Issue 
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 1, March 2022



Page(s)  35  41  
DOI  https://doi.org/10.1051/wujns/2022271035  
Published online  16 March 2022 
Mathematics
CLC number: O178;O18
ComplexValued Valuations on L^{p} Spaces
School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, Hunan, China
Received: 26 September 2021
All continuous translation invariant complexvalued valuations on Lebesgue measurable functions are completely classified. And all continuous rotation invariant complexvalued valuations on spherical Lebesgue measurable functions are also completely classified.
Key words: convex body / valuation / translation invariance / rotation invariance
Biography: LIU Lijuan, female, Ph. D., research direction: convex geometry. Email:lijuanliu@hnust.edu.cn
Foundation item: Supported by the Natural Science Foundation of Hunan Province (2019JJ50172)
© Wuhan University 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
A function z defined on a lattice and taking values in an Aabelian semigroup is called a valuation if(1)for all . A function z defined on some subset of L is called a valuation on if (1) holds whenever . For the set of convex bodies, , in with denoting union and intersection. Valuation on convex bodies is a classical concept. Probably the most famous result on valuations is Hadwiger’s classification theorem of continuous rigid motion invariant valuations^{[1]}. For the more recent contributions on valuations on convex bodies readers can refer to Refs. [233].
Valuations on convex bodies can be considered as valuations on suitable function spaces. Recently, valuations on functions have been rapidly growing (see Refs. [3455]). For a space of realvalued functions, the operations and are defined as pointwise maximum and minimum, respectively. A complete classification of valuations intertwining with the SL(n) on Sobolev space ^{[4649]} and space ^{[45,5052,54]} were established, respectively. Valuations on convex functions ^{[34,36,3942]}, quasiconcave functions ^{[37,38]}, Lipschitz functions ^{[43,44]}, and functions of Bounded variations ^{[53]} were introduced and classified.
Recently, Wang and Liu ^{[55]} showed that the Fourier transform is the only valuation which is a continuous, positive GL(n) covariant and logarithmic translation covariant complexvalued valuation on integral functions. This motivates the study of complexvalued valuations on functions.
Let L be a lattice of complexvalued functions. For , let and denote the real and imaginary parts of f, respectively. The pointwise maximum of f and g, and the pointwise minimum of f and g are defined by(2)and(3)If f, g are realvalued functions, then (2) and (3) coincide with the real cases. A function is called a valuation iffor all and if . It is called continuous ifIt is called translation invariant iffor every . It is called rotation invariant iffor every , where denotes the inverse of .
Let . If is a measure space, then the space, is the collection of µmeasurable complexvalued functions that satisfiesA measure space is called nonatomic if for every with , there exists with and . Let denote the characteristic function of the measurable set E, i.e.
Theorem 1 Let be a nonatomic measure space and let be a continuous valuation. If there exist continuous functions with (k=1,2,3,4) such that for all and all with , then there exist constants such that for , andfor all . In addition, (k = 1,2,3,4) if .
Let denote the space of Lebesgue measurable complexvalued functions on .
Theorem 2 A function is a continuous translation invariant valuation if and only if there exist continuous functions with the property that there exist constants such that for all (k = 1,2,3,4), and(4)for all .
Let be the unit sphere in and let denote the space of spherical Lebesgue measurable complexvalued functions on .
Theorem 3 A function is a continuous rotation invariant valuation if and only if there exist continuous functions with the properties that and there exist constants such that for all (k = 1,2,3,4), and(5)for all .
1 Notation and Preliminary Results
We collect some properties of complexvalued functions. If f is a complexvalued function on , thenwhere and denote the real part and imaginary part of f, respectively. The absolute value of f which is also called modulus is defined byLet . For a measure space , define as the space of µmeasurable complexvalued functions that satisfiesLet , then . The functional is a seminorm. If functions in that are equal almost everywhere with respect to are identified, then becomes a norm. Obviously, is a lattice of complexvalued functions. Let denote the subset of , where the functions take real values. For , if , then in . Moreover,
The following characterizations of realvalued valuations on functions which were established by Tsang ^{[51]} will play key role in our proof. Let and denote the space of Lebesgue measurable realvalued functions on and the space of spherical Lebesgue measurable realvalued functions on ，respectively.
Theorem 4 ^{[51]} A function is a continuous translation invariant valuation if and only if there exists a continuous function with the property that there exists a constant such that for all , andfor all .
