Norm and Essential Norm of an Integral-Type Operator from the Dirichlet Space to the Bloch-Type Space on the Unit Ball

Abstract

Operator norm and essential norm of an integral-type operator, recently introduced by this author, from the Dirichlet space to the Bloch-type space on the unit ball in ℂ^𝑛 are calculated here.

1. Introduction

Let 𝔹^𝑛=𝔹 be the open unit ball in ℂ^𝑛, 𝔹¹=𝔻 the open unit disk in ℂ, 𝐻(𝔹) the class of all holomorphic functions on 𝔹, and 𝐻^∞(𝔹), the space consisting of all 𝑓∈𝐻(𝔹) such that ‖𝑓‖_∞=sup_𝑧∈𝔹|𝑓(𝑧)|<∞.

For an 𝑓∈𝐻(𝔹) with the Taylor expansion 𝑓(𝑧)=∑_𝛼𝑎_𝛼𝑧^𝛼, let ℜ𝑓(𝑧)=_𝛼|𝛼|𝑎_𝛼𝑧^𝛼 (1.1) be the radial derivative of 𝑓, where 𝛼=(𝛼₁,…,𝛼_𝑛) is a multi-index, |𝛼|=𝛼₁+⋯+𝛼_𝑛 and 𝑧^𝛼=𝑧^𝛼1₁⋯𝑧^𝛼𝑛_𝑛. Let 𝛼!=𝛼₁!⋯𝛼_𝑛!.

The Dirichlet space 𝒟²(𝔹)=𝒟² contains all 𝑓(𝑧)=∑_𝛼𝑎_𝛼𝑧^𝛼∈𝐻(𝔹), such that ‖𝑓‖²_𝒟2∶=|𝑓(0)|²+_𝛼|𝛼|𝛼!|𝛼|!|𝑎_𝛼|²<∞. (1.2) The quantity ‖𝑓‖_𝒟² is a norm on 𝒟² which for 𝑛=1 is equal to usual norm ‖𝑓‖_𝒟²_(𝔻)=(|𝑓(0)|²+_𝔻|𝑓^′(𝑧)|²𝑑𝐴(𝑧))^1/2, (1.3) where 𝑑𝐴(𝑧)=(1/𝜋)𝑟𝑑𝑟𝑑𝜃 is the normalized area measure on 𝔻.

The inner product, between two functions 𝑓(𝑧)=_𝛼𝑎_𝛼𝑧^𝛼,𝑔(𝑧)=_𝛼𝑏_𝛼𝑧^𝛼, (1.4) on 𝒟² is defined by ⟨𝑓,𝑔⟩∶=𝑓(0)𝑔(0)+_𝛼|𝛼|𝛼!𝑎|𝛼|!^𝛼𝑏_𝛼. (1.5)

For 𝛼≠0, let 𝑒_𝛼(𝑧)=|𝛼|!𝑧|𝛼|𝛼!^𝛼,𝑧∈𝔹, (1.6) and 𝑒₀(𝑧)≡1, then it is easy to see that the family {𝑒_𝛼} is an orthonormal basis for 𝒟², and hence the reproducing kernel 𝐾_𝑤(𝑧) for 𝒟² is given by ([1]) as follows:𝐾_𝑤(𝑧)=1+_𝛼≠0|𝛼|!𝑧|𝛼|𝛼!^𝛼𝑤^𝛼1=1+ln,1−⟨𝑧,𝑤⟩ (1.7) where ⟨𝑧,𝑤⟩=∑^𝑛_𝑗=1𝑧_𝑗𝑤_𝑗 is the inner product in ℂ^𝑛.

Clearly for each 𝑓∈𝒟² and 𝑤∈𝔹, the next reproducing formula holds: 𝑓(𝑤)=⟨𝑓,𝐾_𝑤⟩. (1.8) Note that for 𝑓=𝐾_𝑤 from (1.8), we obtain 𝐾_𝑤(𝑤)=‖𝐾_𝑤‖²_𝒟2𝑒=ln1−|𝑤|². (1.9)

Also, by the Cauchy-Schwarz inequality and (1.9), we have that, for each 𝑓∈𝒟² and 𝑤∈𝔹, |𝑓(𝑤)|=|⟨𝑓,𝐾_𝑤⟩|≤‖𝑓‖_𝒟²‖𝐾_𝑤‖_𝒟²=‖𝑓‖_𝒟²𝑒(ln1−|𝑤|²)^1/2. (1.10) Note that inequality (1.10) is exact since it is attained for 𝑓=𝐾_𝑤.

