aa r X i v : . [ m a t h . M G ] J un Complex-valued valuations on L p spaces Lijuan LiuSchool of Mathematics and Computational Science, Hunan University of Scienceand Technology, Xiangtan, 411201, P.R.China.E-mail: [email protected]
Abstract:
All continuous translation invariant complex-valued valua-tions on Lebesgue measurable functions are completely classified. Andall continuous rotation invariant complex-valued valuations on sphericalLebesgue measurable functions are also completely classified.
Key words:
Convex body, valuation, translation invariance, rotationinvariance.
MR(2010) Subject Classification §
1. Introduction
A function z defined on a lattice ( L , ∨ , ∧ ) and taking values in an abelian semigroupis called a valuation if z ( f ∨ g ) + z ( f ∧ g ) = z ( f ) + z ( g ) (1 . f, g ∈ L . A function z defined on some subset L of L is called a valuation on L if (1.1) holds whenever f, g, f ∨ g, f ∧ g ∈ S . For L the set of convex bodies, K n ,in R n with ∨ denoting union and ∧ intersection. Valuations on convex bodies are aclassical concept going back to Dehn’s solution in 1900 of Hilbert’s Third Problem.Probably the most famous result on valuations is Hadwiger’s classification theoremof continuous rigid motion invariant valuations. For the more recent contributionson valuations convex bodies can see [1,2,4,6,12-24,26-33,38,39,43-45].Valuations on convex bodies can be considered as valuations on suitable func-tion spaces. Recently, valuations on functions has been rapidly growing (see [3,5,7-11,25,34-37,40-42,46-49]). For a space of real-valued functions, the operations ∨ and ∧ are defined as pointwise maximum and minimum, respectively. A complete clas-sification of valuations intertwining with the SL(n) on Sobolev spaces [34,36,40] and L p spaces [25,35,37,42,46,47,49] was established, respectively. Valuations on convexfunctions [3,7,10,11], log-concave functions [41], quasi-concave functions [8,9] andfunctions of Bounded variations [48] were introduced and classified.Recently, Wang and the author [49] showed that the Fourier transform is theonly valuation which is a continuous, positively GL(n) covariant and logarithmic1ranslation covariant complex-valued valuation on integral functions. This motivatesthe study of complex-valued valuations on functions.Let L be a lattice of complex-valued functions. For f ∈ L , let ℜ f and ℑ f denotethe real and imaginary parts of f , respectively. The pointwise maximum of f and g , f ∨ g and the pointwise minimum f ∧ g are defined by f ∨ g = ℜ f ∨ ℜ g + i ( ℑ f ∨ ℑ g ) , (1 . f ∧ g = ℜ f ∧ ℜ g + i ( ℑ f ∧ ℑ g ) . (1 . f, g are real-valued functions, then (1.2) and (1.3) coincide with the real cases. Afunction Φ : L → C is called a valuation ifΦ( f ∨ g ) + Φ( f ∧ g ) = Φ( f ) + Φ( g )for all f, g ∈ L and Φ(0) = 0 if 0 ∈ L . It is called continuous ifΦ( f i ) → Φ( f ) , as f i → f in L . It is called translation invariant ifΦ( f ( · − t )) = Φ( f )for every t ∈ R n . It is called rotation invariant ifΦ( f ◦ θ − ) = Φ( f )for every θ ∈ O( n ), where θ − denotes the inverse of θ .Let p ≥
1. If ( X, F , µ ) is a measure space, then the space L p -space, L p ( C , µ ) isthe collection of µ -measurable complex-valued functions f : X → C that satisfies Z X | f | p dµ < ∞ . A measure space ( X, F , µ ) is called non-atomic if for every E ∈ F with µ ( E ) > F ∈ F with F ⊆ E and 0 < µ ( F ) < µ ( E ). Let χ E denote thecharacteristic function of the measurable set E , i.e. χ E ( x ) = (cid:26) , x ∈ E ;0 , x / ∈ E . Theorem 1.1
Let ( X, F , µ ) be a non-atomic measure space and let Φ : L p ( C , µ ) → C be a continuous valuation. If there exist continuous functions h k : R → R with h k (0) = 0 ( k = 1 , , ,
4) such that Φ( cχ E ) = ( h ( ℜ c ) + h ( ℑ c )) µ ( E ) + i ( h ( ℜ c ) + h ( ℑ c )) µ ( E ) for all c ∈ C and all E ∈ F with µ ( E ) < ∞ , then there exist constants γ k , δ k ≥ | h k ( a ) | ≤ γ k | a | p + δ k for a ∈ R , andΦ( f ) = Z X ( h ◦ ℜ f + h ◦ ℑ f ) dµ + i Z X ( h ◦ ℜ f + h ◦ ℑ f ) dµ f ∈ L p ( C , µ ). In addition, δ k = 0 ( k = 1 , , ,
4) if µ ( X ) = ∞ .Let L p ( C , R n ) denote the L p space of Lebesgue measurable complex-valued func-tions on R n . Theorem 1.2
