# Multiplicative Isometries onF-Algebras of Holomorphic Functions

Authors: Osamu Hatori; Yasuo Iida; Stevo Stević; Sei-Ichiro Ueki
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study multiplicative isometries on the followingF-algebras of holomorphic functions: Smirnov class N*(X), Privalov class Np(X), Bergman-Privalov class ANαp(X), and Zygmund F-algebra NlogβN(X), where X is the open unit ball 𝔹n or the open unit polydisk 𝔻n in ℂn.


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## Body

## 1. Introduction

Complex-linear isometries on function spaces of holomorphic functions have been studied for almost five decades by many mathematicians. In this paper we study multiplicative isometries on certainF-algebras of holomorphic functions. Recall that an F-algebra is a topological algebra in which the topology arises from a complete metric. For a positive integer n let 𝔹n denote the open unit ball in the n-dimensional complex vector space ℂn and 𝔻n the unit polydisk in ℂn. We characterize multiplicative isometries on the Smirnov class, the Privalov class, the Bergman-Privalov class and the Zygmund F-algebras on 𝔹n or 𝔻n. Surjective multiplicative maps on the Smirnov class, and the Bergman-Privalov class have already been correspondingly characterized in [1, 2].


## 2. Preliminaries

In studying surjective isometries in [1, 2] we applied the Mazur-Ulam theorem for surjective maps on certain subspaces, which themselves are Banach spaces, of the given F-algebras. Generally we do not assume surjectivity of the isometries in this paper, so instead of the Mazur-Ulam theorem we use Lemma 2.1. Recall that a normed real-linear space L is uniformly convex if for any ɛ>0 there exists a δ>0 such that the inequality ∥a+b∥≤2-δ holds for every pair of a,b∈L with ∥a∥≤1, ∥b∥≤1, and ∥a-b∥≥ɛ. It is well known that Hilbert spaces and Lp-spaces for 1<p<∞ are uniformly convex.Lemma 2.1.
LetL1 and L2 be normed real-linear spaces with L2 uniformly convex. Let S be an isometry from L1 into L2 such that S(0)=0. Then S is real-linear.The lemma might be well known, but we give a sketch of the proof for the completeness and the benefit of the reader.Proof of Lemma2.1.
Leta, b be arbitrary elements of L1. Put 2r=∥a-b∥. Then since S is an isometry, ∥S(a)-S(b)∥=2r and ∥S(a)-S((a+b)/2)∥=∥S(b)-S((a+b)/2)∥=r. We also have ∥S(a)-(S(a)+S(b))/2∥=∥S(b)-(S(a)+S(b))/2∥=r.
Suppose thatS((a+b)/2)≠(S(a)+S(b))/2. Set
(2.1)ɛ=‖S(a+b2)-S(a)+S(b)2‖.
Since L2 is uniformly convex and ɛ is positive there exists a δ>0 such that
(2.2)‖(S(a)-S(a+b2))+(S(a)-S(a)+S(b)2)‖≤2r-δ,‖(S(b)-S(a+b2))+(S(b)-S(a)+S(b)2)‖≤2r-δ.
Then by the triangle inequality
(2.3)‖2S(a)-2S(b)‖≤4r-2δ
holds, which contradicts to ∥S(a)-S(b)∥=2r. Thus we get S((a+b)/2)=(S(a)+S(b))/2, from which for b=0 we obtain S(a/2)=S(a)/2. Substituting a by a+b in the last equality we get
(2.4)S(a+b)2=S(a+b2)=S(a)+S(b)2,
so that S(a+b)=S(a)+S(b). A routine argument yields S(ta)=tS(a), t∈ℝ.ForX∈{𝔹n,𝔻n}, we denote by ∂X its distinguished boundary. For X=𝔹n, this is the topological boundary ∂𝔹n, and for the polydisk 𝔻n, it is the torus 𝕋n. Denote the normalized Lebesgue measure on ∂X by σ. A holomorphic map ψ is inner if limr→1-0ψ(rz) exists and lies in ∂X for almost all z∈∂X with respect to σ. We say that limr→1-0ψ(rz) is the boundary map of ψ and denote it by ψ*. We say that ψ* is measure preserving if σ((ψ*)-1(E))=σ(E) for every Borel set E⊂∂X.Now we recall definitions and some properties of the Smirnov class, the Privalov class, the Bergman-Privalov class, and the ZygmundF-algebra on 𝔹n or 𝔻n. The space of all holomorphic functions on X=𝔹n or 𝔻n is denoted by H(X). For each 0<p≤∞, the Hardy space is denoted by Hp(X) with the norm ∥·∥p.


### 2.1. Smirnov ClassN*(X)

LetX∈{𝔹n,𝔻n}. The Nevanlinna class N(X) on X is defined as the set of all holomorphic functions f on X such that(2.5)sup⁡0≤r<1∫∂Xln⁡(1+|f(rζ)|)dσ(ζ)<∞
holds. It is known that every f∈N(X) has a finite nontangential limit, denoted by f*, almost everywhere on ∂X.The Smirnov classN*(X) is defined as(2.6)N*(X)={f∈N(X):sup⁡0≤r<1∫∂Xln⁡(1+|f(rζ)|)dσ(ζ)=∫∂Xln⁡(1+|f*(ζ)|)dσ(ζ)}.
Define a metric(2.7)dN*(X)(f,g)=∫∂Xln⁡(1+|f*(ζ)-g*(ζ)|)dσ(ζ)
for f,g∈N*(X). With the metric dN*(X)(·,·) the Smirnov class N*(X) becomes an F-algebra and(2.8)⋃q>0Hq(X)⊂N*(X),
in particular, H∞(X) is a dense subalgebra of N*(X). The convergence in the metric is stronger than uniform convergence on compact subsets of X.Complex-linear isometries on the Smirnov class were characterized by Stephenson in [3].


### 2.2. Privalov ClassNp(X)

LetX∈{𝔹n,𝔻n}. The Privalov class Np(X), 1<p<∞, is defined as (for the original source see [4, 5])(2.9)Np(X)={f∈H(X):sup⁡0≤r<1∫∂X(ln⁡(1+|f(rζ)|))pdσ(ζ)<∞}.It is well known thatNp(X) is a subalgebra of N*(X), hence every f∈Np(X) has a finite nontangential limit almost everywhere on ∂X. Define a metric(2.10)dp(f,g)=(∫∂X(ln⁡(1+|f*(ζ)-g*(ζ)|))pdσ(ζ))1/p
for f,g∈Np(X). With this metric Np(X) is an F-algebra (cf. [6, 7]) and(2.11)⋃q>0Hq(X)⊂Np(X)⊂N*(X).
The Hardy algebra H∞(X) is dense in Np(X). The convergence on the metric is stronger than uniform convergence on compacts of X.Complex-linear isometries onNp(X) are investigated by Iida and Mochizuki [8] for one-dimensional case, and by Subbotin [7] for a general case.


### 2.3. Bergman-Privalov ClassANαp(X)

Let1≤p<∞ and α>-1. The Bergman-Privalov class on the unit ball 𝔹n and the polydisk 𝔻n are defined, respectively, as(2.12)ANαp(Bn)={f∈H(Bn):‖f‖ANαp(Bn)p=∫Bn(ln⁡(1+|f(z)|))pdVα,n(z)<∞},ANαp(Dn)={f∈H(Dn):‖f‖ANαp(Dn)p=∫Dn(ln⁡(1+|f(z)|))p∏j=1ndVα,1(zj)<∞},
where dVα,n(z)=cα,n(1-|z|2)αdV(z) for the normalized Lebesgue volume measure dV on 𝔹n and cα,n is a normalization constant, that is Vα,n(𝔹n)=1. Let X∈{𝔹n,𝔻n}. In what follows dVα(z) denotes dVα,n(z) for X=𝔹n and ∏j=1ndVα,1(zj) for X=𝔻n, respectively. The Bergman-Privalov class ANαp(X) is an F-algebra with respect to the metric(2.13)dANαp(X)(f,g)=‖f-g‖ANαp(X)
for f,g∈ANαp(X). For some results in the case p=1 see [9].The weighted Bergman space forq>0 and α>-1 on the unit ball 𝔹n and the polydisk 𝔻n are defined, respectively, as(2.14)Aαq(Bn)={f∈H(Bn):‖f‖Aαq(Bn)q=∫Bn|f(z)|qdVα,n(z)<∞},Aαq(Dn)={f∈H(Dn):‖f‖Aαq(Dn)q=∫Dn|f(z)|q∏j=1ndVα,1(zj)<∞}.
It is known that(2.15)⋃q>0Aαq(X)⊂ANαp(X).Complex-linear isometries on the Bergman-Privalov class on the unit ball were characterized by Matsugu and Ueki in [10] and on the polydisk by Stević in [2].