The weighted-type space 𝐻^∞_𝜇(𝔹)=𝐻^∞_𝜇 ([2, 3]) consists of all 𝑓∈𝐻(𝔹) such that ‖𝑓‖_𝐻^∞_𝜇∶=sup_𝑧∈𝔹𝜇(𝑧)|𝑓(𝑧)|<∞, (1.11) where 𝜇 is a positive continuous function on 𝔹 (weight).

The Bloch-type space ℬ_𝜇(𝔹)=ℬ_𝜇 consists of all 𝑓∈𝐻(𝔹) such that ‖𝑓‖_ℬ𝜇∶=|𝑓(0)|+sup_𝑧∈𝔹𝜇(𝑧)|ℜ𝑓(𝑧)|<∞, (1.12) where 𝜇 is a weight.

Let 𝑔∈𝐻(𝔻), 𝑔(0)=0, and 𝜑 be a holomorphic self-map of 𝔹, then the following integral-type operator: 𝑃^𝑔_𝜑(𝑓)(𝑧)=¹₀𝑓(𝜑(𝑡𝑧))𝑔(𝑡𝑧)𝑑𝑡𝑡,𝑧∈𝔹,𝑓∈𝐻(𝔹) (1.13) has been recently introduced in [4] and considerably studied (see, e.g., [5–8] and the related references therein). For some related operators, see also [9–16] and the references therein.

Usual problem in this research area is to provide function-theoretic characterizations for when 𝜑 and 𝑔 induce bounded or compact integral-type operator on spaces of holomorphic functions. Majority of papers only find asymptotics of operator norm of linear operators. Somewhat concrete but perhaps more interesting problem is to calculate operator norm of these operators between spaces of holomorphic functions on various domains. Some results on this problem can be found, for example, in [3, 17–26] (for related results see also [7, 27–32]). Having published paper [3], we started with systematic investigation of methods for calculating operator norms of concrete operators between spaces of holomorphic function.

Here, we calculate the operator norm as well as the essential norm of the operator 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇, considerably extending our recent result in note [33].

2. Auxiliary Results

In this section, we quote several auxiliary results which are used in the proofs of the main results.

Lemma 2.1 (see [4]). Let 𝑔∈𝐻(𝔹), 𝑔(0)=0, and 𝜑 be a holomorphic self-map of 𝔹, then ℜ𝑃^𝑔_𝜑(𝑓)(𝑧)=𝑔(𝑧)𝑓(𝜑(𝑧)). (2.1)

The next Schwartz-type lemma ([34]) can be proved in a standard way. Hence, we omit its proof.

Lemma 2.2. Assume that 𝑔∈𝐻(𝔹), 𝑔(0)=0, 𝜇 is a weight, and 𝜑 is an analytic self-map of 𝔹, then 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇 is compact if and only if 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇 is bounded and for any bounded sequence (𝑓_𝑘)_𝑘∈ℕ in 𝒟² converging to zero uniformly on compacts of 𝔹 as 𝑘→∞, one has lim_𝑘→∞‖𝑃^𝑔_𝜑(𝑓_𝑘)‖_ℬ𝜇=0. (2.2)

Lemma 2.3. Assume that 𝑔∈𝐻(𝔹), 𝑔(0)=0, 𝜇 is a weight, 𝜑 is an analytic self-map of 𝔹 such that ‖𝜑‖_∞<1, and the operator 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇 is bounded, then 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇 is compact.

Proof. First note that since 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇 is bounded and 𝑓₀(𝑧)≡1∈𝒟² by Lemma 2.1, it follows that ℜ𝑃^𝑔_𝜑(𝑓₀)=𝑔∈𝐻^∞_𝜇. Now, assume that (𝑓_𝑘)_𝑘∈ℕ is a bounded sequence in 𝒟² converging to zero on compacts of 𝔹 as 𝑘→∞, then we have ‖𝑃^𝑔_𝜑(𝑓_𝑘)‖_ℬ𝜇≤‖𝑔‖_𝐻^∞_𝜇sup_{𝑤∈𝜑(𝔹)}|𝑓_𝑘(𝑤)|→0, (2.3) as 𝑘→∞, since 𝜑(𝔹) is contained in the ball |𝑤|≤‖𝜑‖_∞ which is a compact subset of 𝔹, according to the assumption ‖𝜑‖_∞<1. Hence, by Lemma 2.2, the operator 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇 is compact.

