Isometries of a Bergman-Privalov-Type Space on the Unit Ball

Abstract

We introduce a new space 𝐴𝑁_log,𝛼(𝔹) consisting of all holomorphic functions on the unit ball 𝔹⊂ℂ^𝑛 such that ‖𝑓‖_{𝐴𝑁log,𝛼}∶=∫_𝔹𝜑_𝑒(ln(1+|𝑓(𝑧)|))𝑑𝑉_𝛼(𝑧)<∞, where 𝛼>−1, 𝑑𝑉_𝛼(𝑧)=𝑐_𝛼,𝑛(1−|𝑧|²)^𝛼𝑑𝑉(𝑧) (𝑑𝑉(𝑧) is the normalized Lebesgue volume measure on 𝔹, and 𝑐_𝛼,𝑛 is a normalization constant, that is, 𝑉_𝛼(𝔹)=1), and 𝜑_𝑒(𝑡)=𝑡ln(𝑒+𝑡) for 𝑡∈[0,∞). Some basic properties of this space are presented. Among other results we proved that 𝐴𝑁_log,𝛼(𝔹) with the metric 𝑑(𝑓,𝑔)=‖𝑓−𝑔‖_{𝐴𝑁log,𝛼} is an 𝐹-algebra with respect to pointwise addition and multiplication. We also prove that every linear isometry 𝑇 of 𝐴𝑁_log,𝛼(𝔹) into itself has the form 𝑇𝑓=𝑐(𝑓∘𝜓) for some 𝑐∈ℂ such that |𝑐|=1 and some 𝜓 which is a holomorphic self-map of 𝔹 satisfying a measure-preserving property with respect to the measure 𝑑𝑉_𝛼. As a consequence of this result we obtain a complete characterization of all linear bijective isometries of 𝐴𝑁_log,𝛼(𝔹).

1. Introduction

Let 𝔹 denote the open unit ball in the 𝑛-dimensional complex vector space ℂ^𝑛, 𝐻(𝔹) the space of all holomorphic functions on 𝔹, 𝑑𝑉(𝑧) the normalized Lebesgue measure on 𝔹, 𝑑𝜎 the normalized surface measure on the boundary 𝜕𝔹 of the unit ball, and 𝑑𝑉_𝛼(𝑧)=𝑐_𝛼,𝑛(1−|𝑧|²)^𝛼𝑑𝑉(𝑧), where 𝛼>−1 and 𝑐_𝛼,𝑛 is a normalization constant, that is, 𝑉_𝛼(𝔹)=1. For each 𝛼>−1 we define the holomorphic function space 𝐴𝑁_log,𝛼(𝔹)=𝐴𝑁_log,𝛼 as follows:

𝐴𝑁_log,𝛼(𝔹)={𝑓∈𝐻(𝔹)∣‖𝑓‖_{𝐴𝑁log,𝛼}∶=_𝔹𝜑_𝑒(ln(1+|𝑓(𝑧)|))𝑑𝑉_𝛼(𝑧)<∞}, (1.1) where 𝜑_𝑒(𝑡)=𝑡ln(𝑒+𝑡) for 𝑡∈[0,∞).

Since

𝜑^′_𝑒𝑡(𝑡)=ln(𝑒+𝑡)+𝑒+𝑡≥1,𝜑_𝑒^′′(𝑡)=2𝑒+𝑡(𝑒+𝑡)²>0,𝑡∈[0,∞), (1.2) it follows that 𝜑_𝑒(𝑡) is an increasing convex function which is obviously nonnegative.

Let 𝑠(𝑡)=𝜑_𝑒(ln(1+𝑡)), then

𝑠^′′𝜑(𝑡)=^𝑒′′(ln(1+𝑡))−𝜑^′_𝑠(ln(1+𝑡))(1+𝑡)². (1.3) Let 𝐹(𝑢)=𝜑^′_𝑠(𝑢)−𝜑_𝑒^′′(𝑢), then 𝐹(0)=1−(2/𝑒), 𝐹^′(𝑢)>0, so that 𝜑_𝑒^′′(𝑢)≤𝜑^′_𝑠(𝑢) for 𝑢≥0. Thus 𝑠(𝑡) is a nonnegative concave function on the interval [0,∞), so that

𝑠(𝑥+𝑦)−𝑠(𝑥)𝑦≤𝑠(𝑦)−𝑠(0)𝑦 (1.4) for 𝑥≥0 and 𝑦>0, and consequently 𝑠(𝑥+𝑦)≤𝑠(𝑥)+𝑠(𝑦),𝑥,𝑦≥0, which implies the following inequality:

𝜑_𝑒(ln(1+𝑥+𝑦))≤𝜑_𝑒(ln(1+𝑥))+𝜑_𝑒(ln(1+𝑦)) (1.5) for all 𝑥,𝑦∈[0,∞). It is easy to see that ‖𝑐𝑓‖_{𝐴𝑁log,𝛼} need not be equal to |𝑐|‖𝑓‖_{𝐴𝑁log,𝛼}, for every 𝑐∈ℂ and 𝑓∈𝐴𝑁_log,𝛼(𝔹). These facts imply that ‖⋅‖_{𝐴𝑁log,𝛼} is not a norm on 𝐴𝑁_log,𝛼(𝔹) but satisfies the triangle inequality

‖𝑓+𝑔‖_{𝐴𝑁log,𝛼}≤‖𝑓‖_{𝐴𝑁log,𝛼}+‖𝑔‖_{𝐴𝑁log,𝛼}. (1.6) Furthermore if we define 𝑑(𝑓,𝑔)=‖𝑓−𝑔‖_{𝐴𝑁log,𝛼} for any 𝑓,𝑔∈𝐴𝑁_log,𝛼(𝔹), then we see that 𝐴𝑁_log,𝛼(𝔹) is a metric space with respect to 𝑑(𝑓,𝑔).

