Abstract
In (Iida and Kasuga 2013), the authors described multiplicative (but not necessarily linear) isometries of onto in the case of positive integer , where is included in the Smirnov class . In this paper, we will generalize the result to arbitrary (not necessarily positive integer) value of the exponents .
1. Introduction
Let be a positive integer. The space of -complex variables is denoted by . The unit polydisk , is denoted by and the distinguished boundary is , . The unit ball is denoted by and is its boundary. In this paper denotes the unit polydisk or the unit ball for and denotes for or for . The normalized (in the sense that ) Lebesgue measure on is denoted by .
The Hardy space on is denoted by and denotes the norm on .
The Nevanlinna class on is defined as the set of all holomorphic functions on such thatholds. It is known that has a finite nontangential limit, also denoted by , almost everywhere on .
The Smirnov class is defined as the set of all which satisfy the equalityDefine a metricfor . With the metric the Smirnov class is an -algebra. Recall that an -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 class , , is defined as the set of all holomorphic functions on such thatholds. It is well-known that is a subalgebra of ; hence, every has a finite nontangential limit almost everywhere on . Under the metric defined byfor , becomes an -algebra (cf. [2]). Complex-linear isometries on are investigated by Iida and Mochizuki [3] for one-dimensional case and by Subbotin [2, 4] for a general case.
Now we define the class . For , the class is defined as the set of all holomorphic functions on such thatThe class with in the case where was introduced by Kim in [5]. As for and , the class was considered in [6, 7]. For , define a metricwhere . With this metric is also an -algebra (see [2]). Complex-linear surjective isometries on are investigated by Subbotin [2, 4, 8].
It is well-known that the following inclusion relations hold:Moreover, it is known that [8]. As shown in [6], for any the class coincides with the class and the metrics and are equivalent. Therefore, the topologies induced by these metrics are identical on the set . But we note that in [4, Theorems 1 and 4] it is implied that the sets of linear isometries on and are different. It is known that is a dense subalgebra of . The convergence in the metric is stronger than uniform convergence on compact subsets of .
In [9], the authors described multiplicative (but not necessarily linear) isometries of onto in the case of positive integer . In this paper, we will generalize the result to arbitrary (not necessarily positive integer) value of the exponents .
2. The Results
Proposition 1. Let be a positive integer and let be either or . Let and suppose that is a surjective isometry. If is 2-homogeneous in the sense that holds for every , then eitherorwhere is a complex number with the unit modulus and for , is a unitary transformation; for , , where , and is some permutation of the integers from through .
To prove Proposition 1, we need the following lemmas.
Lemma 2 (see [4]). Let , . Then the normis equivalent to the standard norm in .
Lemma 3 (see [4]). Let . Then
We recall that the function on the interval is said to be completely monotone if it is infinitely differentiable on and
Lemma 4 (see [8]). The functions are completely monotone for all .
Lemma 5 (see [8]). If a completely monotone function on can be continued to an infinitely differentiable function on , then the inequalityholds for any and if only is not constant.
Lemma 6. Let be a cone of measurable functions on a measurable space with a measure , and let be a mapping from to the set of measurable functions. Suppose that is homogeneous with positive coefficients andfor some . Thenfor all , where .
Proof. We follow [2, Lemma 2]. is homogeneous with positive coefficients, so we have, using (15),for any . Letting , we obtainNext we argue by induction. Let and suppose that (16) holds for , . Then, for any and any function , we havewhere are the Taylor coefficients of the function at zero.
Assume first that . It is easy to see that the integrand on the right-hand side of (19) converges to as ; this integrand is of fixed sign by Lemmas 4 and 5 and is dominated by the function with some constant . By the Lebesgue theorem on dominated convergence, the right-hand side of (19) converges to the integral of . Therefore, the left-hand side of (19) has a finite limit and, by the Fatou theorem, the function is integrable. Since for any by Lemmas 4 and 5, we deduce that , and repeating the above arguments, we see that the left-hand side of (19) converges to the integral of as . Therefore, passing to the limit in (19) as and dividing the result by , we obtain (16) for . The case of can be considered in a similar way. For , relation (16) with is trivial.
Proof of Proposition 1. Suppose first that . Let . We easily confirm that, by utilizing Lemma 2 and the celebrated theorem of Mazur and Ulam [10], is a real-linear isometry in a way similar to [9, Proposition 1].
Next suppose that . We define the class as the set of all holomorphic functions on such thatand define a metric for . If is a surjective isometry and is 2-homogeneous in the sense that holds for every , we confirm that is also a surjective isometry.
For , let . We have since . Then the following equality holds:Therefore, it follows thatUsing the elementary inequalities , and , , , we confirm that the integrand on the left-hand side of (22) is dominated by an -function. The integrand on the right-hand side of (22) is also dominated by an -function in the same way. Applying the Lebesgue theorem on dominated convergence and Lemma 3 on both sides of (22), we have the equalityHence, we obtainThe equivalence of the norms and guarantees that is a surjective isometry. By using Mazur-Ulam theorem again, is a real-linear isometry since is a normed vector space and .
We consider an arbitrary function and the cone generated by . Here is the radial maximal function for . Moreover, consider the following mapping on this cone:Since is isometric with respect to the metric , it follows that the assumptions of Lemma 6 hold on the cone .
is a surjective isometry, so the equation guarantees the equalitiesfor all . Therefore, is isometric in the norm , .
Since is a finite measure, we verify thatholds for every , and it is clear that . Moreover, for every and , so we have and for every . Similarly we see that if belongs to . Therefore, is a surjective isometry with respect to from onto itself. We may suppose that 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 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 from onto itself. Recall that is multiplicative if for every .
The following theorem is proved by the same method as [9, Theorem 2]; therefore, we do not prove it here.
Theorem 7. Let and be a multiplicative (not necessarily linear) isometry from onto itself. Then there exists a holomorphic automorphism on such that either of the following holds:or where is a unitary transformation for ; for , where for every and is some permutation of the integers from through .
Remark 8. We note that surjective multiplicative isometries of the class have the same form as surjective multiplicative isometries of [9, Theorem 2], the Smirnov class [12, Theorem 2.2], and the Privalov class [13, Corollary 3.4].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors wish to express their sincere gratitude to Professor Takahiko Nakazi and Professor Osamu Hatori, who introduced this subject and kindly directed them. The authors also would like to thank the referee for detailed comments and valuable suggestions.