3. Operator Norm of 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇

In this section, we calculate the operator norm of 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇.

Theorem 3.1. Assume that 𝑔∈𝐻(𝔹), 𝑔(0)=0, and 𝜑 is a holomorphic self-map of 𝔹, then ‖𝑃^𝑔_𝜑‖_𝒟²_→ℬ𝜇=sup_𝑧∈𝔹𝑒𝜇(𝑧)|𝑔(𝑧)|(ln1−|𝜑(𝑧)|²)^1/2=∶𝐿. (3.1)

Proof. Using Lemma 2.1, reproducing formula (1.8), the Cauchy-Schwarz inequality, and finally (1.9), we get that, for each 𝑓∈𝒟² and 𝑤∈𝔹, 𝜇(𝑤)|ℜ𝑃^𝑔_𝜑(𝑤)|=𝜇(𝑤)|𝑔(𝑤)||𝑓(𝜑(𝑤))|=𝜇(𝑤)|𝑔(𝑤)||⟨𝑓,𝐾_𝜑(𝑤)⟩|≤𝜇(𝑤)|𝑔(𝑤)|‖𝑓‖_𝒟²‖𝐾_𝜑(𝑤)‖_𝒟²=‖𝑓‖_𝒟²𝑒𝜇(𝑤)|𝑔(𝑤)|(ln1−|𝜑(𝑤)|²)^1/2. (3.2) Taking the supremum in (3.2) over 𝑤∈𝔹 as well as the supremum over the unit ball in 𝒟² and using the fact 𝑃^𝑔_𝜑(𝑓)(0)=0, for each 𝑓∈𝐻(𝔹), which follows from the assumption 𝑔(0)=0, we get ‖𝑃^𝑔_𝜑‖_𝒟²_→ℬ𝜇≤𝐿. (3.3)
Now assume that the operator 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇 is bounded. From (1.9), we obtain that, for each 𝑤∈𝔹, 𝑒(ln1−|𝜑(𝑤)|²)^1/2‖𝑃^𝑔_𝜑‖_𝒟²_→ℬ𝜇=‖𝐾_𝜑(𝑤)‖_𝒟²‖𝑃^𝑔_𝜑‖_𝒟²_→ℬ𝜇≥‖𝑃^𝑔_𝜑(𝐾_𝜑(𝑤))‖_ℬ𝜇=sup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||𝐾_𝜑(𝑤)(𝜑(𝑧))|≥𝜇(𝑤)|𝑔(𝑤)||𝐾_𝜑(𝑤)(𝜑(𝑤))|. (3.4) From (1.9) and (3.4), it follows that 𝐿≤‖𝑃^𝑔_𝜑‖_𝒟²_→ℬ𝜇. (3.5) Hence, if 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇 is bounded, then from (3.3) and (3.5) we obtain (3.1).
In the case 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇 is unbounded, the result follows from inequality (3.3).

4. Essential Norm of 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇

Let 𝑋 and 𝑌 be Banach spaces, and let 𝐿∶𝑋→𝑌 be a bounded linear operator. The essential norm of the operator 𝐿∶𝑋→𝑌, ‖𝐿‖_{𝑒,𝑋→𝑌}, is defined as follows: ‖𝐿‖_{𝑒,𝑋→𝑌}=inf{‖𝐿+𝐾‖_𝑋→𝑌∶𝐾iscompactfrom𝑋to𝑌}, (4.1) where ‖⋅‖_𝑋→𝑌 denote the operator norm.

From this and since the set of all compact operators is a closed subset of the set of bounded operators, it follows that 𝐿 is compact if and only if ‖𝐿‖_{𝑒,𝑋→𝑌}=0. (4.2)

Here, we calculate the essential norm of the operator 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇.

Theorem 4.1. Assume that 𝑔∈𝐻(𝔹), 𝑔(0)=0,𝜇 is a weight, 𝜑 is a holomorphic self-map of 𝔹, and 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇 is bounded. If ‖𝜑‖_∞<1, then ‖𝑃^𝑔_𝜑‖_𝑒,𝒟²_→ℬ𝜇=0, and if ‖𝜑‖_∞=1, then ‖𝑃^𝑔_𝜑‖_𝑒,𝒟²_→ℬ𝜇=limsup_{|𝜑(𝑧)|→1}𝑒𝜇(𝑧)|𝑔(𝑧)|(ln1−|𝜑(𝑧)|²)^1/2. (4.3)