Let 𝑋 be a space of all holomorphic functions on some domain and 𝑇 a linear isometry of 𝑋 into 𝑋 [1]. When 𝑋 is the Hardy space 𝐻^𝑝(𝔹)(0<𝑝<∞,𝑝≠2), Forelli [2, 3] and Rudin [4] have determined the injective and/or surjective isometries of 𝐻^𝑝(𝔹). For the case when 𝑋 is the weighted Bergman spaces 𝐴^𝑝_𝛼(𝔹)(0<𝑝<∞,𝑝≠2), the isometries were completely characterized in a sequence of papers by Kolaski [5–7]. By these works we see that the isometries on these holomorphic function spaces are described as weighted composition operators, which is one of the reasons why these operators have been investigated so much recently in the settings of the unit ball or the unit polydisk (see, e.g., monograph [8], recent papers [9–19], and references therein). See also paper [20] for integral-type operators closely related with weighted composition operators. The case when 𝑋 is not a Banach space has also been studied by many authors. The Smirnov class 𝑁^∗ and the Privalov space 𝑁^𝑝(1<𝑝<∞) which are contained in the Nevanlinna class 𝑁 are examples of such spaces. These type of spaces are 𝐹-spaces with respect to a suitable metric on them. For properties of these space, we can refer [21]. Stephenson [22], Iida and Mochizuki [23] and Subbotin [24–26] have studied linear isometries on these spaces. Their works showed that the injective isometries are weighted composition operators induced by some inner functions and inner maps of 𝔹 whose radial limit maps satisfy a measure-preserving property. Recently Matsugu and the second author of the present paper have studied the weighted Bergman-Privalov space 𝐴𝑁^𝑝_𝛼(𝔹)(1≤𝑝<∞) and characterized the isometries of this space in [27]. They showed that in this case, 𝑇 has the form of a constant multiple composition operator which satisfies a measure-preserving property with respect to the measure 𝑑𝑉_𝛼.

Motivated by paper [28], in this paper, we investigate the space 𝐴𝑁_log,𝛼(𝔹). Some basic properties of the space 𝐴𝑁_log,𝛼(𝔹) are presented in Section 2; among other results it was proved that 𝐴𝑁_log,𝛼(𝔹) with the metric 𝑑(𝑓,𝑔)=‖𝑓−𝑔‖_{𝐴𝑁log,𝛼} is an 𝐹-algebra with respect to pointwise addition and multiplication. Also an estimate for the point evaluation functional on 𝐴𝑁_log,𝛼(𝔹) is given. In Section 3, we will prove that every linear isometry 𝑇 of 𝐴𝑁_log,𝛼(𝔹) has the form 𝑇𝑓=𝑐(𝑓∘𝜓), where 𝑐∈ℂ such that |𝑐|=1 and 𝜓 is a holomorphic self-map of 𝔹 satisfying a measure-preserving property with respect to the measure 𝑑𝑉_𝛼. As a consequence of this result we show that every surjective isometry 𝑇 of 𝐴𝑁_log,𝛼(𝔹) is of the form 𝑇𝑓=𝑐(𝑓∘𝑈) for any 𝑓∈𝐴𝑁_log,𝛼(𝔹), where 𝑐∈ℂ with |𝑐|=1 and 𝑈 is a unitary operator on ℂ^𝑛.

2. Basic Properties of 𝐴𝑁_log,𝛼(𝔹)

This section is devoted to collecting fundamental results on 𝐴𝑁_log,𝛼(𝔹) which will be used in the proofs of the main results.

Recall that the weighted Bergman space 𝐴¹_𝛼(𝔹) is defined as follows:

𝐴¹_𝛼(𝔹)={𝑓∈𝐻(𝔹)∣‖𝑓‖_𝐴1𝛼∶=_𝔹|𝑓(𝑧)|𝑑𝑉_𝛼(𝑧)<∞}. (2.1)

First we prove an elementary inequality, which has the main role in determining the relationship between 𝐴𝑁_log,𝛼(𝔹) space and the weighted Bergman space 𝐴¹_𝛼(𝔹).

Lemma 2.1. The following inequality holds: 𝜑_𝑒(ln(1+𝑥))≤𝑥,𝑥∈[0,∞). (2.2)

Proof. Note that inequality (2.2) is transformed into 𝑒^𝑡−1−𝜑_𝑒(𝑡)≥0,𝑡∈[0,∞) (2.3) by the change 𝑥=𝑒^𝑡−1.
Let 𝐹(𝑡)=𝑒^𝑡−1−𝜑_𝑒(𝑡). Then 𝐹(0)=0 and 𝐹^′(𝑡)=𝑒^𝑡𝑡−ln(𝑒+𝑡)−𝑒+𝑡, (2.4) which implies 𝐹^′(0)=0 and 𝐹^′′(𝑡)=𝑒^𝑡−1−𝑒𝑒+𝑡(𝑒+𝑡)²2≥1−𝑒>0. (2.5) From all these relations it follows that inequality (2.3) holds and consequently inequality (2.2).

From Lemma 2.1 we obtain the following corollary.

Corollary 2.2. For each 𝛼>−1, 𝐴¹_𝛼(𝔹)⊂𝐴𝑁_log,𝛼(𝔹).

Proof. From Lemma 2.1 it follows that 𝜑_𝑒(ln(1+|𝑓(𝑧)|))≤|𝑓(𝑧)| (2.6) for every 𝑓∈𝐴¹_𝛼(𝔹) and 𝑧∈𝔹. Multiplying this inequality by 𝑑𝑉_𝛼(𝑧), then integrating such obtained inequality over 𝔹 we obtain ‖𝑓‖_{𝐴𝑁log,𝛼}≤‖𝑓‖_𝐴1𝛼, (2.7) from which the result follows.

Remark 2.3. Note that from inequality (2.7) it follows that the inclusion 𝑖∶𝐴¹_𝛼(𝔹)→𝐴𝑁_log,𝛼(𝔹) has “norm" less than one, if we define operator norm as usual by ‖𝑖‖_𝐴¹_{𝛼→𝐴𝑁log,𝛼}=sup_{‖𝑓‖𝐴1𝛼≤1}‖𝑖(𝑓)‖_{𝐴𝑁log,𝛼}. (2.8)

Lemmas 2.5 and 2.7 are based on the following lemma.

Lemma 2.4. Let Ψ(𝑥)=ln(1+𝑒^𝑥)ln{𝑒+ln(1+𝑒^𝑥)}(𝑥∈(−∞,∞)). Then Ψ is a positive continuous, increasing, and convex function on (−∞,∞).