### 2.4. ZygmundF-Algebra NlogβN(X)

Letβ>0 and φβ(t)=t(ln(γβ+t))β, where γβ=max{e,eβ}. Let X∈{𝔹n,𝔻n}. The Zygmund F-algebra NlogβN(X) on X is defined as(2.16)Nlog⁡βN(X)={f∈H(X):sup⁡0≤r<1∫∂Xφβ(ln⁡(1+|f(rζ)|))dσ(ζ)<∞}.
It is known that(2.17)Nlog⁡βN(X)={f∈H(X):sup⁡0≤r<1∫∂Xφβ(ln⁡+|f(rζ)|)dσ(ζ)<∞},(2.18)⋃p>0Hp(X)⊂Nlog⁡βN(X)⊂N*(X).
This implies that the finite nontangential limit f* exists almost everywhere on ∂X, for any f∈NlogβN. For f,g∈NlogβN(2.19)dNlog⁡βN(X)(f,g)=∫∂Xφβ(ln⁡(1+|f*(ζ)-g*(ζ)|))dσ(ζ)
defines a complete metric on NlogβN(X) and NlogβN(X) is an F-algebra with this metric (cf. [11]).Ueki [12] characterized the complex-linear isometries on the Zygmund F-algebra on the balls.


## 2.1. Smirnov ClassN*(X)

LetX∈{𝔹n,𝔻n}. The Nevanlinna class N(X) on X is defined as the set of all holomorphic functions f on X such that(2.5)sup⁡0≤r<1∫∂Xln⁡(1+|f(rζ)|)dσ(ζ)<∞
holds. It is known that every f∈N(X) has a finite nontangential limit, denoted by f*, almost everywhere on ∂X.The Smirnov classN*(X) is defined as(2.6)N*(X)={f∈N(X):sup⁡0≤r<1∫∂Xln⁡(1+|f(rζ)|)dσ(ζ)=∫∂Xln⁡(1+|f*(ζ)|)dσ(ζ)}.
Define a metric(2.7)dN*(X)(f,g)=∫∂Xln⁡(1+|f*(ζ)-g*(ζ)|)dσ(ζ)
for f,g∈N*(X). With the metric dN*(X)(·,·) the Smirnov class N*(X) becomes an F-algebra and(2.8)⋃q>0Hq(X)⊂N*(X),
in particular, H∞(X) is a dense subalgebra of N*(X). The convergence in the metric is stronger than uniform convergence on compact subsets of X.Complex-linear isometries on the Smirnov class were characterized by Stephenson in [3].


## 2.2. Privalov ClassNp(X)

LetX∈{𝔹n,𝔻n}. The Privalov class Np(X), 1<p<∞, is defined as (for the original source see [4, 5])(2.9)Np(X)={f∈H(X):sup⁡0≤r<1∫∂X(ln⁡(1+|f(rζ)|))pdσ(ζ)<∞}.It is well known thatNp(X) is a subalgebra of N*(X), hence every f∈Np(X) has a finite nontangential limit almost everywhere on ∂X. Define a metric(2.10)dp(f,g)=(∫∂X(ln⁡(1+|f*(ζ)-g*(ζ)|))pdσ(ζ))1/p
for f,g∈Np(X). With this metric Np(X) is an F-algebra (cf. [6, 7]) and(2.11)⋃q>0Hq(X)⊂Np(X)⊂N*(X).
The Hardy algebra H∞(X) is dense in Np(X). The convergence on the metric is stronger than uniform convergence on compacts of X.Complex-linear isometries onNp(X) are investigated by Iida and Mochizuki [8] for one-dimensional case, and by Subbotin [7] for a general case.


## 2.3. Bergman-Privalov ClassANαp(X)

Let1≤p<∞ and α>-1. The Bergman-Privalov class on the unit ball 𝔹n and the polydisk 𝔻n are defined, respectively, as(2.12)ANαp(Bn)={f∈H(Bn):‖f‖ANαp(Bn)p=∫Bn(ln⁡(1+|f(z)|))pdVα,n(z)<∞},ANαp(Dn)={f∈H(Dn):‖f‖ANαp(Dn)p=∫Dn(ln⁡(1+|f(z)|))p∏j=1ndVα,1(zj)<∞},
where dVα,n(z)=cα,n(1-|z|2)αdV(z) for the normalized Lebesgue volume measure dV on 𝔹n and cα,n is a normalization constant, that is Vα,n(𝔹n)=1. Let X∈{𝔹n,𝔻n}. In what follows dVα(z) denotes dVα,n(z) for X=𝔹n and ∏j=1ndVα,1(zj) for X=𝔻n, respectively. The Bergman-Privalov class ANαp(X) is an F-algebra with respect to the metric(2.13)dANαp(X)(f,g)=‖f-g‖ANαp(X)
for f,g∈ANαp(X). For some results in the case p=1 see [9].The weighted Bergman space forq>0 and α>-1 on the unit ball 𝔹n and the polydisk 𝔻n are defined, respectively, as(2.14)Aαq(Bn)={f∈H(Bn):‖f‖Aαq(Bn)q=∫Bn|f(z)|qdVα,n(z)<∞},Aαq(Dn)={f∈H(Dn):‖f‖Aαq(Dn)q=∫Dn|f(z)|q∏j=1ndVα,1(zj)<∞}.
It is known that(2.15)⋃q>0Aαq(X)⊂ANαp(X).Complex-linear isometries on the Bergman-Privalov class on the unit ball were characterized by Matsugu and Ueki in [10] and on the polydisk by Stević in [2].


## 2.4. ZygmundF-Algebra NlogβN(X)

Letβ>0 and φβ(t)=t(ln(γβ+t))β, where γβ=max{e,eβ}. Let X∈{𝔹n,𝔻n}. The Zygmund F-algebra NlogβN(X) on X is defined as(2.16)Nlog⁡βN(X)={f∈H(X):sup⁡0≤r<1∫∂Xφβ(ln⁡(1+|f(rζ)|))dσ(ζ)<∞}.
It is known that(2.17)Nlog⁡βN(X)={f∈H(X):sup⁡0≤r<1∫∂Xφβ(ln⁡+|f(rζ)|)dσ(ζ)<∞},(2.18)⋃p>0Hp(X)⊂Nlog⁡βN(X)⊂N*(X).
This implies that the finite nontangential limit f* exists almost everywhere on ∂X, for any f∈NlogβN. For f,g∈NlogβN(2.19)dNlog⁡βN(X)(f,g)=∫∂Xφβ(ln⁡(1+|f*(ζ)-g*(ζ)|))dσ(ζ)
defines a complete metric on NlogβN(X) and NlogβN(X) is an F-algebra with this metric (cf. [11]).Ueki [12] characterized the complex-linear isometries on the Zygmund F-algebra on the balls.


## 3.  Main Results

In this section we formulate and prove the main results in this paper.


