# Multiplicative Isometries on ClassesMp(X) of Holomorphic Functions

Authors: Yasuo Iida; Kazuhiro Kasuga
Journal: Journal of Function Spaces
 (2015)

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## Abstract

In (Iida and Kasuga 2013), the authors described multiplicative (but not necessarily linear) isometries ofMp(X) onto Mp(X) in the case of positive integer p∈N, where Mp(X)  (p≥1) is included in the Smirnov class N∗(X). In this paper, we will generalize the result to arbitrary (not necessarily positive integer) value of the exponents 0<p<∞.


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## Body

## 1. Introduction

Letn be a positive integer. The space of n-complex variables z=(z1,…,zn) is denoted by Cn. The unit polydisk {z∈Cn:|zj|<1, 1≤j≤n} is denoted by Un and the distinguished boundary Tn is {z∈Cn:|zj|=1, 1≤j≤n}. The unit ball {z∈Cn:∑j=1n‍|zj|2<1} is denoted by Bn and Sn is its boundary. In this paper X denotes the unit polydisk or the unit ball for n≥1 and ∂X denotes Tn for X=Un or Sn for X=Bn. The normalized (in the sense that σ(∂X)=1) Lebesgue measure on ∂X is denoted by dσ.The Hardy space onX is denoted by Hq(X)(0<q≤∞) and ·q denotes the norm on Hq(X)(1≤q≤∞).The Nevanlinna classN(X) on X is defined as the set of all holomorphic functions f on X such that(1)sup0≤r<1⁡∫∂Xlog⁡1+frzdσz<∞holds. It is known that f∈N(X) has a finite nontangential limit, also denoted by f, almost everywhere on ∂X.The Smirnov classN∗(X) is defined as the set of all f∈N(X) which satisfy the equality(2)sup0≤r<1⁡∫∂Xlog⁡1+frzdσz=∫∂Xlog⁡1+fzdσz.Define a metric(3)dN∗Xf,g=∫∂Xlog⁡1+fz-gzdσzfor f,g∈N∗(X). With the metric dN∗(X)(·,·) the Smirnov class N∗(X) is an F-algebra. Recall that an F-algebra is a topological algebra in which the topology arises from a complete metric. Complex-linear isometries on the Smirnov class are characterized by Stephenson in [1].The Privalov classNp(X), 1<p<∞, is defined as the set of all holomorphic functions f on X such that(4)sup0≤r<1⁡∫∂Xlog⁡1+frzpdσz<∞holds. It is well-known that Np(X) is a subalgebra of N∗(X); hence, every f∈Np(X) has a finite nontangential limit almost everywhere on ∂X. Under the metric defined by(5)dNpXf,g=∫∂Xlog⁡1+fz-gzpdσz1/pfor f,g∈Np(X), Np(X) becomes an F-algebra (cf. [2]). Complex-linear isometries on Np(X) are investigated by Iida and Mochizuki [3] for one-dimensional case and by Subbotin [2, 4] for a general case.Now we define the classMp(X). For 0<p<∞, the class Mp(X) is defined as the set of all holomorphic functions f on X such that(6)∫∂Xlog⁡1+sup0≤r<1⁡frzpdσz<∞.The class Mp(X) with p=1 in the case where n=1 was introduced by Kim in [5]. As for p>0 and n>1, the class was considered in [6, 7]. For f,g∈Mp(X), define a metric(7)dMpXf,g=∫∂Xlog⁡1+sup0≤r<1⁡frz-grzpdσzαp/p,where αp=min(1,p). With this metric Mp(X) is also an F-algebra (see [2]). Complex-linear surjective isometries on Mp(X) are investigated by Subbotin [2, 4, 8].It is well-known that the following inclusion relations hold:(8)HqX⊊NpX⊊M1X⊊N∗X0<q≤∞,p>1.Moreover, it is known that N(X)⊊Mp(X)(0<p<1) [8]. As shown in [6], for any p>1 the class Mp(X) coincides with the class Np(X) and the metrics dMp(X) and dNp(X) are equivalent. Therefore, the topologies induced by these metrics are identical on the set Mp(X)=Np(X). But we note that in [4, Theorems 1 and 4] it is implied that the sets of linear isometries on Mp(X) and Np(X) are different. It is known that H∞(X) is a dense subalgebra of Mp(X). The convergence in the metric is stronger than uniform convergence on compact subsets of X.In [9], the authors described multiplicative (but not necessarily linear) isometries of Mp(X) onto Mp(X) in the case of positive integer p∈N. In this paper, we will generalize the result to arbitrary (not necessarily positive integer) value of the exponents 0<p<∞.