Proof. It is clear that Ψ is a positive and continuous function on the interval (−∞,∞). Now we prove that it is increasing and convex on (−∞,∞). Note that Ψ(𝑥)=𝜑_𝑒(ln(1+𝑒^𝑥)). Hence Ψ^′(𝑥)=𝜑^′_𝑒(ln(1+𝑒^𝑥𝑒))^𝑥1+𝑒^𝑥,Ψ^′′(𝑥)=𝜑_𝑒^′′(ln(1+𝑒^𝑥𝑒))(^𝑥1+𝑒^𝑥)²+𝜑^′_𝑒(ln(1+𝑒^𝑥𝑒))^𝑥(1+𝑒^𝑥)². (2.9) From (1.2) and (2.9), and since ln(1+𝑒^𝑥)∈(0,∞), when 𝑥∈(−∞,∞), the lemma follows.

Lemma 2.5. Let 𝛼>−1. If 𝑓∈𝐴𝑁_log,𝛼(𝔹), then it holds that lim_𝑟→1−‖𝑓_𝑟−𝑓‖_{𝐴𝑁log,𝛼}=0, (2.10) where 𝑓_𝑟(𝑧)=𝑓(𝑟𝑧) for each 𝑟∈(0,1) and 𝑧∈𝔹.

Proof. Take an 𝑓∈𝐴𝑁_log,𝛼(𝔹) and 𝜀>0. Then we can choose an 𝑟₀∈(0,1) such that _{𝔹⧵𝑟0𝔹}𝜑_𝑒(ln(1+|𝑓(𝑧)|))𝑑𝑉_𝛼𝜀(𝑧)<3, (2.11) where 𝑟₀𝔹={𝑧∈𝔹∶|𝑧|≤𝑟₀}. Since 𝜑_𝑒(ln(1+|𝑓(𝑧)|))=Ψ(ln|𝑓(𝑧)|), from Lemma 2.4 it follows that 𝜑_𝑒(ln(1+|𝑓(𝑧)|)) is a positive plurisubharmonic function in 𝔹. Hence we have _𝜕𝔹𝜑_𝑒(ln(1+|𝑓(𝑟𝑡𝜁)|))𝑑𝜎(𝜁)≤_𝜕𝔹𝜑_𝑒(ln(1+|𝑓(𝑡𝜁)|))𝑑𝜎(𝜁) (2.12) for any 𝑟,𝑡∈(0,1). This inequality implies that _{𝔹⧵𝑟0𝔹}𝜑_𝑒(ln(1+|𝑓_𝑟(𝑧)|))𝑑𝑉_𝛼(𝑧)≤_{𝔹⧵𝑟0𝔹}𝜑_𝑒(ln(1+|𝑓(𝑧)|))𝑑𝑉_𝛼𝜀(𝑧)<3 (2.13) for any 𝑟∈(0,1).
Now we choose an 𝜀₀>0 such that 𝜑_𝑒(ln(1+𝜀₀))=𝜀/3. By the continuity of 𝑓 on the compact subset 𝑟₀𝔹, we see that there exists a 𝛿∈(0,1) such that if 𝑧,𝑤∈𝑟₀𝔹 with |𝑧−𝑤|<𝛿, then |𝑓(𝑧)−𝑓(𝑤)|<𝜀₀. (2.14)
Set 𝑟₁=1−𝛿. If 𝑟₁<𝑟<1, then _𝑟0𝔹𝜑_𝑒(ln(1+|𝑓_𝑟(𝑧)−𝑓(𝑧)|))𝑑𝑉_𝛼(𝑧)≤𝜑_𝑒(ln(1+𝜀₀𝜀))=3. (2.15) By (2.13), (2.15), and (1.5), we obtain ‖𝑓_𝑟−𝑓‖_{𝐴𝑁log,𝛼}=(_𝑟0𝔹+_{𝔹⧵𝑟0𝔹})𝜑_𝑒(ln(1+|𝑓_𝑟(𝑧)−𝑓(𝑧)|))𝑑𝑉_𝛼≤(𝑧)_𝑟0𝔹𝜑_𝑒(ln(1+|𝑓_𝑟(𝑧)−𝑓(𝑧)|))𝑑𝑉_𝛼+(𝑧)_{𝔹⧵𝑟0𝔹}{𝜑_𝑒(ln(1+|𝑓_𝑟(𝑧)|))+𝜑_𝑒(ln(1+|𝑓(𝑧)|))}𝑑𝑉_𝛼<𝜀(𝑧)3+𝜀3+𝜀3=𝜀. (2.16) This completes the proof.

Corollary 2.6. For each 𝛼>−1 the space 𝐴𝑁_log,𝛼(𝔹) is separable.

Proof. Since the dilated function 𝑓_𝑟 is approximated by the 𝑛th partial sum of its Taylor expansion uniformly on 𝔹, Lemma 2.5 implies that polynomials are dense in 𝐴𝑁_log,𝛼(𝔹). Since the polynomials with rational coefficients approximate any polynomial on the close unit ball 𝔹, the corollary follows.

Lemma 2.7. Let 𝛼>−1, 𝑓∈𝐻(𝔹) and 𝑧∈𝔹. Then it holds that ln(1+|𝑓(𝑧)|)ln{𝑒+ln(1+|𝑓(𝑧)|)}≤(1+|𝑧|)1−|𝑧|^𝑛+𝛼+1‖𝑓‖_{𝐴𝑁log,𝛼}. (2.17) Moreover, if 𝑓∈𝐴𝑁_log,𝛼(𝔹), then 𝑓 satisfies ln(1+|𝑓(𝑧)|)ln{𝑒+ln(1+|𝑓(𝑧)|)}=𝑜((1−|𝑧|)^{−(𝑛+𝛼+1)})(as|𝑧|→1⁻). (2.18)