### 3.1.  Multiplicative Isometries onN*(X)

Our first result concerns the Smirnov class.Theorem 3.1.
LetX∈{𝔹n,𝔻n}. Suppose that T:N*(X)→N*(X) is a (not necessarily linear) multiplicative isometry. Then there is an inner map ψ on X whose boundary map ψ* is measure preserving and such that either of the following formulas holds:
(3.1)T(f)=f∘ψforeveryf∈N*(X),T(f)=f∘ψ¯¯foreveryf∈N*(X).Proof.
First we claim thatT(1)=1. Since T(1)=T(1)2 and T(1) is a holomorphic function on the connected open set X we get T(1)=0 or T(1)=1. But T(1)=0 is impossible because if it were T(1)=0, then 0=T(f)T(1)=T(f), for each f∈N*(X), which contradicts with the assumption that T is an isometry. As T(0)=T(0)2 and T is injective, we obtain T(0)=0. Similarly T(-1)=-1 is also observed by making use of T(-1)2=T(1)=1. Then T(i)2=T(i2)=-1 assert that T(i)=i or T(i)=-i. If T(i)=i, then the first formula of the conclusion will follow and the second one will follow from T(i)=-i.
Next we showT(1/2)=1/2. Put r=1/2. Suppose that |T(r)*|>r on a set of positive measure on ∂X. Then there exists a subset E of positive measure and ɛ>0 with |T(r)*|≥(1+ɛ)r on E. Since
(3.2)lim⁡n→∞ln⁡⁡(1+(1+ɛ)nrn)ln⁡(1+rn)=∞,
there is a positive integer n0 such that
(3.3)∫Eln⁡⁡(1+(1+ɛ)n0rn0)dσ>∫∂Xln⁡(1+rn0)dσ.
From this and since T is a multiplicative isometry on N*(X) we have that
(3.4)∫∂Xln⁡(1+rn0)dσ=∫∂Xln⁡⁡(1+|T(r)*|n0)dσ≥∫Eln⁡⁡(1+(1+ɛ)n0rn0)dσ>∫∂Xln⁡(1+rn0)dσ,
which is a contradiction proving |T(r)*|≤r almost everywhere on ∂X. Hence |T(1/r)*|≥1/r holds almost everywhere on ∂X as T(r)T(1/r)=T(1)=1 almost everywhere on ∂X. Since
(3.5)ln⁡(1+1r)=∫∂Xln⁡(1+1r)dσ=∫∂Xln⁡(1+|T(1r)*|)dσ,
we have that |T(1/r)*|=1/r and |T(r)*|=r almost everywhere on ∂X.
Sinceln(1+(1-r))=d(r,1)=d(T(r),1) and
(3.6)d(T(r),1)=∫∂Xln⁡⁡(1+|1-T(r)*|)dσ,
it is easy to check that T(1/2)*=1/2 almost everywhere on ∂X. Hence T(1/2)=1/2 holds. As T is multiplicative, T is 1/2-homogeneous in the sense that T(f/2)=T(f)/2 holds for every f∈N*(X).
Letf,g∈H1(X). It requires only elementary calculation applying the 1/2-homogeneity of T to check that
(3.7)∫∂Xln⁡(1+|f*2m-g*2m|)dσ=∫∂Xln⁡(1+|T(f)*2m-T(g)*2m|)dσ
holds. Multiplying (3.7) by 2m and then letting m→∞ we get
(3.8)∫∂X|f*-g*|dσ=∫∂X|T(f)*-T(g)*|dσ
by the monotone convergence theorem, since 2mln(1+(t/2m)) nondecreases monotonically to t as m→∞ for any t≥0, which can be easily proved by considering the function gt(x)=xln(1+(t/x)). From (3.8) for g=0, we obtain T(H1(X))⊆H1(X) and the restricted map T|H1(X) is an isometry with respect to the metric induced by the H1-norm ∥·∥1.
Let the functionθ on the interval [0,∞) be defined as
(3.9)θ(x)={12,x=0x-ln⁡(1+x)x2,x>0.
It is easy to check that θ is positive and continuous on [0,∞) and limx→∞θ(x)=0. Hence θ is bounded on [0,∞), so that
(3.10)Mθ∶=sup⁡x≥0θ(x)<∞.
We claim that the inclusionT(H2(X))⊆H2(X) and T|H2(X) is isometric with respect to the metric induced by the H2-norm. For this purpose let f,g∈H2(X). Now note that since H2(X)⊂H1(X), equality (3.7) holds and as well as the next equality
(3.11)∫∂X|f*2m-g*2m|dσ=∫∂X|T(f)*2m-T(g)*2m|dσ.
By subtracting (3.7) from (3.11) and then multiplying such obtained equation by 2m we obtain
(3.12)∫∂X|f*-g*|2θ(|f*2m-g*2m|)dσ=∫∂X|T(f)*-T(g)*|2θ(|T(f)*2m-T(g)*2m|)dσ.
As θ is bounded the function Mθ|f*-g*|2 is an integrable function dominating the integrand in the left-hand side integral in (3.12). Letting m→∞ and applying the Lebesgue theorem on dominated convergence to the left-hand side and Fatou’s lemma to the right-hand side (as θ is positive on [0,∞)) we obtain
(3.13)∫∂X|f*-g*|2θ(0)dσ≥∫∂X|T(f)*-T(g)*|2θ(0)dσ.
From this and since θ(0)=1/2 we get that the function |T(f)*-T(g)*|2 is integrable. Letting again m→∞ in (3.12) we have that
(3.14)∫∂X|f*-g*|2dσ=∫∂X|T(f)*-T(g)*|2dσ
by the Lebesgue theorem on dominated convergence now applied to both integrals in (3.12). Hence ∥f-g∥2=∥T(f)-T(g)∥2 for every pair of f,g∈H2(X). For g=0, we get ∥f∥2=∥T(f)∥2 and consequently T(H2(X))⊆H2(X), as claimed.
SinceH2(X) is a Hilbert space, it is uniformly convex. Hence by Lemma 2.1 the restriction T|H2(X) is real-linear. Since the operations of scalar multiplication and addition on N* are continuous and H2(X) is dense in N*(X) we see that T is real-linear on N*(X).
First assumeT(i)=i. As T is real-linear and multiplicative, T is complex-linear in this case. Then by [3, Theorem 2.2] and since T(1)=1, there is an inner map ψ such that T(f)=f∘ψ for every f∈N*(X).
Now assumeT(i)=-i. Let T̃:N*(X)→N*(X) be defined as T̃(f)=T(f̃) for every f∈N*(X), where
(3.15)f̃(z1,…,zn)=f(z¯1,…,z¯n)¯
for f∈N*(X). Then T̃ is well defined and a complex-linear isometry from N*(X) into itself. Again by [3, Theorem 2.2] we have that there is an inner map ψ on X whose boundary map ψ* is measure preserving such that T̃(f)=f∘ψ for every f∈N*. This implies that T(f)=f∘ψ¯¯ for every f∈N*(X).Corollary 3.2 (see [1]).
LetX∈{𝔹n,𝔻n}. Suppose that T:N*(X)→N*(X) is a (not necessarily linear) surjective multiplicative isometry. Then there is a holomorphic automorphism ψ on X such that either of the following formulas holds:
(3.16)T(f)=f∘ψforeveryf∈N*(X),T(f)=f∘ψ¯¯foreveryf∈N*(X),
where ψ is a unitary transformation for X=𝔹n, while for X=𝔻n, ψ(z1,…,zn)=(eiθ1zj1,…,eiθnzjn) for some real numbers θj for j=1,…,n and a permutation (j1,…,jn) of the integers from 1 to n.Proof.
By Theorem3.1, T is complex-linear or conjugate linear. If T is complex-linear, then the result holds by [3, Corollary 2.3]. If T is conjugate linear, then put T̃(f)=T(f̃) for f∈N*(X), where f̃ is defined as in (3.15). Then T̃(f)=f∘ψ, for every f∈N*(X), and for an inner map ψ on X whose boundary map ψ* is measure preserving. Since T̃ is a surjective isometry, the desired property of ψ again follows from [3, Corollary 2.3].


