾,0 is compact. (d)𝐼_𝜑,𝑔∶ℬ^𝛼0→𝑄_𝐾,0 is compact. (e)𝐼_𝜑,𝑔∶ℬ^𝛼0→𝑄_𝐾,0 is weakly compact. (f)lim_|𝑎|→1∫_𝔻𝐾(1−|𝜂_𝑎(𝑧)|²)|𝑔(𝑧)|²𝑑𝑚(𝑧)=0. (g)𝑑𝜇₁(𝑧)=|𝑔(𝑧)|²𝑑𝑚(𝑧) is a vanishing 𝐾-Carleson measure.

Acknowledgment

This work is partially supported by the National Board of Higher Mathematics (NBHM)/DAE, India (Grant no. 48/4/2009/R&D-II/426) and by the Serbian Ministry of Science (Projects III41025 and III44006).