Proof. Fix 𝑓∈𝐴𝑁_log,𝛼(𝔹) and 𝑧∈𝔹. Let 𝜑_𝑧 be the biholomorphic involution of 𝔹 described in [29, page 25]. Since 𝜑_𝑒(ln(1+|𝑓∘𝜑_𝑧|)) is a positive plurisubharmonic function in 𝔹 by Lemma 2.4, we have 𝜑_𝑒(ln(1+|𝑓(𝑧)|))=𝜑_𝑒(ln(1+|𝑓∘𝜑_𝑧(0)|))=2𝑛𝑐_𝛼,𝑛¹₀𝑟^2𝑛−1(1−𝑟²)^𝛼𝜑_𝑒(ln(1+|𝑓(𝜑_𝑧(0))|))𝑑𝑟≤2𝑛𝑐_𝛼,𝑛¹₀𝑟^2𝑛−1(1−𝑟²)^𝛼𝑑𝑟_𝜕𝔹𝜑_𝑒(ln(1+|𝑓(𝜑_𝑧(𝑟𝜁))|))𝑑𝜎(𝜁)=𝑐_{𝛼,𝑛𝔹}𝜑_𝑒(ln(1+|𝑓(𝜑_𝑧(𝑤))|))(1−|𝑤|²)^𝛼𝑑𝑉(𝑤)=𝑐_{𝛼,𝑛𝔹}𝜑_𝑒(ln(1+|𝑓(𝑤)|))(𝐽_ℝ𝜑_𝑧)(𝑤)(1−|𝜑_𝑧(𝑤)|²)^𝛼𝑑𝑉(𝑤), (2.19) where (𝐽_ℝ𝜑_𝑧)(𝑤) denotes the real Jacobian of 𝜑_𝑧 at 𝑤. By [29, Theorem 2.2.2 and 2.2.6], for 𝑤∈𝔹 we have (𝐽_ℝ𝜑_𝑧)(𝑤)(1−|𝜑_𝑧(𝑤)|²)^𝛼≤(1+|𝑧|)1−|𝑧|^𝑛+𝛼+1(1−|𝑤|²)^𝛼. (2.20) By (2.19) and (2.20), we obtain 𝜑_𝑒(ln(1+|𝑓(𝑧)|))≤(1+|𝑧|)1−|𝑧|^𝑛+𝛼+1‖𝑓‖_{𝐴𝑁log,𝛼}, (2.21) which completes the proof of the first claim.
Next we prove the second claim. Fix an 𝜀>0. By Lemma 2.5, there exists an 𝑟∈(0,1) such that ‖𝑓_𝑟−𝑓‖_{𝐴𝑁log,𝛼}<𝜀. From, (1.5) and (2.21) applied to the function 𝑓_𝑟−𝑓 we have that 𝜑_𝑒(ln(1+|𝑓(𝑧)|))(1−|𝑧|)^𝑛+𝛼+1≤2^𝑛+𝛼+1‖𝑓_𝑟−𝑓‖_{𝐴𝑁log,𝛼}+(1−|𝑧|)^𝑛+𝛼+1𝑀_𝑟<2^𝑛+𝛼+1𝜀+(1−|𝑧|)^𝑛+𝛼+1𝑀_𝑟, (2.22) where 𝑀_𝑟=max_{𝑤∈𝑟𝔹}𝜑_𝑒(ln(1+|𝑓(𝑤)|))<∞. Letting |𝑧|→1⁻ in (12) and using the fact that 𝜀 is an arbitrary positive number we obtain 𝜑_𝑒(ln(1+|𝑓(𝑧)|))(1−|𝑧|)^𝑛+𝛼+1→0,as|𝑧|→1⁻. (2.23) This completes the proof of this lemma.

Lemma 2.8. 𝐴𝑁_log,𝛼(𝔹) is a complete metric space.

Proof. In the introduction we have seen that 𝐴𝑁_log,𝛼(𝔹) is a metric space. Since the convergence in 𝐴𝑁_log,𝛼(𝔹) implies the uniform convergence on compact subsets of 𝔹 by Lemma 2.7, a usual normal family argument along with Fatou's lemma shows that every Cauchy sequence in 𝐴𝑁_log,𝛼(𝔹) converges to an element of 𝐴𝑁_log,𝛼(𝔹). Hence the space 𝐴𝑁_log,𝛼(𝔹) equipped with the metric 𝑑(𝑓,𝑔)=‖𝑓−𝑔‖_{𝐴𝑁log,𝛼} is a complete metric space.

Recall that a metric space (𝑋,𝑑) is called an 𝐹-space if it is a complete and

(i)𝑑(𝑥,𝑦)=𝑑(𝑥−𝑦,0) for every 𝑥,𝑦∈𝑋; (ii)for each sequence {𝑥_𝑘}⊂𝑋 such that 𝑑(𝑥_𝑘,0)→0 as 𝑘→∞ it follows that 𝑑(𝑐𝑥_𝑘,0)→0 for every 𝑐∈ℂ; (iii)for each sequence {𝑐_𝑘}⊂ℂ such that 𝑐_𝑘→0 as 𝑘→∞, 𝑑(𝑐_𝑘𝑥,0)→0 for each 𝑥∈𝑋.

If in an 𝐹-space (𝑋,𝑑) is introduced an operation of pointwise multiplication such that 𝑋 becomes an algebra and the operation of multiplication is continuous in the metric 𝑑, then the 𝐹-space (𝑋,𝑑) is called an 𝐹-algebra.

Before we prove our next result we will prove two technical lemmas.

Lemma 2.9. The following inequality holds: 𝜑_𝑒(ln(1+𝑐𝑥))≤max{1,𝑐²}𝜑_𝑒(ln(1+𝑥)) (2.24) for 𝑥,𝑐≥0.

Proof. It is enough to prove the case 𝑐>1. Since 𝜑_𝑒 is an increasing function and ln(1+𝑐𝑥)≤𝑐ln(1+𝑥), we have 𝜑_𝑒(ln(1+𝑐𝑥))≤𝜑_𝑒(𝑐ln(1+𝑥))=𝑐ln(1+𝑥)ln(𝑒+𝑐ln(1+𝑥)). (2.25) Let 𝜂(𝑡)=(𝑒+𝑡)^𝑐−(𝑒+𝑐𝑡),for𝑡∈[0,∞). (2.26) Then from 𝜂^′(𝑡)=𝑐((𝑒+𝑡)^𝑐−1−1), (2.27) we see that 𝜂 is an increasing function on the interval [0,∞), so that 𝜂(𝑡)≥𝜂(0)=𝑒^𝑐−𝑒>0. Hence (𝑒+𝑐𝑡)<(𝑒+𝑡)^𝑐. This implies that ln(𝑒+𝑐𝑡)<𝑐ln(𝑒+𝑡). (2.28) So we obtain 𝜑_𝑒(ln(1+𝑐𝑥))≤𝑐²ln(1+𝑥)ln(𝑒+ln(1+𝑥))=𝑐²𝜑_𝑒(ln(1+𝑥)), (2.29) completing the proof of the lemma.