### 3.2. Multiplicative Isometries onNp(X)

The next result concerns the Privalov class.Theorem 3.3.
LetX∈{𝔹n,𝔻n} and 1<p<∞. Suppose that T:Np(X)→Np(X) is a (not necessarily linear) multiplicative isometry. Then there is an inner map ψ on X whose boundary map ψ* is measure preserving and such that either of the following formulas holds:
(3.17)T(f)=f∘ψforeveryf∈Np(X),T(f)=f∘ψ¯¯foreveryf∈Np(X).Proof.
SinceT is multiplicative we see by the same way as in the proof of Theorem 3.1 that T(0)=0, T(1)=1 and T(i)=i or T(i)=-i. Also we see that T(1/2)=1/2. It follows by the proof of Theorem 3.1 that for every pair f and g in Hp(X),
(3.18)∫∂X(ln⁡(1+|f*2m-g*2m|))pdσ=∫∂X(ln⁡(1+|T(f)*2m-T(g)*2m|))pdσ
holds. Multiplying (3.18) by 2mp and then letting m→∞ we get
(3.19)∫∂X|f*-g*|pdσ=∫∂X|T(f)*-T(g)*|pdσ.
Thus T(Hp(X))⊆Hp(X). The Hardy space Hp(X) can be seen as a subspace of Lp(∂X). Since Lp(∂X) is uniformly convex, so is Hp(X) for 1<p<∞. Then by Lemma 2.1 the operator T is real-linear on Hp(X). Since Hp(X) is a dense subspace of Np(X) we see that T is real-linear on Np(X). As we have already learnt that T(i)=i or T(i)=-i, we obtain that T is complex-linear or conjugate linear on Np(X). The rest of the proof is similar to the last part of the proof of Theorem 3.1 applying [7, Theorem 1] instead of [3, Theorem 2.2]. We omit the details.Corollary 3.4.
LetX∈{𝔹n,𝔻n} and 1<p<∞. Suppose that T:Np(X)→Np(X) is a (not necessarily linear) surjective multiplicative isometry. Then there is a holomorphic automorphism ψ on X such that either of the following formulas holds:
(3.20)T(f)=f∘ψforeveryf∈Np(X),T(f)=f∘ψ¯¯foreveryf∈Np(X),
where ψ is a unitary transformation for X=𝔹n, while for X=𝔻n, ψ(z1,…,zn)=(eiθ1zj1,…,eiθnzjn) for some real numbers θj for j=1,…,n and a permutation (j1,…,jn) of the integers from 1 to n.Proof.
By Theorem3.3, T is complex-linear or conjugate linear. If T is complex-linear, then the result follows directly from [7, Corollary and Remark 3]. If T is conjugate linear, then put T̃(f)=T(f̃) for f∈Np(X), where f̃ is defined as in (3.15). Then T̃ is a complex-linear isometric surjection from Np(X) onto itself. Hence by [7, Corollary and Remark 3] there is a desired automorphism on X such that T(f)=f∘ψ¯¯ for every f∈Np(X).


### 3.3. Multiplicative Isometries onANαp(X)

The next result concerns the Bergman-Privalov class.Theorem 3.5.
LetX∈{𝔹n,𝔻n}, 1≤p<∞ and α>-1. Suppose that T:ANαp(X)→ANαp(X) is a (not necessarily linear) multiplicative isometry. Then there is a holomorphic self-map ψ on X with the property that
(3.21)∫Xh∘ψ(z)dVα(z)=∫Xh(z)dVα(z)
for every bounded or positive Borel function h on X such that either of the following formulas holds:
(3.22)T(f)=f∘ψforeveryf∈ANαp(X),T(f)=f∘ψ¯¯foreveryf∈ANαp(X).Proof.
We can prove the theorem in a way similar to that in the proofs of Theorem3.1 for p=1 and Theorem 3.3 for 1<p<∞. For the case of p=1, instead of using the Hardy spaces H1(X) and H2(X) we make use of the weighted Bergman spaces Aα1(X) and Aα2(X). For the case of 1<p<∞, instead of using the Hardy space Hp(X) we make use of the weighted Bergman space Aαp(X). We also apply [10, Theorem 1] for X=𝔹n and [2, Theorem 2] for X=𝔻n to represent complex-linear isometries instead of [3, Theorem 2.2].Corollary 3.6 (see [2]).
LetX∈{𝔹n,𝔻n}, 1≤p<∞ and α>-1. Suppose that T:ANαp(X)→ANαp(X) is a (not necessarily linear) surjective multiplicative isometry. Then there is a holomorphic automorphism ψ on X such that either of the following formulas holds:
(3.23)T(f)=f∘ψforeveryf∈ANαp(X),T(f)=f∘ψ¯¯foreveryf∈ANαp(X),
where ψ is a unitary transformation for X=𝔹n, while for X=𝔻n, ψ(z1,…,zn)=(eiθ1zj1,…,eiθnzjn) for some real numbers θj for j=1,…,n and a permutation (j1,…,jn) of the integers from 1 to n.Proof.
By Theorem3.5, T is complex-linear or conjugate linear. Suppose that T is complex-linear. If X=𝔹n, then the conclusion follows by [10, Theorem 2], while for X=𝔻n the conclusion follows similar to the corresponding part of the proof of [2, Theorem 3]. If T is conjugate linear, then the conclusion follows from the similar argument in the proof of Corollary 3.2.