Proof. Since 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇 is bounded, for the test function 𝑓(𝑧)≡1, we get 𝑔∈𝐻^∞_𝜇. If ‖𝜑‖_∞<1, then from Lemma 2.3 it follows that 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇 is compact which is equivalent with ‖𝑃^𝑔_𝜑‖_𝑒,𝒟²_→ℬ𝜇=0. On the other hand, it is clear that in this case the condition |𝜑(𝑧)|→1 is vacuous, so that (4.3) is vacuously satisfied.
Now assume that ‖𝜑‖_∞=1, and that (𝜑(𝑧_𝑘))_𝑘∈ℕ is a sequence in 𝔹 such that |𝜑(𝑧_𝑘)|→1 as 𝑘→∞. For 𝑤∈𝔹 fixed, set 𝑓_𝑤(𝑧)=ln(𝑒/(1−⟨𝑧,𝑤⟩))(ln(𝑒/(1−|𝑤|²)))^1/2,𝑧∈𝔹. (4.4) By (1.9), we have that ‖𝑓_𝑤‖_𝒟²=1, for each 𝑤∈𝔹. Hence, the sequence (𝑓_{𝜑(𝑧𝑘)})_𝑘∈ℕ is such that ‖𝑓_{𝜑(𝑧𝑘)}‖_𝒟²=1, for each 𝑘∈ℕ, and clearly it converges to zero uniformly on compacts of 𝔹. From this and by [35, Theorems 6.19], it easily follows that 𝑓_{𝜑(𝑧𝑘)}→0 weakly in 𝒟², as 𝑘→∞. Hence, for every compact operator 𝐾∶𝒟²→ℬ_𝜇, we have that lim_𝑘→∞‖𝐾𝑓_{𝜑(𝑧𝑘)}‖_ℬ𝜇=0. (4.5) Thus, for every such sequence and for every compact operator 𝐾∶𝒟²→ℬ_𝜇, we have that ‖𝑃^𝑔_𝜑+𝐾‖_𝒟²_→ℬ𝜇≥limsup_𝑘→∞‖𝑃^𝑔_𝜑𝑓_{𝜑(𝑧𝑘)}‖_ℬ𝜇−‖𝐾𝑓_{𝜑(𝑧𝑘)}‖_ℬ𝜇‖𝑓_{𝜑(𝑧𝑘)}‖_𝒟²=limsup_𝑘→∞‖𝑃^𝑔_𝜑𝑓_{𝜑(𝑧𝑘)}‖_ℬ𝜇≥limsup_𝑘→∞𝜇(𝑧_𝑘)|𝑔(𝑧_𝑘)𝑓_{𝜑(𝑧𝑘)}(𝜑(𝑧_𝑘))|=limsup_𝑛→∞𝜇(𝑧_𝑘)|𝑔(𝑧_𝑘𝑒)|(ln1−|𝜑(𝑧_𝑘)|²)^1/2. (4.6)
Taking the infimum in (4.6) over the set of all compact operators 𝐾∶𝒟²→ℬ_𝜇, we obtain ‖𝑃^𝑔_𝜑‖_𝑒,𝒟²_→ℬ𝜇≥limsup_𝑛→∞𝜇(𝑧_𝑘)|𝑔(𝑧_𝑘𝑒)|(ln1−|𝜑(𝑧_𝑘)|²)^1/2, (4.7) from which an inequality in (4.3) follows.
In the sequel, we prove the reverse inequality. Assume that (𝑟_𝑙)_𝑙∈ℕ is a sequence of positive numbers which increasingly converges to 1. Consider the operators defined by (𝑃^𝑔_𝑟𝑙𝜑𝑓)(𝑧)=¹₀𝑓(𝑟_𝑙𝜑(𝑡𝑧))𝑔(𝑡𝑧)𝑑𝑡𝑡,𝑙∈ℕ. (4.8) Since ‖𝑟_𝑙𝜑‖_∞<1, by Lemma 2.3, we have that these operators are compact.