The next lemma improves inequality (19) in [28].

Lemma 2.10. The following inequality holds: 𝜑_𝑒(𝑥+𝑦)≤2(𝜑_𝑒(𝑥)+𝜑_𝑒(𝑦)) (2.30) for 𝑥,𝑦≥0.

Proof. First note that the function 𝑓(𝑥,𝑦)=2(𝜑_𝑒(𝑥)+𝜑_𝑒(𝑦))−𝜑_𝑒(𝑥+𝑦) (2.31) satisfies the following condition: 𝑓(𝑥,𝑦)=𝑓(𝑦,𝑥). Hence it is enough to prove inequality (2.30) for the case 𝑥≥𝑦≥0. Note that 𝑓(0,𝑦)=𝜑_𝑒(𝑦)≥0 and 𝜕𝑓𝑥𝜕𝑥(𝑥,𝑦)=2(ln(𝑒+𝑥)+𝑒+𝑥)−ln(𝑒+𝑥+𝑦)−𝑥+𝑦𝑥𝑒+𝑥+𝑦≥2(ln(𝑒+𝑥)+𝑒+𝑥)−ln(𝑒+2𝑥)−2𝑥𝑒+2𝑥≥ln(𝑒+𝑥)²−ln(𝑒+2𝑥)>0. (2.32) Hence, for each fixed 𝑦 we have that 𝑓(𝑥,𝑦)≥𝑓(0,𝑦)≥0, from which inequality (2.30) follows.

Theorem 2.11. 𝐴𝑁_log,𝛼(𝔹) is an 𝐹-algebra with respect to pointwise addition and multiplication.

Proof. By Lemma 2.8  𝐴𝑁_log,𝛼(𝔹) is a complete metric space satisfying the condition 𝑑(𝑓,𝑔)=𝑑(𝑓−𝑔,0). From Lemma 2.9 we have the inequality 𝜑_𝑒(ln(1+𝑠𝑡))≤max{1,𝑠²}𝜑_𝑒(ln(1+𝑡)),𝑡,𝑠≥0, (2.33) which implies that ‖𝑐𝑓‖_{𝐴𝑁log,𝛼}≤max{1,|𝑐|²}‖𝑓‖_{𝐴𝑁log,𝛼} (2.34) for each 𝑐∈ℂ, so that the operation of the multiplication by complex numbers is closed in 𝐴𝑁_log,𝛼(𝔹). On the other hand, by inequality (1.5) it follows that the pointwise addition is also closed in 𝐴𝑁_log,𝛼(𝔹). From (2.34) it follows that the condition 𝑑(𝑓_𝑘,0)→0 implies 𝑑(𝑐𝑓_𝑘,0)→0 for every 𝑐∈ℂ, since 𝑑(𝑐𝑓_𝑘,0)=‖𝑐𝑓_𝑘‖_{𝐴𝑁log,𝛼}≤max{1,|𝑐|²}‖𝑓_𝑘‖_{𝐴𝑁log,𝛼}=max{1,|𝑐|²}𝑑(𝑓_𝑘,0). (2.35) Now assume that {𝑐_𝑘}_𝑘∈ℕ⊂ℂ is a sequence such that 𝑐_𝑘→0 as 𝑘→∞. Note that for each such sequence there is 𝑘₀∈ℕ such that |𝑐_𝑘|≤1 for 𝑘≥𝑘₀. Hence we have that for each 𝑓∈𝐴𝑁_log,𝛼(𝔹) and 𝑧∈𝔹𝜑_𝑒(ln(1+|𝑐_𝑘𝑓(𝑧)|))≤𝜑_𝑒(ln(1+|𝑓(𝑧)|)),𝑘≥𝑘₀. (2.36) From (2.36), since 𝜑_𝑒(ln(1+|𝑐_𝑘𝑓(𝑧)|))→0as𝑘→∞ (2.37) for each 𝑧∈𝔹, and by the Lebesgue dominated convergence theorem, we get 𝑑(𝑐_𝑘𝑓,0)=‖𝑐_𝑘𝑓‖_{𝐴𝑁log,𝛼}→0 as 𝑘→∞, which proves (iii) and that 𝐴𝑁_log,𝛼(𝔹) is an 𝐹-space.
Now we prove that 𝐴𝑁_log,𝛼(𝔹) is closed with respect to the pointwise multiplication. From Lemma 2.10 we have that 𝜑_𝑒(𝑡+𝑠)≤2(𝜑_𝑒(𝑡)+𝜑_𝑒(𝑠)),𝑡,𝑠≥0, (2.38) from which it follows that ‖𝑓𝑔‖_{𝐴𝑁log,𝛼}≤2(‖𝑓‖_{𝐴𝑁log,𝛼}+‖𝑔‖_{𝐴𝑁log,𝛼}), (2.39) so that 𝐴𝑁_log,𝛼(𝔹) is closed with respect to the pointwise multiplication.
Finally, we prove that the pointwise multiplication is continuous with respect to the metric 𝑑, that is, that from 𝑑(𝑓_𝑘,𝑓)→0 as 𝑘→∞ and 𝑑(𝑔_𝑙,𝑔)→0 as 𝑙→∞ it follows that 𝑑(𝑓_𝑘𝑔_𝑙,𝑓𝑔)→0 as 𝑘,𝑙→∞. Note that from triangle inequality and by (2.39) it follows that ‖𝑓_𝑘𝑔_𝑙−𝑓𝑔‖_{𝐴𝑁log,𝛼}≤‖(𝑓_𝑘−𝑓)(𝑔_𝑙−𝑔)‖_{𝐴𝑁log,𝛼}+‖(𝑓_𝑘−𝑓)𝑔‖_{𝐴𝑁log,𝛼}+‖𝑓(𝑔_𝑙−𝑔)‖_{𝐴𝑁log,𝛼}≤2(‖𝑓_𝑘−𝑓‖_{𝐴𝑁log,𝛼}+‖𝑔_𝑙−𝑔‖_{𝐴𝑁log,𝛼})+‖(𝑓_𝑘−𝑓)𝑔‖_{𝐴𝑁log,𝛼}+‖𝑓(𝑔_𝑙−𝑔)‖_{𝐴𝑁log,𝛼}. (2.40) From (2.40) and the symmetry ‖𝑓𝑔‖_{𝐴𝑁log,𝛼}=‖𝑔𝑓‖_{𝐴𝑁log,𝛼} we see that it is enough to prove that from 𝑑(𝑓_𝑘,𝑓)→0 as 𝑘→∞, for each 𝑔∈𝐴𝑁_log,𝛼(𝔹), it follows that 𝑑(𝑓_𝑘𝑔,𝑓𝑔)→0 as 𝑘→∞. Fix a 𝑔∈𝐴𝑁_log,𝛼(𝔹) and put 𝐸_𝑠={𝑧∈𝔹∶|𝑔(𝑧)|≥𝑠} for each 𝑠∈[0,∞). Hence we see that for every 𝛿>0 there is an 𝑠₀>0 such that for every 𝑠≥𝑠₀𝑉_𝛼(𝐸_𝑠1)≤𝜑_𝑒(ln(1+𝑠))_𝔹𝜑_𝑒(ln(1+|𝑔(𝑧)|))𝑑𝑉_𝛼(𝑧)<𝛿. (2.41) Fix an 𝜀>0. Since 𝜑_𝑒(ln(1+|𝑔|))∈𝐿¹(𝑑𝑉_𝛼), we have that for every 𝜀>0 there is a 𝛿₁>0 such that _𝐸𝜑_𝑒(ln(1+|𝑔(𝑧)|))𝑑𝑉_𝛼(𝑧)<𝜀 (2.42) for every set 𝐸 such that 𝑉_𝛼(𝐸)<𝛿₁. From (2.41) with 𝛿=𝛿₁ we have that 𝑉_𝛼(𝐸_𝑠)<𝛿 for sufficiently large 𝑠. From this and inequalities (2.34) and (2.39) we have that ‖𝑔(𝑓_𝑘−𝑓)‖_{𝐴𝑁log,𝛼}=_𝔹𝜑_𝑒(ln(1+|𝑔(𝑧)(𝑓_𝑘(𝑧)−𝑓(𝑧))|))𝑑𝑉_𝛼(𝑧)=(_{𝔹⧵𝐸𝑠}+_𝐸𝑠)𝜑_𝑒(ln(1+|𝑔(𝑧)(𝑓_𝑘(𝑧)−𝑓(𝑧))|))𝑑𝑉_𝛼(𝑧)≤2_𝐸𝑠𝜑_𝑒(ln(1+|𝑔(𝑧)|))𝑑𝑉_𝛼(𝑧)+2_𝐸𝑠𝜑_𝑒(ln(1+|𝑓_𝑘(𝑧)−𝑓(𝑧)|))𝑑𝑉_𝛼+(𝑧)_{𝔹⧵𝐸𝑠}𝜑_𝑒(ln(1+𝑠|𝑓_𝑘(𝑧)−𝑓(𝑧)|))𝑑𝑉_𝛼(𝑧)≤2𝜀+max{3,2+𝑠²}‖𝑓_𝑘−𝑓‖_{𝐴𝑁log,𝛼}. (2.43) Letting 𝑘→∞ in (2.43) and since 𝜀 is an arbitrary positive number, it follows that ‖𝑔(𝑓_𝑘−𝑓)‖_{𝐴𝑁log,𝛼}→0 as 𝑘→∞, finishing the proof of the theorem.