### 3.4. Isometries onNlogβN(X)

In [12] Ueki characterized complex-linear isometries on the Zygmund F-algebra on 𝔹n. For 𝔻n the following result is proved similar to [12, Theorem 1]. Hence it is omitted.Theorem 3.7.
Letβ>0. If T is a complex-linear isometry of NlogβN(𝔻n) into itself, then there exist an inner function Ψ and an inner map ψ on 𝔻n whose boundary map ψ* is measure preserving on 𝕋n such that
(3.24)T(f)=ΨCψ(f)=Ψ(f∘ψ)foreveryf∈Nlog⁡βN(Dn).
Conversely, for given such Ψ and ψ, the weighted composition operator ΨCψ is an injective linear isometry of NlogβN(𝔻n).For the surjective isometries the result is as follows.Corollary 3.8.
An isometryT of NlogβN(𝔻n) is surjective if and only if T=aC𝒰 where a∈ℂ with |a|=1 and 𝒰(z1,…,zn)=(eiθ1zj1,…,eiθnzjn) for some real numbers θj, j=1,…,n and a permutation (j1,…,jn) of the integers from 1 to n.To prove Corollary3.8 we need the next auxiliary result.Lemma 3.9.
For any functionf∈N(𝔻n), f∈NlogβN(𝔻n) if and only if φβ(ln+|f*|)∈L1(𝕋n) and
(3.25)φβ(ln⁡+|f(z)|)≤∫TnP(z,ζ)φβ(ln⁡+|f*(ζ)|)dσ(ζ)forz∈Dn,
where P(z,ζ) denotes the Poisson kernel for 𝔻n;
(3.26)P(z,ζ)=Pr1(θ1-ϕ1)⋯Prn(θn-ϕn)
for z=(r1eiθ1,…,rneiθn), ζ=(eiϕ1,…,eiϕn) and
(3.27)Pr(θ)=1-r21-2rcos⁡θ+r2
is the Poisson kernel for the unit disk 𝔻.Proof.
Iff∈NlogβN(𝔻n), then Fatou’s lemma shows that φβ(ln+|f*|)∈L1(𝕋n). The inclusion (2.18) implies f∈N*(𝔻n), and so we see that ln+|f| has the least n-harmonic majorant. Since the least n-harmonic majorant of ln+|f| is the Poisson integral P[ln+|f*|], we obtain the following inequality:
(3.28)ln+|f(z)|≤∫TnP(z,ζ)ln+|f*(ζ)|dσ(ζ)forz∈Dn.
Note that φβ(t) is strictly increasing and convex on [0,∞), and the measures dμz(ζ)=P(z,ζ)dσ(ζ) are normalized on 𝕋n, which follows from the well-known equality
(3.29)∫TnP(z,ζ)dσ(ζ)=1.
Applying Jensen’s inequality to (3.28), we obtain the desired inequality (3.25).
Conversely we putz=rη(0≤r<1,η∈𝕋n) in (3.25). By integrating with respect to η and applying Fubini’s theorem, we have that
(3.30)∫Tnφβ(ln+|f(rη)|)dσ(η)≤∫Tnφβ(ln+|f*(ζ)|)dσ(ζ)∫TnP(rη,ζ)dσ(η).
By the symmetric property P(rη,ζ)=P(rζ,η) and the normalization property of the Poisson kernel, we obtain that
(3.31)sup⁡0≤r<1∫Tnφβ(ln+|f(rη)|)dσ(η)≤∫Tnφβ(ln+|f*(ζ)|)dσ(ζ).
Hence the condition φβ(ln+|f*|)∈L1(𝕋n) implies that f∈NlogβN(𝔻n).Now we give a proof of Corollary3.8.Proof of Corollary3.8.
Suppose thatT is surjective. Then Theorem 3.7 gives that T=ΨCψ. A standard argument shows that ψ is an automorphism of 𝔻n. So there are conformal maps φj (j=1,…,n) of 𝔻 onto 𝔻 and there is a permutation (j1,…,jn) of the integers from 1 to n such that
(3.32)ψ(z1,…,zn)=(φ1(zj1),…,φn(zjn)).
The mean value theorem shows that
(3.33)∫Tnφk(ζjk)dσ(ζ)=∫Tφk(ζjk)dσ1(ζjk)=φk(0)
for each k∈{1,…,n}. Here dσ1 denotes the one-dimensional normalized Lebesgue measure on the unit circle 𝕋.
On the other hand, the measure-preserving property ofψ* gives that
(3.34)∫Tnφk(ζjk)dσ(ζ)=∫Tn〈ψ*(ζ),ek〉dσ(ζ)=∫Tn〈ζ,ek〉dσ(ζ)=∫Tnζkdσ(ζ)=0.
By (3.33) and (3.34) we see that ψ fixes the origin, and so each φk is the rotation transform.
Next we prove thatΨ is a unimodular constant. If f∈NlogβN(𝔻n) is such that 1=T(f)=ΨCψ(f), then 1/Ψ=f∘ψ∈NlogβN(𝔻n). Inequality (3.25) in Lemma 3.9 gives that
(3.35)φβ(ln⁡+1|Ψ(z)|)≤∫TnP(z,ζ)φβ(ln+1|Ψ*(ζ)|)dσ(ζ)=0,
and so we have 1/|Ψ|≤1 on 𝔻n. Since Ψ is inner, Ψ is a unimodular constant.Now we show results on multiplicative isometries on the ZygmundF-algebras on 𝔹n and 𝔻n.Theorem 3.10.
LetX∈{𝔹n,𝔻n}. Suppose that T:NlogβN(X)→NlogβN(X) is a (not necessarily linear) multiplicative isometry. Then there exists an inner map ψ on X whose boundary map ψ* is measure preserving on ∂X, such that either of the following formulas holds:
(3.36)T(f)=f∘ψforeveryf∈Nlog⁡βN(X),T(f)=f∘ψ¯¯foreveryf∈Nlog⁡βN(X).Note that multiplicative isometries of the Privalov class and the ZygmundF-algebra have the same form as multiplicative isometries of the Smirnov class.Proof of Theorem3.10.
AsT is multiplicative we obtain T(1)=1, T(0)=0, T(-1)=-1 and T(i)=i or T(i)=-i. Since
(3.37)lim⁡n→∞φβ(((1+ɛ)/2)n)φβ((1/2n))=∞
holds for every ɛ>0, the equation T(1/2)=1/2 is proved similarly as in Theorem 3.1.
Letf,g∈H1(X). Then we can prove that
(3.38)∫∂X2mφβ(ln⁡(1+|f*2m-g*2m|))dσ=∫∂X2mφβ(ln⁡(1+|T(f)*2m-T(g)*2m|))dσ,
following the lines of the corresponding part of the proof in Theorem 3.1. By some calculation we see that
(3.39)φβ(ln⁡(1+x))≤(ln⁡γβ)βx
holds for every x≥0. Hence we get
(3.40)2mφβ(ln⁡(1+|f*2m-g*2m|))≤(ln⁡γβ)β|f*-g*|,
almost everywhere on ∂X and (lnγβ)β|f*-g*| is an integrable function dominating 2mφβ(ln(1+|(f*/2m)-(g*/2m)|)). We get
(3.41)lim⁡m→∞∫∂X2mφβ(ln⁡(1+|f*2m-g*2m|))dσ=(ln⁡γβ)β∫∂X|f*-g*|dσ
by the Lebesgue dominated convergence theorem since
(3.42)lim⁡m→∞2mφβ(ln⁡(1+|f*2m-g*2m|))=(ln⁡γβ)β|f*-g*|.
On the other hand, applying Fatou’s lemma we get
(3.43)(ln⁡γβ)β∫∂X|T(f)*-T(g)*|dσ≤lim⁡⁡inf⁡m→∞∫∂X2mφβ(ln⁡(1+|T(f)*2m-T(g)*2m|))dσ=lim⁡⁡inf⁡m→∞∫∂X2mφβ(ln⁡(1+|f*2m-g*2m|))dσ=(ln⁡γβ)β∫∂X|f*-g*|dσ<∞,
from which for g=0 we get T(H1(X))⊆H1(X). Since
(3.44)2mφβ(ln⁡(1+|T(f)*2m-T(g)*2m|))≤(ln⁡γβ)β|T(f)*-T(g)*|
follows from (3.40), the function (lnγβ)β|T(f)*-T(g)*| is an integrable function dominating 2mφβ(ln(1+|T(f)*2m-T(g)*2m|)). Hence
(3.45)(ln⁡γβ)β∫∂X|T(f)*-T(g)*|dσ=lim⁡m→∞∫∂X2mφβ(ln⁡(1+|T(f)*2m-T(g)*2m|))dσ
holds by the Lebesgue dominated convergence theorem. Consequently
(3.46)∫∂X|f*-g*|dσ=∫∂X|T(f)*-T(g)*|dσ
holds. As f and g are arbitrary elements of H1(X) we obtain that T|H1(X) is isometric on H1(X) with respect to the metric induced by the H1-norm.
We also obtain that there exists a bounded positive continuous functionθ1 on [0,∞) such that θ1(0)≠0 and
(3.47)x2θ1(x)={ln⁡γβ}βx-φβ(ln⁡(1+x)).
Applying this equality we obtain that T(H2(X))⊆H2(X) and T|H2(X) is a real-linear isometry on H2(X), hence T is a complex-linear (if T(i)=i) or conjugate linear isometry (if T(i)=-i) on NlogβN(X), similar as in the proof of Theorem 3.1. The rest of the proof is similar to the last part of the proof of Theorem 3.1 applying [12, Theorem 1] for X=𝔹n and Theorem 3.7 for X=𝔻n instead of [3, Theorem 2.2]. We omit the details.Corollary 3.11.
LetX∈{𝔹n,𝔻n}. Suppose that T:NlogβN(X)→NlogβN(X) is a (not necessarily linear) surjective multiplicative isometry. Then there exists a holomorphic automorphism ψ on X such that either of the following formulas holds:
(3.48)T(f)=f∘ψforeveryf∈Nlog⁡βN(X),T(f)=f∘ψ¯¯foreveryf∈Nlog⁡βN(X),
where ψ is a unitary transformation for X=𝔹n, while for X=𝔻n, ψ(z1,…,zn)=(eiθ1zj1,…,eiθnzjn) for some real numbers θj, j=1,…,n and a permutation (j1,…,jn) of the integers from 1 to n.Note that surjective multiplicative isometries of the Privalov class, the Bergman-Privalov class, and the ZygmundF-algebra have the same form as surjective multiplicative isometries of the Smirnov class.Proof of Corollary3.11.
By Theorem3.10, T is complex-linear or conjugate linear. Suppose that T is complex-linear. Applying [12, Corollary 1] for X=𝔹n and Corollary 3.8 for X=𝔻n the result follows in this case. If T is conjugate linear, then the result follows by similar arguments as in the proof of Corollary 3.2.