## 2. The Results

Proposition 1.
Letn be a positive integer and let X be either Bn or Un. Let 0<p<∞ and suppose that A:Mp(X)→Mp(X) is a surjective isometry. If A is 2-homogeneous in the sense that A(2f)=2A(f) holds for every f∈Mp(X), then either(9)Af=αf∘Φforeveryf∈MpXor(10)Af=αf∘Φ¯¯foreveryf∈MpX,where α is a complex number with the unit modulus and for X=Bn, Φ is a unitary transformation; for X=Un, Φ(z1,…,zn)=(λ1zi1,…,λnzin), where |λj|=1, 1≤j≤n and (i1,…,in) is some permutation of the integers from 1 through n.To prove Proposition1, we need the following lemmas.Lemma 2 (see [4]).
Letf∈Hp(X), p≥1. Then the norm(11)fHmp≔∫∂Xsup0≤r<1⁡frzpdσz1/pis equivalent to the standard norm in Hp(X).Lemma 3 (see [4]).
Let0<p<∞. Then(12)limε→0+⁡1εp+1εtp-log⁡1+εtp=p2tp+1,t≥0.We recall that the functionφ(x) on the interval (0,+∞) is said to becompletely monotone if it is infinitely differentiable on (0,+∞) and(13)-1nφnx≥0,x>0,n∈Z+.Lemma 4 (see [8]).
The functions(log(1+x))/xp are completely monotone for all p>0.Lemma 5 (see [8]).
If a completely monotone functionφ(x) on (0,+∞) can be continued to an infinitely differentiable function on [0,+∞), then the inequality(14)-1nxnφx-φ0-φ′0x-⋯-φn-10n-1!xn-1≥0,x>0,holds for any n∈N and φ(n)(0)≠0 if only φ is not constant.Lemma 6.
LetC be a cone of measurable functions on a measurable space with a measure (G,μ), and let A be a mapping from C to the set of measurable functions. Suppose that A is homogeneous with positive coefficients and(15)∫Glog⁡1+Afxpμdx=∫Glog⁡1+fxpμdx,f∈C,for some p>0. Then(16)∫GAfxqμdx=∫Gfxqμdx,f∈C,for all q=p+l, where l∈Z+.Proof.
We follow [2, Lemma 2]. A is homogeneous with positive coefficients, so we have, using (15),(17)∫Glog⁡1+tAfxptpμdx=∫Glog⁡1+tfxptpμdx,f∈C,for any t>0. Letting t→0+, we obtain(18)∫GAfxpμdx=∫Gfxpμdx,f∈C.Next we argue by induction. Let k∈N and suppose that (16) holds for q=p+1, p+2,…,p+k-1. Then, for any t>0 and any function f∈C, we have(19)∫G1tp+klog⁡1+tAfxp-∑l=0k-1cltAfxp+lμdx=∫G1tp+klog⁡1+tfxp-∑l=0k-1cltfxp+lμdx,where cl are the Taylor coefficients of the function ((log(1+x))/x)p at zero.
Assume first thatf∈Lp+k(G,μ). It is easy to see that the integrand on the right-hand side of (19) converges to ck|f(x)|p+k as t→0+; this integrand is of fixed sign by Lemmas 4 and 5 and is dominated by the function Ck|f(x)|p+k with some constant Ck. By the Lebesgue theorem on dominated convergence, the right-hand side of (19) converges to the integral of ck|f(x)|p+k. Therefore, the left-hand side of (19) has a finite limit and, by the Fatou theorem, the function ck|Af(x)|p+k is integrable. Since ck≠0 for any k by Lemmas 4 and 5, we deduce that Af∈Lp+k(G,μ), and repeating the above arguments, we see that the left-hand side of (19) converges to the integral of ck|Af(x)|p+k as t→0+. Therefore, passing to the limit in (19) as t→0+ and dividing the result by ck≠0, we obtain (16) for q=p+k. The case of Af∈Lp+k(G,μ) can be considered in a similar way. For f,Af∉Lp+k(G,μ), relation (16) with q=p+k is trivial.Proof of Proposition1.
Suppose first thatp≥1. Let f,g∈Hp(X). We easily confirm that, by utilizing Lemma 2 and the celebrated theorem of Mazur and Ulam [10], AHp(X) is a real-linear isometry in a way similar to [9, Proposition 1].
Next suppose that0<p<1. We define the class Hmp(X)(0<p<1) as the set of all holomorphic functions f on X such that(20)dHmpf,0≔∫∂Xsup0≤r<1⁡frzpdσz<∞and define a metric dHmp(f,g)=dHmp(f-g,0) for f,g∈Hmp(X). If A:Mp(X)→Mp(X) is a surjective isometry and A is 2-homogeneous in the sense that A(2f)=2A(f) holds for every f∈Mp(X), we confirm that A:Hmp(X)→Hmp(X) is also a surjective isometry.