3. Linear Isometries of 𝐴𝑁_log,𝛼(𝔹)

In this section, we investigate linear isometries of 𝐴𝑁_log,𝛼(𝔹) into 𝐴𝑁_log,𝛼(𝔹), by modifying ideas from some earlier papers in this area (see, e.g., [23, 27, 28]).

The following two lemmas play an important role in the proofs of the main results in this section.

Lemma 3.1. If 𝑇 is a linear isometry of 𝐴𝑁_log,𝛼(𝔹) into itself, then the restriction of 𝑇 to 𝐴¹_𝛼(𝔹) is also a linear isometry of 𝐴¹_𝛼(𝔹) into itself.

Proof. Take an 𝑓∈𝐴¹_𝛼(𝔹) and put 𝑔=𝑇𝑓. For each 𝑚∈ℕ we have 𝑔/𝑚=𝑇(𝑓/𝑚), and so the assumption 𝑇 which is an isometry of 𝐴𝑁_log,𝛼(𝔹) gives _𝔹𝜑_𝑒(ln(1+|𝑓(𝑧)|𝑚))𝑑𝑉_𝛼(𝑧)=_𝔹𝜑_𝑒(ln(1+|𝑔(𝑧)|𝑚))𝑑𝑉_𝛼(𝑧). (3.1) By inequality (2.2), we have 𝑚𝜑_𝑒(ln(1+|𝑓(𝑧)|𝑚))≤|𝑓(𝑧)| (3.2) for any 𝑚∈ℕ and 𝑧∈𝔹. Also, it is easy to see that lim_𝑚→∞𝑚𝜑_𝑒(ln(1+|𝑓(𝑧)|𝑚))=|𝑓(𝑧)| (3.3) for each 𝑧∈𝔹. Hence the Lebesgue-dominated convergence theorem gives lim_{𝑚→∞𝔹}𝑚𝜑_𝑒(ln(1+|𝑓(𝑧)|𝑚))𝑑𝑉_𝛼(𝑧)=_𝔹|𝑓(𝑧)|𝑑𝑉_𝛼(𝑧). (3.4) On the other hand, Fatou's lemma (3.1) and (3.4) show that _𝔹|𝑔(𝑧)|𝑑𝑉_𝛼(𝑧)≤liminf_{𝑚→∞𝔹}𝑚𝜑_𝑒(ln(1+|𝑔(𝑧)|𝑚))𝑑𝑉_𝛼(𝑧)=liminf_{𝑚→∞𝔹}𝑚𝜑_𝑒(ln(1+|𝑓(𝑧)|𝑚))𝑑𝑉_𝛼=(𝑧)_𝔹|𝑓(𝑧)|𝑑𝑉_𝛼(𝑧), (3.5) and so 𝑔∈𝐴¹_𝛼(𝔹). By applying the Lebesgue-dominated convergence theorem once more again, we have lim_{𝑚→∞𝔹}𝑚𝜑_𝑒(ln(1+|𝑔(𝑧)|𝑚))𝑑𝑉_𝛼(𝑧)=_𝔹|𝑔(𝑧)|𝑑𝑉_𝛼(𝑧). (3.6) By (3.1), (3.4), and (3.6), we see that 𝑇 is a linear isometry of 𝐴¹_𝛼(𝔹) into 𝐴¹_𝛼(𝔹).