## 3.1.  Multiplicative Isometries onN*(X)

Our first result concerns the Smirnov class.Theorem 3.1.
LetX∈{𝔹n,𝔻n}. Suppose that T:N*(X)→N*(X) is a (not necessarily linear) multiplicative isometry. Then there is an inner map ψ on X whose boundary map ψ* is measure preserving and such that either of the following formulas holds:
(3.1)T(f)=f∘ψforeveryf∈N*(X),T(f)=f∘ψ¯¯foreveryf∈N*(X).Proof.
First we claim thatT(1)=1. Since T(1)=T(1)2 and T(1) is a holomorphic function on the connected open set X we get T(1)=0 or T(1)=1. But T(1)=0 is impossible because if it were T(1)=0, then 0=T(f)T(1)=T(f), for each f∈N*(X), which contradicts with the assumption that T is an isometry. As T(0)=T(0)2 and T is injective, we obtain T(0)=0. Similarly T(-1)=-1 is also observed by making use of T(-1)2=T(1)=1. Then T(i)2=T(i2)=-1 assert that T(i)=i or T(i)=-i. If T(i)=i, then the first formula of the conclusion will follow and the second one will follow from T(i)=-i.
Next we showT(1/2)=1/2. Put r=1/2. Suppose that |T(r)*|>r on a set of positive measure on ∂X. Then there exists a subset E of positive measure and ɛ>0 with |T(r)*|≥(1+ɛ)r on E. Since
(3.2)lim⁡n→∞ln⁡⁡(1+(1+ɛ)nrn)ln⁡(1+rn)=∞,
there is a positive integer n0 such that
(3.3)∫Eln⁡⁡(1+(1+ɛ)n0rn0)dσ>∫∂Xln⁡(1+rn0)dσ.
From this and since T is a multiplicative isometry on N*(X) we have that
(3.4)∫∂Xln⁡(1+rn0)dσ=∫∂Xln⁡⁡(1+|T(r)*|n0)dσ≥∫Eln⁡⁡(1+(1+ɛ)n0rn0)dσ>∫∂Xln⁡(1+rn0)dσ,
which is a contradiction proving |T(r)*|≤r almost everywhere on ∂X. Hence |T(1/r)*|≥1/r holds almost everywhere on ∂X as T(r)T(1/r)=T(1)=1 almost everywhere on ∂X. Since
(3.5)ln⁡(1+1r)=∫∂Xln⁡(1+1r)dσ=∫∂Xln⁡(1+|T(1r)*|)dσ,
we have that |T(1/r)*|=1/r and |T(r)*|=r almost everywhere on ∂X.
Sinceln(1+(1-r))=d(r,1)=d(T(r),1) and
(3.6)d(T(r),1)=∫∂Xln⁡⁡(1+|1-T(r)*|)dσ,
it is easy to check that T(1/2)*=1/2 almost everywhere on ∂X. Hence T(1/2)=1/2 holds. As T is multiplicative, T is 1/2-homogeneous in the sense that T(f/2)=T(f)/2 holds for every f∈N*(X).
Letf,g∈H1(X). It requires only elementary calculation applying the 1/2-homogeneity of T to check that
(3.7)∫∂Xln⁡(1+|f*2m-g*2m|)dσ=∫∂Xln⁡(1+|T(f)*2m-T(g)*2m|)dσ
holds. Multiplying (3.7) by 2m and then letting m→∞ we get
(3.8)∫∂X|f*-g*|dσ=∫∂X|T(f)*-T(g)*|dσ
by the monotone convergence theorem, since 2mln(1+(t/2m)) nondecreases monotonically to t as m→∞ for any t≥0, which can be easily proved by considering the function gt(x)=xln(1+(t/x)). From (3.8) for g=0, we obtain T(H1(X))⊆H1(X) and the restricted map T|H1(X) is an isometry with respect to the metric induced by the H1-norm ∥·∥1.
Let the functionθ on the interval [0,∞) be defined as
(3.9)θ(x)={12,x=0x-ln⁡(1+x)x2,x>0.
It is easy to check that θ is positive and continuous on [0,∞) and limx→∞θ(x)=0. Hence θ is bounded on [0,∞), so that
(3.10)Mθ∶=sup⁡x≥0θ(x)<∞.
We claim that the inclusionT(H2(X))⊆H2(X) and T|H2(X) is isometric with respect to the metric induced by the H2-norm. For this purpose let f,g∈H2(X). Now note that since H2(X)⊂H1(X), equality (3.7) holds and as well as the next equality
(3.11)∫∂X|f*2m-g*2m|dσ=∫∂X|T(f)*2m-T(g)*2m|dσ.
By subtracting (3.7) from (3.11) and then multiplying such obtained equation by 2m we obtain
(3.12)∫∂X|f*-g*|2θ(|f*2m-g*2m|)dσ=∫∂X|T(f)*-T(g)*|2θ(|T(f)*2m-T(g)*2m|)dσ.
As θ is bounded the function Mθ|f*-g*|2 is an integrable function dominating the integrand in the left-hand side integral in (3.12). Letting m→∞ and applying the Lebesgue theorem on dominated convergence to the left-hand side and Fatou’s lemma to the right-hand side (as θ is positive on [0,∞)) we obtain
(3.13)∫∂X|f*-g*|2θ(0)dσ≥∫∂X|T(f)*-T(g)*|2θ(0)dσ.
From this and since θ(0)=1/2 we get that the function |T(f)*-T(g)*|2 is integrable. Letting again m→∞ in (3.12) we have that
(3.14)∫∂X|f*-g*|2dσ=∫∂X|T(f)*-T(g)*|2dσ
by the Lebesgue theorem on dominated convergence now applied to both integrals in (3.12). Hence ∥f-g∥2=∥T(f)-T(g)∥2 for every pair of f,g∈H2(X). For g=0, we get ∥f∥2=∥T(f)∥2 and consequently T(H2(X))⊆H2(X), as claimed.
SinceH2(X) is a Hilbert space, it is uniformly convex. Hence by Lemma 2.1 the restriction T|H2(X) is real-linear. Since the operations of scalar multiplication and addition on N* are continuous and H2(X) is dense in N*(X) we see that T is real-linear on N*(X).
First assumeT(i)=i. As T is real-linear and multiplicative, T is complex-linear in this case. Then by [3, Theorem 2.2] and since T(1)=1, there is an inner map ψ such that T(f)=f∘ψ for every f∈N*(X).
Now assumeT(i)=-i. Let T̃:N*(X)→N*(X) be defined as T̃(f)=T(f̃) for every f∈N*(X), where
(3.15)f̃(z1,…,zn)=f(z¯1,…,z¯n)¯
for f∈N*(X). Then T̃ is well defined and a complex-linear isometry from N*(X) into itself. Again by [3, Theorem 2.2] we have that there is an inner map ψ on X whose boundary map ψ* is measure preserving such that T̃(f)=f∘ψ for every f∈N*. This implies that T(f)=f∘ψ¯¯ for every f∈N*(X).Corollary 3.2 (see [1]).
LetX∈{𝔹n,𝔻n}. Suppose that T:N*(X)→N*(X) is a (not necessarily linear) surjective multiplicative isometry. Then there is a holomorphic automorphism ψ on X such that either of the following formulas holds:
(3.16)T(f)=f∘ψforeveryf∈N*(X),T(f)=f∘ψ¯¯foreveryf∈N*(X),
where ψ is a unitary transformation for X=𝔹n, while for X=𝔻n, ψ(z1,…,zn)=(eiθ1zj1,…,eiθnzjn) for some real numbers θj for j=1,…,n and a permutation (j1,…,jn) of the integers from 1 to n.Proof.
By Theorem3.1, T is complex-linear or conjugate linear. If T is complex-linear, then the result holds by [3, Corollary 2.3]. If T is conjugate linear, then put T̃(f)=T(f̃) for f∈N*(X), where f̃ is defined as in (3.15). Then T̃(f)=f∘ψ, for every f∈N*(X), and for an inner map ψ on X whose boundary map ψ* is measure preserving. Since T̃ is a surjective isometry, the desired property of ψ again follows from [3, Corollary 2.3].