Since 𝑃^𝑔_𝜑∶𝒟²→ℬ_𝜇 is bounded, then 𝑔∈𝐻^∞_𝜇. Let 𝜌∈(0,1) be fixed for a moment. By Lemma 2.1, we get ‖𝑃^𝑔_𝜑−𝑃^𝑔_𝑟𝑙𝜑‖_𝒟²_→ℬ𝜇=sup_{‖𝑓‖𝒟2≤1}sup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||𝑓(𝜑(𝑧))−𝑓(𝑟_𝑙𝜑(𝑧))|≤sup_{‖𝑓‖𝒟2≤1}sup_{|𝜑(𝑧)|≤𝜌}𝜇(𝑧)|𝑔(𝑧)||𝑓(𝜑(𝑧))−𝑓(𝑟_𝑙𝜑(𝑧))|+sup_{‖𝑓‖𝒟2≤1}sup_{|𝜑(𝑧)|>𝜌}𝜇(𝑧)|𝑔(𝑧)||𝑓(𝜑(𝑧))−𝑓(𝑟_𝑙𝜑(𝑧))|≤‖𝑔‖_𝐻^∞_𝜇sup_{‖𝑓‖𝒟2≤1}sup_{|𝜑(𝑧)|≤𝜌}|𝑓(𝜑(𝑧))−𝑓(𝑟_𝑙𝜑(𝑧))|+sup_{‖𝑓‖𝒟2≤1}sup_{|𝜑(𝑧)|>𝜌}𝜇(𝑧)|𝑔(𝑧)||𝑓(𝜑(𝑧))−𝑓(𝑟_𝑙𝜑(𝑧))|. (4.9)(4.10)
Further, we have ‖𝑓−𝑓_𝑟‖²_𝒟2=_𝛼|𝛼|𝛼!|𝛼|!|𝑎_𝛼|²(1−𝑟^|𝛼|)²≤_𝛼|𝛼|𝛼!|𝛼|!|𝑎_𝛼|²≤‖𝑓‖_𝒟². (4.11)
From (1.10), (4.11), and the fact 𝑓(𝑧)−𝑓(𝑟𝑧)∈𝒟², we obtain |𝑓(𝑧)−𝑓(𝑟𝑧)|≤‖𝑓‖_𝒟²𝑒(ln1−|𝑧|²)^1/2, (4.12) in particular |𝑓(𝜑(𝑧))−𝑓(𝑟_𝑙𝜑(𝑧))|≤‖𝑓‖_𝒟²𝑒(ln1−|𝜑(𝑧)|²)^1/2. (4.13)
Let 𝐼_𝑙∶=sup_{‖𝑓‖𝒟2≤1}sup_{|𝜑(𝑧)|≤𝜌}|𝑓(𝜑(𝑧))−𝑓(𝑟_𝑙𝜑(𝑧))|. (4.14) The mean value theorem along with the subharmonicity of the moduli of partial derivatives of 𝑓, well-known estimates among the partial derivatives of analytic functions, Theorem 6.2, and Proposition 6.2 in [35], yield 𝐼_𝑙≤sup_{‖𝑓‖𝒟2≤1}sup_{|𝜑(𝑧)|≤𝜌}(1−𝑟_𝑙)|𝜑(𝑧)|sup_{|𝑤|≤𝜌}|∇𝑓(𝑤)|≤𝐶_𝜌(1−𝑟_𝑙)sup_{‖𝑓‖𝒟2≤1}(^[𝑛/𝑝]_𝑗=1|∇^𝑗𝑓(0)|+sup_{|𝑤|≤(1+𝜌)/2}|∇^{[𝑛/𝑝]+1}𝑓(𝑤)|)≤𝐶_𝜌(1−𝑟_𝑙)sup_{‖𝑓‖𝒟2≤1}(^[𝑛/𝑝]_𝑗=1|∇^𝑗𝑓(0)|+(_{|𝑤|≤(3+𝜌)/4}|∇^{[𝑛/𝑝]+1}𝑓(𝑤)|²(1−|𝑤|²)^{2([𝑛/𝑝]+1)}𝑑𝜏(𝑤))^1/2)≤𝐶_𝜌(1−𝑟_𝑙)sup_{‖𝑓‖𝒟2≤1}(^[𝑛/𝑝]_𝑗=1|∇^𝑗𝑓(0)|+_𝔹|∇^{[𝑛/𝑝]+1}𝑓(𝑤)|²(1−|𝑤|²)^{2([𝑛/𝑝]+1)}𝑑𝜏(𝑤))^1/2≤𝐶_𝜌(1−𝑟_𝑙)→0,as𝑙→∞, (4.15) where 𝑑𝜏(𝑧)=𝑑𝑉(𝑧)/(1−|𝑧|²)^𝑛+1 and 𝑑𝑉(𝑧) is the Lebesgue volume measure on 𝔹.
Using (4.13) in (4.10), letting 𝑙→∞ in (4.9), using (4.15), and then letting 𝜌→1, the reverse inequality follows, finishing the proof of the theorem.