Lemma 3.2. There exists a bounded continuous function 𝜃(𝑥) on [0,∞) such that (12−1𝑒)𝑥²−𝑥³𝜃(𝑥)=𝑥−𝜑_𝑒(ln(1+𝑥))≥0𝑥∈[0,∞). (3.7)

Proof. Set 𝜑𝜃(𝑥)∶=^𝑒(ln(1+𝑥))−𝑥−(−1/2+1/𝑒)𝑥²𝑥³ (3.8) for 𝑥∈(0,∞). Since 𝜃(𝑥) is a continuous function on (0,∞) such that 𝜃(𝑥)→0 as 𝑥→+∞, it is enough to prove that 𝜃(𝑥) has a finite limit as 𝑥→0⁺. By the application of Taylor's theorem to 𝜑_𝑒(ln(1+𝑥)), we have 𝜑_𝑒1(ln(1+𝑥))=𝑥+(−2+1𝑒)𝑥²1+(3−1𝑒−12𝑒²)𝑥³+𝑅₄(𝑥), (3.9) where 𝑅₄(𝑥) denotes the remainder term of order 4. Since 𝑅₄(𝑥)/𝑥³→0 as 𝑥→0⁺, we obtain lim_𝑥→0+1𝜃(𝑥)=3−1𝑒−12𝑒², (3.10) which completes the proof.

Theorem 3.3. Every linear isometry 𝑇 of 𝐴𝑁_log,𝛼(𝔹) into itself has the form 𝑇𝑓=𝑐(𝑓∘𝜓) for any 𝑓∈𝐴𝑁_log,𝛼(𝔹), where 𝑐∈ℂ with |𝑐|=1 and 𝜓∶𝔹→𝔹 is a holomorphic map which satisfies the condition _𝔹ℎ(𝜓(𝑧))𝑑𝑉_𝛼(𝑧)=_𝔹ℎ(𝑧)𝑑𝑉_𝛼(𝑧) (3.11) for every bounded or positive Borel function ℎ in 𝔹.

Proof. First suppose that 𝑇∶𝐴𝑁_log,𝛼(𝔹)→𝐴𝑁_log,𝛼(𝔹) is a linear isometry. By Lemma 3.1, the restriction of 𝑇 to 𝐴¹_𝛼(𝔹) is a linear isometry of 𝐴¹_𝛼(𝔹) into 𝐴¹_𝛼(𝔹). Hence Kolaski's theorem [6, page 911, Theorem 2.11] implies that 𝑇 has the following form: 𝑇𝑓(𝑧)=𝑔(𝑧)𝑓(𝜓(𝑧))(𝑓∈𝐴¹_𝛼(𝔹),𝑧∈𝔹), (3.12) where 𝑔=𝑇1 and 𝜓 is a holomorphic self-map of 𝔹 such that _𝔹ℎ(𝜓(𝑧))|𝑔(𝑧)|𝑑𝑉_𝛼(𝑧)=_𝔹ℎ(𝑧)𝑑𝑉_𝛼(𝑧) (3.13) for every bounded or positive Borel function ℎ in 𝔹. Fix an 𝑓∈𝐴𝑁_log,𝛼(𝔹). Since {𝑓_𝑟}_0<𝑟<1⊂𝐴¹_𝛼(𝔹), the representation (3.12) and Lemmas 2.5 and 2.7 give 𝑇𝑓(𝑧)=lim_𝑟→1−𝑇𝑓_𝑟(𝑧)=lim_𝑟→1−𝑔(𝑧)𝑓_𝑟(𝜓(𝑧))=𝑔(𝑧)𝑓(𝜓(𝑧)) (3.14) for all 𝑧∈𝔹. Since 𝑔=𝑇1∈𝐴¹_𝛼(𝔹) and 𝑉_𝛼(𝔹)=1, Hölder's inequality gives 1=‖1‖_𝐴¹_𝛼=‖𝑔‖_𝐴1𝛼≤‖𝑔‖_𝐴2𝛼, (3.15) where ‖𝑔‖²_𝐴2𝛼=∫_𝔹|𝑔|²𝑑𝑉_𝛼. On the other hand, from Lemma 3.2 it follows that _𝔹1{(2−1𝑒)|𝑡𝑔(𝑧)|²−|𝑡𝑔(𝑧)|³𝜃(|𝑡𝑔(𝑧)|)}𝑑𝑉_𝛼(𝑧)=_𝔹{|𝑡𝑔(𝑧)|−𝜑_𝑒(ln(1+|𝑡𝑔(𝑧)|))}𝑑𝑉_𝛼=(𝑧)_𝔹|𝑇(𝑡)(𝑧)|𝑑𝑉_𝛼(𝑧)−_𝔹𝜑_𝑒(ln(1+|𝑇(𝑡)(𝑧)|))𝑑𝑉_𝛼=(𝑧)_𝔹𝑡𝑑𝑉_𝛼(𝑧)−_𝔹𝜑_𝑒(ln(1+|𝑡|))𝑑𝑉_𝛼1(𝑧)=(2−1𝑒)𝑡²−𝑡³𝜃(𝑡). (3.16) By Lemma 3.2 and Fatou's lemma we have _𝔹(12−1𝑒)|𝑔(𝑧)|²𝑑𝑉_𝛼(𝑧)≤liminf_{𝑡→0+𝔹}1{(2−1𝑒)|𝑔(𝑧)|²−𝑡|𝑔(𝑧)|³𝜃(|𝑡𝑔(𝑧)|)}𝑑𝑉_𝛼(𝑧)=liminf_𝑡→0+1{(2−1𝑒1)−𝑡𝜃(𝑡)}=2−1𝑒, (3.17) and so ‖𝑔‖_𝐴2𝛼≤1. Combining this with (3.15), we obtain ‖𝑔‖_𝐴1𝛼=‖𝑔‖_𝐴2𝛼=1. (3.18) This implies that 𝑔≡𝑐 in 𝔹 for some 𝑐∈ℂ with |𝑐|=1. Hence (3.13) and (3.14) show that 𝑇𝑓=𝑐(𝑓∘𝜓) for any 𝑓∈𝐴𝑁_log,𝛼(𝔹) and 𝜓 satisfies the condition in (3.11).
Conversely, for some complex number 𝑐 with |𝑐|=1 and holomorphic self-map 𝜓 of 𝔹 which satisfies (3.11) we define an operator 𝑇 on 𝐴𝑁_log,𝛼(𝔹) by 𝑇𝑓=𝑐(𝑓∘𝜓) for 𝑓∈𝐴𝑁_log,𝛼(𝔹). Since 𝜑_𝑒(ln(1+|𝑓|)) is a positive Borel function in 𝔹, condition (3.11) implies that 𝑇 is a linear isometry of 𝐴𝑁_log,𝛼(𝔹) into 𝐴𝑁_log,𝛼(𝔹). This completes the proof.