Theorem 5 ^{[51]} A function is a continuous rotation invariant valuation if and only if there exists a continuous function with the properties that and there exist constants such that for all , andfor all .
2 Main Results
Lemma 1 Let be continuous functions with the properties that and there exist such that for all (k =1,...,4). If the function is defined by(6)then is a continuous valuation provided that if .
Proof For , letBy (6) and (2), we obtainSimilarly, by (6) and (3), we haveNote that and that E, F, G, H are pairwise disjoints. Thus,Hence is a valuation.
It remains to show that is continuous. Let and let be a sequence in with in . Next, we will show that converges to by showing that every subsequence of has a subsequence, which converges to . Set and with such that and in . Let be a subsequence of , then converges to f in . Then there exists a subsequence of which converges to f in , where with such that and in . Since is continuous, we haveSince for all , we getIf , apply in to getAnd we take in the above equation if . By a modification of Lebesgue’s Dominated Convergence Theorem (see Ref.[51], Proposition 2.2), we have and(7)Similarly,(8)(9)(10)Thus,From (7)(10), we conclude . Hence, is continuous.
Lemma 2 If the function is a valuation, thenfor all .
Proof If or , then, by (2) and (3), we have(11)If or , then, by (2) and (3), we have(12)Note that , apply (11) and (12), then getfor all .
If we restrict f to , then it is obvious that is a valuation on . Also, we can construct another valuation on which is related to .
Lemma 3 Let be a valuation. If the functions is defined byfor all , then is a valuation on .
Proof For , by (2) and (3), we have(13)By (13) and the valuation property of , it follows thatfor all . Thus, is a valuation on .
Lemma 4 Let be a valuation. If the functions are defined byfor all , then both are realvalued valuations on .
Proof Since is a valuation, we havefor all . Thus,andfor all .
Therefore, both are realvalued valuations on .
In order to establish a representation theorem for continuous complexvalued valuations on , we will use the corresponding representation theorem for real case which was obtained by Tsang ^{[51]}.
Theorem 6 ^{[51]} Let be a nonatomic measure space and let be a continuous translation invariant valuation. If there exists a continuous function with such that for all and all with , then there exist constants such that for all , andfor all . In addition, if .
Proof of Theorem 1
Let be a continuous valuation. For , by Lemma 2, Lemma 3 and Lemma 4, we havewhere and .
Since . Moreover, Lemma 3 and Lemma 4 imply that are realvalued valuations on .
If we restrict to , then the continuity of implies that are continuous on .
If we consider with , then the continuity of implies that are continuous on . Thus, are continuous realvalued valuations on . It follows from Theorem 6 that there exist continuous functions with the properties that and there exist constants such that for all (k = 1,2,3,4), andfor all . In addition, (k = 1,2,3,4) if .
If µ is Lebesgue measure, then becomes the space of Lebesgue measurable complexvalued functions. We usually write as .
Lemma 5 Let be continuous functions with the property that there exists such that for all (k = 1,...,4). If the function is defined byThen is a continuous translation invariant valuation.
Proof Let M enote the collection of Lebesgue measurable sets in . Take and µ Lebesgue measure in Lemma 1 to conclude that is a continuous valuation on .
For every and every , we havewhich means that is translation invariant.
Proof of Theorem 2
It follows from Lemma 5 that (4) determines a continuous translation invariant valuation on . Conversely, let be a continuous translation invariant valuation. Taking and µ Lebesgue measure in the proof of Theorem 1, we obtainwhere are realvalued valuations on . Theorem 4 implies that there exist continuous functions with the property that there exist constants such that for all (k = 1,2,3,4), andfor all .
Let W denote the algebra defined as
Also denote by the spherical Lebesgue measure. If µ is the spherical Lebesgue measure, then denotes the space of spherical Lebesgue measurable complex valued functions. We usually write as .
Lemma 6 Let be continuous functions with the properties that and there exists such that for all (k=1,...,4). If the function is defined bythen is a continuous rotation invariant valuation.
Proof Take and in Lemma 1 to conclude that is a continuous valuation on .
Note that for every and every . Since the spherical Lebesgue measure is rotation invariant, we havefor all , which completes the proof.
Proof of Theorem 3
It follows from Lemma 6 that (5) determines a continuous rotation invariant valuation on .
Conversely, let be a continuous rotation invariant valuation. Taking and in the proof of Theorem 1, we obtainwhere are realvalued valuations on .
Theorem 5 implies that there exist continuous functions with the properties that and there exist constants such that for all (k = 1,2,3,4), andfor all .
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