## 3.2. Multiplicative Isometries onNp(X)

The next result concerns the Privalov class.Theorem 3.3.
LetX∈{𝔹n,𝔻n} and 1<p<∞. Suppose that T:Np(X)→Np(X) is a (not necessarily linear) multiplicative isometry. Then there is an inner map ψ on X whose boundary map ψ* is measure preserving and such that either of the following formulas holds:
(3.17)T(f)=f∘ψforeveryf∈Np(X),T(f)=f∘ψ¯¯foreveryf∈Np(X).Proof.
SinceT is multiplicative we see by the same way as in the proof of Theorem 3.1 that T(0)=0, T(1)=1 and T(i)=i or T(i)=-i. Also we see that T(1/2)=1/2. It follows by the proof of Theorem 3.1 that for every pair f and g in Hp(X),
(3.18)∫∂X(ln⁡(1+|f*2m-g*2m|))pdσ=∫∂X(ln⁡(1+|T(f)*2m-T(g)*2m|))pdσ
holds. Multiplying (3.18) by 2mp and then letting m→∞ we get
(3.19)∫∂X|f*-g*|pdσ=∫∂X|T(f)*-T(g)*|pdσ.
Thus T(Hp(X))⊆Hp(X). The Hardy space Hp(X) can be seen as a subspace of Lp(∂X). Since Lp(∂X) is uniformly convex, so is Hp(X) for 1<p<∞. Then by Lemma 2.1 the operator T is real-linear on Hp(X). Since Hp(X) is a dense subspace of Np(X) we see that T is real-linear on Np(X). As we have already learnt that T(i)=i or T(i)=-i, we obtain that T is complex-linear or conjugate linear on Np(X). The rest of the proof is similar to the last part of the proof of Theorem 3.1 applying [7, Theorem 1] instead of [3, Theorem 2.2]. We omit the details.Corollary 3.4.
LetX∈{𝔹n,𝔻n} and 1<p<∞. Suppose that T:Np(X)→Np(X) is a (not necessarily linear) surjective multiplicative isometry. Then there is a holomorphic automorphism ψ on X such that either of the following formulas holds:
(3.20)T(f)=f∘ψforeveryf∈Np(X),T(f)=f∘ψ¯¯foreveryf∈Np(X),
where ψ is a unitary transformation for X=𝔹n, while for X=𝔻n, ψ(z1,…,zn)=(eiθ1zj1,…,eiθnzjn) for some real numbers θj for j=1,…,n and a permutation (j1,…,jn) of the integers from 1 to n.Proof.
By Theorem3.3, T is complex-linear or conjugate linear. If T is complex-linear, then the result follows directly from [7, Corollary and Remark 3]. If T is conjugate linear, then put T̃(f)=T(f̃) for f∈Np(X), where f̃ is defined as in (3.15). Then T̃ is a complex-linear isometric surjection from Np(X) onto itself. Hence by [7, Corollary and Remark 3] there is a desired automorphism on X such that T(f)=f∘ψ¯¯ for every f∈Np(X).


## 3.3. Multiplicative Isometries onANαp(X)

The next result concerns the Bergman-Privalov class.Theorem 3.5.
LetX∈{𝔹n,𝔻n}, 1≤p<∞ and α>-1. Suppose that T:ANαp(X)→ANαp(X) is a (not necessarily linear) multiplicative isometry. Then there is a holomorphic self-map ψ on X with the property that
(3.21)∫Xh∘ψ(z)dVα(z)=∫Xh(z)dVα(z)
for every bounded or positive Borel function h on X such that either of the following formulas holds:
(3.22)T(f)=f∘ψforeveryf∈ANαp(X),T(f)=f∘ψ¯¯foreveryf∈ANαp(X).Proof.
We can prove the theorem in a way similar to that in the proofs of Theorem3.1 for p=1 and Theorem 3.3 for 1<p<∞. For the case of p=1, instead of using the Hardy spaces H1(X) and H2(X) we make use of the weighted Bergman spaces Aα1(X) and Aα2(X). For the case of 1<p<∞, instead of using the Hardy space Hp(X) we make use of the weighted Bergman space Aαp(X). We also apply [10, Theorem 1] for X=𝔹n and [2, Theorem 2] for X=𝔻n to represent complex-linear isometries instead of [3, Theorem 2.2].Corollary 3.6 (see [2]).
LetX∈{𝔹n,𝔻n}, 1≤p<∞ and α>-1. Suppose that T:ANαp(X)→ANαp(X) is a (not necessarily linear) surjective multiplicative isometry. Then there is a holomorphic automorphism ψ on X such that either of the following formulas holds:
(3.23)T(f)=f∘ψforeveryf∈ANαp(X),T(f)=f∘ψ¯¯foreveryf∈ANαp(X),
where ψ is a unitary transformation for X=𝔹n, while for X=𝔻n, ψ(z1,…,zn)=(eiθ1zj1,…,eiθnzjn) for some real numbers θj for j=1,…,n and a permutation (j1,…,jn) of the integers from 1 to n.Proof.
By Theorem3.5, T is complex-linear or conjugate linear. Suppose that T is complex-linear. If X=𝔹n, then the conclusion follows by [10, Theorem 2], while for X=𝔻n the conclusion follows similar to the corresponding part of the proof of [2, Theorem 3]. If T is conjugate linear, then the conclusion follows from the similar argument in the proof of Corollary 3.2.