For0<p<1, let f,g∈Hp(X). We have A(f)/2m=A(f/2m)(m∈N) since A(2f)=2A(f). Then the following equality holds:(21)∫∂Xsup0≤r<1⁡frz2mpdσz-∫∂Xlog⁡1+sup0≤r<1⁡frz2mpdσz=∫∂Xsup0≤r<1⁡Afrz2mpdσz-∫∂Xlog⁡1+sup0≤r<1⁡Afrz2mpdσz.Therefore, it follows that(22)∫∂X11/2mp+112msup0≤r<1⁡frzp-log⁡1+12msup0≤r<1⁡frzpdσz=∫∂X11/2mp+112msup0≤r<1⁡Afrzp-log1+12msup0≤r<1⁡Afrzpdσz.Using the elementary inequalities log(1+xy)≤log(1+x)+log(1+y)(x≥0, y≥0) and (a+b)p≤2p(ap+bp)(a≥0, b≥0, p>0), we confirm that the integrand on the left-hand side of (22) is dominated by an L1(∂X)-function. The integrand on the right-hand side of (22) is also dominated by an L1(∂X)-function in the same way. Applying the Lebesgue theorem on dominated convergence and Lemma 3 on both sides of (22), we have the equality(23)∫∂Xp2sup0≤r<1⁡frzp+1dσz=∫∂Xp2sup0≤r<1⁡Afrzp+1dσz.Hence, we obtain(24)∫∂Xsup0≤r<1⁡frzp+1dσz=∫∂Xsup0≤r<1⁡Afrzp+1dσz.The equivalence of the norms ·p+1 and ·Hmp+1 guarantees that A:Hp+1(X)→Hp+1(X) is a surjective isometry. By using Mazur-Ulam theorem again, AHp+1(X) is a real-linear isometry since Hp+1(X) is a normed vector space and A(0)=0.
We consider an arbitrary functionf∈Mp(X)(0<p<∞) and the cone Cf≔{λMf(ζ)∣λ>0} generated by f. Here Mf(ζ)=sup0≤r<1|f(rζ)| is the radial maximal function for f. Moreover, consider the following mapping on this cone:(25)A~:λMfζ⟼λMAfζ=MAλfζ,ζ∈∂X,λ≥0.Since A is isometric with respect to the metric dMp, it follows that the assumptions of Lemma 6 hold on the cone Cf.
A
:
M
p
(
X
)
→
M
p
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X
) is a surjective isometry, so the equation (26)∫∂Xlog⁡1+sup0≤r<1⁡frzpdσz=∫∂Xlog⁡1+sup0≤r<1⁡Afrzpdσzguarantees the equalities(27)∫∂Xsup0≤r<1⁡frzp+ldσz=∫∂Xsup0≤r<1⁡Afrzp+ldσzfor all l∈Z+. Therefore, A is isometric in the norm Hmp+l(p>0, l=0,1,2,…).
Sincedσ is a finite measure, we verify that(28)liml→∞⁡fHmp+l=fHm∞holds for every f∈H∞(X), and it is clear that fHm∞=f∞. Moreover, fp=A(f)p for every f∈H∞(X) and limp→∞A(f)p=A(f)∞, so we have A(f)∈H∞(X) and f∞=A(f)∞ for every f∈H∞(X). Similarly we see that f∈H∞(X) if A(f) belongs to H∞(X). Therefore, A|H∞(X) is a surjective isometry with respect to ·∞ from H∞(X) onto itself. We may suppose that H∞(X) is a uniform algebra on the maximal ideal space and the maximal ideal space is connected by the Šilov idempotent theorem; hence, we see that AH∞(X) is complex-linear or conjugate linear by [11, Theorem]. As for the rest of this proof, we follow the proof of [9, Proposition 1].Finally we consider multiplicative isometries fromMp(X)(0<p<∞) onto itself. Recall that A:Mp(X)→Mp(X) ismultiplicative if A(fg)=A(f)A(g) for every f,g∈Mp(X).The following theorem is proved by the same method as [9, Theorem 2]; therefore, we do not prove it here.Theorem 7.
Let0<p<∞ and A be a multiplicative (not necessarily linear) isometry from Mp(X) onto itself. Then there exists a holomorphic automorphism Φ on X such that either of the following holds:(29)Af=f∘Φforeveryf∈MpX
or(30)Af=f∘Φ¯¯foreveryf∈MpX,where Φ is a unitary transformation for X=Bn; Φ(z1,…,zn)=(λ1zi1,…,λnzin) for X=Un, where |λj|=1 for every 1≤j≤n and (i1,…,in) is some permutation of the integers from 1 through n.Remark 8.
We note that surjective multiplicative isometries of the classMp(X)(0<p<∞) have the same form as surjective multiplicative isometries of Mp(X)(p∈N) [9, Theorem 2], the Smirnov class [12, Theorem 2.2], and the Privalov class [13, Corollary 3.4].



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*Source: 598538-2015-06-29.xml*