Corollary 3.4. Every surjective isometry 𝑇 of 𝐴𝑁_log,𝛼(𝔹) is of the form 𝑇𝑓=𝑐(𝑓∘𝑈) for any 𝑓∈𝐴𝑁_log,𝛼(𝔹), where 𝑐∈ℂ with |𝑐|=1 and 𝑈 is a unitary operator on ℂ^𝑛.

Proof. Assume that 𝑇∶𝐴𝑁_log,𝛼(𝔹)→𝐴𝑁_log,𝛼(𝔹) is a surjective isometry. Then Theorem 3.3 implies that 𝑇 is written in the form 𝑇𝑓=𝑐_𝑇(𝑓∘𝜓_𝑇) where 𝑐_𝑇∈ℂ with |𝑐_𝑇|=1 and 𝜓_𝑇 is a holomorphic self-map of 𝔹 which satisfies (3.11). Since 𝑇⁻¹ is also a surjective isometry of 𝐴𝑁_log,𝛼(𝔹), we see that 𝑓=𝑇⁻¹(𝑇(𝑓))=𝑇⁻¹(𝑐_𝑇(𝑓∘𝜓_𝑇))=𝑐_𝑇⁻¹𝑐_𝑇(𝑓∘𝜓_𝑇∘𝜓_𝑇−1), (3.19) so that 𝑐_𝑇⁻¹=𝑐_𝑇⁻¹ and 𝜓_𝑇−1=𝜓_𝑇⁻¹, that is, 𝜓_𝑇 is an automorphism of 𝔹.
We prove that 𝜓∶=𝜓_𝑇 fixes the origin. Let 𝜓_𝑗(1≤𝑗≤𝑛) be the components of 𝜓. For each 𝑗∈{1,…,𝑛} and an 𝑟∈(0,1) we have _𝜕𝔹𝜓_𝑗(𝑟𝜁)𝑑𝜎(𝜁)=_𝜕𝔹1𝑑𝜎(𝜁)2𝜋^𝜋_−𝜋𝜓_𝑗(𝑟𝑒^𝑖𝜃𝜁)𝑑𝜃. (3.20) Since the slice function (𝜓_𝑗)_𝜁 of 𝜓_𝑗 is holomorphic in the open unit disc of ℂ, the mean value theorem gives (1/2𝜋)∫^𝜋_−𝜋𝜓_𝑗(𝑟𝑒^𝑖𝜃𝜁)𝑑𝜃=𝜓_𝑗(0) for each 𝜁∈𝜕𝔹 and 𝑗∈{1,…,𝑛}. Using this fact in formula (3.20), multiplying such obtained equality by 2𝑛𝑐_𝛼,𝑛(1−𝑟²)^𝛼𝑟^2𝑛−1𝑑𝑟, then integrating it from 0 to 1 and using the polar coordinates on the unit ball we get _𝔹𝜓_𝑗(𝑧)𝑑𝑉_𝛼(𝑧)=𝜓_𝑗(0) (3.21) for each 𝑗∈{1,…,𝑛}. Now we fix a 𝑗∈{1,…,𝑛}. By applying the condition (3.11) to a bounded Borel function ℎ(𝑤)=⟨𝑤,𝑒_𝑗⟩, 𝑗∈{1,…,𝑛}, where 𝑒_𝑗 is the standard orthonormal base vector in ℂ^𝑛, we have _𝔹𝜓_𝑗(𝑧)𝑑𝑉_𝛼(𝑧)=_𝔹⟨𝜓(𝑧),𝑒_𝑗⟩𝑑𝑉_𝛼=(𝑧)_𝔹⟨𝑧,𝑒_𝑗⟩𝑑𝑉_𝛼(𝑧)=2𝑛𝑐_𝛼,𝑛¹₀𝑟^2𝑛(1−𝑟²)^𝛼𝑑𝑟_𝜕𝔹⟨𝜁,𝑒_𝑗⟩𝑑𝜎(𝜁). (3.22) From [29, page 15, § 1.4.5(2)] we have that _𝜕𝔹⟨𝜁,𝑒_𝑗⟩𝑑𝜎(𝜁)=𝑛−1𝜋^𝜋_−𝜋¹₀(1−𝑡²)^𝑛−2𝑡²𝑒^𝑖𝜃𝑑𝑡𝑑𝜃=0. (3.23) Hence (3.21), (3.22), and (3.23) show that 𝜓_𝑗(0)=0 for each 𝑗∈{1,…,𝑛}, and so 𝜓 fixes the origin. By a well-known theorem [29] we see that 𝜓 is a unitary operator on ℂ^𝑛. This completes the proof.

Remark 3.5. Let Isom𝐴𝑁_log,𝛼(𝔹) denote the set of all surjective isometries of 𝐴𝑁_log,𝛼(𝔹). Since the transformations 𝜓 in Corollary 3.4 form a group of unitary transformations the corollary implies the following isomorphism: Isom𝐴𝑁_log,𝛼(𝔹)≃𝕋×𝑈(𝑛,ℂ). (3.24) Here 𝕋 denotes the set of all complex numbers with |𝑐|=1 and 𝑈(𝑛,ℂ) being the group of all unitary operators on ℂ^𝑛. For the Smirnov class 𝑁^∗ and the Privalov space 𝑁^𝑝(𝑝>1), analogous results are obtained by Stephenson and Subbotin (see [22, 24–26]). So we see that every surjective isometry of 𝐴𝑁_log,𝛼(𝔹) has the same form as one of the Smirnov class or the Privalov space.

Acknowledgment

The second author (S. Ueki) of the present paper is supported by Grant-in-Aid for Young Scientists (Start-up; no. 20840004).