## 3.4. Isometries onNlogβN(X)

In [12] Ueki characterized complex-linear isometries on the Zygmund F-algebra on 𝔹n. For 𝔻n the following result is proved similar to [12, Theorem 1]. Hence it is omitted.Theorem 3.7.
Letβ>0. If T is a complex-linear isometry of NlogβN(𝔻n) into itself, then there exist an inner function Ψ and an inner map ψ on 𝔻n whose boundary map ψ* is measure preserving on 𝕋n such that
(3.24)T(f)=ΨCψ(f)=Ψ(f∘ψ)foreveryf∈Nlog⁡βN(Dn).
Conversely, for given such Ψ and ψ, the weighted composition operator ΨCψ is an injective linear isometry of NlogβN(𝔻n).For the surjective isometries the result is as follows.Corollary 3.8.
An isometryT of NlogβN(𝔻n) is surjective if and only if T=aC𝒰 where a∈ℂ with |a|=1 and 𝒰(z1,…,zn)=(eiθ1zj1,…,eiθnzjn) for some real numbers θj, j=1,…,n and a permutation (j1,…,jn) of the integers from 1 to n.To prove Corollary3.8 we need the next auxiliary result.Lemma 3.9.
For any functionf∈N(𝔻n), f∈NlogβN(𝔻n) if and only if φβ(ln+|f*|)∈L1(𝕋n) and
(3.25)φβ(ln⁡+|f(z)|)≤∫TnP(z,ζ)φβ(ln⁡+|f*(ζ)|)dσ(ζ)forz∈Dn,
where P(z,ζ) denotes the Poisson kernel for 𝔻n;
(3.26)P(z,ζ)=Pr1(θ1-ϕ1)⋯Prn(θn-ϕn)
for z=(r1eiθ1,…,rneiθn), ζ=(eiϕ1,…,eiϕn) and
(3.27)Pr(θ)=1-r21-2rcos⁡θ+r2
is the Poisson kernel for the unit disk 𝔻.Proof.
Iff∈NlogβN(𝔻n), then Fatou’s lemma shows that φβ(ln+|f*|)∈L1(𝕋n). The inclusion (2.18) implies f∈N*(𝔻n), and so we see that ln+|f| has the least n-harmonic majorant. Since the least n-harmonic majorant of ln+|f| is the Poisson integral P[ln+|f*|], we obtain the following inequality:
(3.28)ln+|f(z)|≤∫TnP(z,ζ)ln+|f*(ζ)|dσ(ζ)forz∈Dn.
Note that φβ(t) is strictly increasing and convex on [0,∞), and the measures dμz(ζ)=P(z,ζ)dσ(ζ) are normalized on 𝕋n, which follows from the well-known equality
(3.29)∫TnP(z,ζ)dσ(ζ)=1.
Applying Jensen’s inequality to (3.28), we obtain the desired inequality (3.25).
Conversely we putz=rη(0≤r<1,η∈𝕋n) in (3.25). By integrating with respect to η and applying Fubini’s theorem, we have that
(3.30)∫Tnφβ(ln+|f(rη)|)dσ(η)≤∫Tnφβ(ln+|f*(ζ)|)dσ(ζ)∫TnP(rη,ζ)dσ(η).
By the symmetric property P(rη,ζ)=P(rζ,η) and the normalization property of the Poisson kernel, we obtain that
(3.31)sup⁡0≤r<1∫Tnφβ(ln+|f(rη)|)dσ(η)≤∫Tnφβ(ln+|f*(ζ)|)dσ(ζ).
Hence the condition φβ(ln+|f*|)∈L1(𝕋n) implies that f∈NlogβN(𝔻n).Now we give a proof of Corollary3.8.Proof of Corollary3.8.
Suppose thatT is surjective. Then Theorem 3.7 gives that T=ΨCψ. A standard argument shows that ψ is an automorphism of 𝔻n. So there are conformal maps φj (j=1,…,n) of 𝔻 onto 𝔻 and there is a permutation (j1,…,jn) of the integers from 1 to n such that
(3.32)ψ(z1,…,zn)=(φ1(zj1),…,φn(zjn)).
The mean value theorem shows that
(3.33)∫Tnφk(ζjk)dσ(ζ)=∫Tφk(ζjk)dσ1(ζjk)=φk(0)
for each k∈{1,…,n}. Here dσ1 denotes the one-dimensional normalized Lebesgue measure on the unit circle 𝕋.
On the other hand, the measure-preserving property ofψ* gives that
(3.34)∫Tnφk(ζjk)dσ(ζ)=∫Tn〈ψ*(ζ),ek〉dσ(ζ)=∫Tn〈ζ,ek〉dσ(ζ)=∫Tnζkdσ(ζ)=0.
By (3.33) and (3.34) we see that ψ fixes the origin, and so each φk is the rotation transform.
Next we prove thatΨ is a unimodular constant. If f∈NlogβN(𝔻n) is such that 1=T(f)=ΨCψ(f), then 1/Ψ=f∘ψ∈NlogβN(𝔻n). Inequality (3.25) in Lemma 3.9 gives that
(3.35)φβ(ln⁡+1|Ψ(z)|)≤∫TnP(z,ζ)φβ(ln+1|Ψ*(ζ)|)dσ(ζ)=0,
and so we have 1/|Ψ|≤1 on 𝔻n. Since Ψ is inner, Ψ is a unimodular constant.Now we show results on multiplicative isometries on the ZygmundF-algebras on 𝔹n and 𝔻n.Theorem 3.10.
LetX∈{𝔹n,𝔻n}. Suppose that T:NlogβN(X)→NlogβN(X) is a (not necessarily linear) multiplicative isometry. Then there exists an inner map ψ on X whose boundary map ψ* is measure preserving on ∂X, such that either of the following formulas holds:
(3.36)T(f)=f∘ψforeveryf∈Nlog⁡βN(X),T(f)=f∘ψ¯¯foreveryf∈Nlog⁡βN(X).Note that multiplicative isometries of the Privalov class and the ZygmundF-algebra have the same form as multiplicative isometries of the Smirnov class.Proof of Theorem3.10.
AsT is multiplicative we obtain T(1)=1, T(0)=0, T(-1)=-1 and T(i)=i or T(i)=-i. Since
(3.37)lim⁡n→∞φβ(((1+ɛ)/2)n)φβ((1/2n))=∞
holds for every ɛ>0, the equation T(1/2)=1/2 is proved similarly as in Theorem 3.1.
Letf,g∈H1(X). Then we can prove that
(3.38)∫∂X2mφβ(ln⁡(1+|f*2m-g*2m|))dσ=∫∂X2mφβ(ln⁡(1+|T(f)*2m-T(g)*2m|))dσ,
following the lines of the corresponding part of the proof in Theorem 3.1. By some calculation we see that
(3.39)φβ(ln⁡(1+x))≤(ln⁡γβ)βx
holds for every x≥0. Hence we get
(3.40)2mφβ(ln⁡(1+|f*2m-g*2m|))≤(ln⁡γβ)β|f*-g*|,
almost everywhere on ∂X and (lnγβ)β|f*-g*| is an integrable function dominating 2mφβ(ln(1+|(f*/2m)-(g*/2m)|)). We get
(3.41)lim⁡m→∞∫∂X2mφβ(ln⁡(1+|f*2m-g*2m|))dσ=(ln⁡γβ)β∫∂X|f*-g*|dσ
by the Lebesgue dominated convergence theorem since
(3.42)lim⁡m→∞2mφβ(ln⁡(1+|f*2m-g*2m|))=(ln⁡γβ)β|f*-g*|.
On the other hand, applying Fatou’s lemma we get
(3.43)(ln⁡γβ)β∫∂X|T(f)*-T(g)*|dσ≤lim⁡⁡inf⁡m→∞∫∂X2mφβ(ln⁡(1+|T(f)*2m-T(g)*2m|))dσ=lim⁡⁡inf⁡m→∞∫∂X2mφβ(ln⁡(1+|f*2m-g*2m|))dσ=(ln⁡γβ)β∫∂X|f*-g*|dσ<∞,
from which for g=0 we get T(H1(X))⊆H1(X). Since
(3.44)2mφβ(ln⁡(1+|T(f)*2m-T(g)*2m|))≤(ln⁡γβ)β|T(f)*-T(g)*|
follows from (3.40), the function (lnγβ)β|T(f)*-T(g)*| is an integrable function dominating 2mφβ(ln(1+|T(f)*2m-T(g)*2m|)). Hence
(3.45)(ln⁡γβ)β∫∂X|T(f)*-T(g)*|dσ=lim⁡m→∞∫∂X2mφβ(ln⁡(1+|T(f)*2m-T(g)*2m|))dσ
holds by the Lebesgue dominated convergence theorem. Consequently
(3.46)∫∂X|f*-g*|dσ=∫∂X|T(f)*-T(g)*|dσ
holds. As f and g are arbitrary elements of H1(X) we obtain that T|H1(X) is isometric on H1(X) with respect to the metric induced by the H1-norm.
We also obtain that there exists a bounded positive continuous functionθ1 on [0,∞) such that θ1(0)≠0 and
(3.47)x2θ1(x)={ln⁡γβ}βx-φβ(ln⁡(1+x)).
Applying this equality we obtain that T(H2(X))⊆H2(X) and T|H2(X) is a real-linear isometry on H2(X), hence T is a complex-linear (if T(i)=i) or conjugate linear isometry (if T(i)=-i) on NlogβN(X), similar as in the proof of Theorem 3.1. The rest of the proof is similar to the last part of the proof of Theorem 3.1 applying [12, Theorem 1] for X=𝔹n and Theorem 3.7 for X=𝔻n instead of [3, Theorem 2.2]. We omit the details.Corollary 3.11.
LetX∈{𝔹n,𝔻n}. Suppose that T:NlogβN(X)→NlogβN(X) is a (not necessarily linear) surjective multiplicative isometry. Then there exists a holomorphic automorphism ψ on X such that either of the following formulas holds:
(3.48)T(f)=f∘ψforeveryf∈Nlog⁡βN(X),T(f)=f∘ψ¯¯foreveryf∈Nlog⁡βN(X),
where ψ is a unitary transformation for X=𝔹n, while for X=𝔻n, ψ(z1,…,zn)=(eiθ1zj1,…,eiθnzjn) for some real numbers θj, j=1,…,n and a permutation (j1,…,jn) of the integers from 1 to n.Note that surjective multiplicative isometries of the Privalov class, the Bergman-Privalov class, and the ZygmundF-algebra have the same form as surjective multiplicative isometries of the Smirnov class.Proof of Corollary3.11.
By Theorem3.10, T is complex-linear or conjugate linear. Suppose that T is complex-linear. Applying [12, Corollary 1] for X=𝔹n and Corollary 3.8 for X=𝔻n the result follows in this case. If T is conjugate linear, then the result follows by similar arguments as in the proof of Corollary 3.2.



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