A function Φ : L p ( C , R n ) → C is a continuous translation invariantvaluation if and only if there exist continuous functions h k : R → R with theproperty that there exist constants γ k ≥ | h k ( a ) | ≤ γ k | a | p for all a ∈ R ( k = 1 , , , f ) = Z R n ( h ◦ ℜ f + h ◦ ℑ f )( x ) dx + i Z R n ( h ◦ ℜ f + h ◦ ℑ f )( x ) dx (1 . f ∈ L p ( C , R n ).Let S n − be the unit sphere in R n and let L p ( C , S n − ) denote the L p space ofspherical Lebesgue measurable complex-valued functions on S n − . Theorem 1.3
A function Φ : L p ( C , S n − ) → C is a continuous rotation invariantvaluation if and only if there exist continuous functions h k : R → R with theproperties that h k (0) = 0 and there exist constants γ k , δ k ≥ | h k ( a ) | ≤ γ k | a | p + δ k for all a ∈ R ( k = 1 , , , f ) = Z S n − ( h ◦ ℜ f + h ◦ ℑ f )( u ) du + i Z S n − ( h ◦ ℜ f + h ◦ ℑ f )( u ) du (1 . f ∈ L p ( C , S n − ). §
2. Notation and preliminary results
We collect some properties of complex-valued functions. If f is a complex-valuedfunction on R n , then f ( x ) = ℜ f + i ℑ f, where ℜ f and ℑ f denote the real part and imaginary part of f , respectively. Theabsolute value of f which is also called modulus is defined by | f | = p ( ℜ f ) + ( ℑ f ) . Let p ≥
1. For a measure space ( X, F , µ ), define L p ( C , µ ) as the space of µ -measurable complex-valued functions f : X → C that satisfies k f k p = ( Z X | f | p dµ ) p < ∞ . Let f, g ∈ L p ( C , µ ), then f ∨ g, f ∧ g ∈ L p ( C , µ ). The functional k · k : L p ( C , µ ) → R is a semi-norm. If functions in L p ( C , µ ) that are equal almost everywhere withrespect to µ (a.e. [ µ ]) are identified, then k · k : L p ( C , µ ) → R becomes a norm.Obviously, L p ( C , µ ) is a lattice of complex-valued functions. Let L p ( R , µ ) denote3he subset of L p ( C , µ ) where the functions take real values. For f i , f ∈ L p ( C , µ ), if k f i − f k p →
0, then f i → f in L p ( C , µ ). Moreover, f i → f ∈ L p ( C , µ ) ⇔ ℜ f i → ℜ f, ℑ f i → ℑ f ∈ L p ( R , µ ) . The following characterizations of real-valued valuations on L p spaces whichwere established by Tsang [46] will play key role in our proof. Let L p ( R , R n ) and L p ( R , S n − ) denote the L p space of Lebesgue measurable real-valued functions on R n and the L p space of spherical Lebesgue measurable real-valued functions on S n − ,respectively. Theorem 2.1 ([46]) A function Φ : L p ( R , R n ) → R is a continuous translationinvariant valuation if and only if there exists a continuous function h : R → R withthe property that there exists a constant γ ≥ | h ( a ) | ≤ γ | a | p for all a ∈ R ,and Φ( f ) = Z R n ( h ◦ f )( x ) dx for all f ∈ L p ( R , R n ). Theorem 2.2 ([46]) A function Φ : L p ( R , S n − ) → R is a continuous rotationinvariant valuation if and only if there exists a continuous function h : R → R with the properties that h (0) = 0 and there exist constants γ, δ ≥ | h ( a ) | ≤ γ | a | p + δ for all a ∈ R , andΦ( f ) = Z S n − ( h ◦ f )( u ) du for all f ∈ L p ( R , S n − ). §
3. Main resultsLemma 3.1
Let h k : R → R be continuous functions with the properties that h k (0) = 0 and there exist γ k , δ k ≥ | h k ( a ) | ≤ γ k | a | p + δ k for all a ∈ R ( k =1 , . . . , L p ( C , µ ) → C is defined byΦ( f ) = Z X ( h ◦ ℜ f + h ◦ ℑ f ) dµ + i Z X ( h ◦ ℜ f + h ◦ ℑ f ) dµ, (3 . δ k = 0 if µ ( X ) = ∞ . Proof.
For f, g ∈ L p ( C , µ ), let E = { x ∈ X : ℜ f ≤ ℜ g, ℑ f ≤ ℑ g } , F = { x ∈ X : ℜ f ≤ ℜ g, ℑ f > ℑ g } ,G = { x ∈ X : ℜ f > ℜ g, ℑ f ≤ ℑ g } , H = { x ∈ X : ℜ f > ℜ g, ℑ f > ℑ g } .
4y (3.1) and (1.2), we obtainΦ( f ∨ g ) = Z X ( h ◦ ℜ ( f ∨ g ) + h ◦ ℑ ( f ∨ g )) dµ + i Z X ( h ◦ ℜ ( f ∨ g ) + h ◦ ℑ ( f ∨ g )) dµ = Z E ( h ◦ ℜ g + h ◦ ℑ g ) dµ + i Z E ( h ◦ ℜ g + h ◦ ℑ g ) dµ + Z F ( h ◦ ℜ g + h ◦ ℑ f ) dµ + i Z F ( h ◦ ℜ g + h ◦ ℑ f ) dµ + Z G ( h ◦ ℜ f + h ◦ ℑ g ) dµ + i Z G ( h ◦ ℜ f + h ◦ ℑ g ) dµ + Z H ( h ◦ ℜ f + h ◦ ℑ f ) dµ + i Z H ( h ◦ ℜ f + h ◦ ℑ f ) dµ. Similarly, by (3.1) and (1.3), we haveΦ( f ∧ g ) = Z E ( h ◦ ℜ f + h ◦ ℑ f ) dµ + i Z E ( h ◦ ℜ f + h ◦ ℑ f ) dµ + Z F ( h ◦ ℜ f + h ◦ ℑ g ) dµ + i Z F ( h ◦ ℜ f + h ◦ ℑ g ) dµ + Z G ( h ◦ ℜ g + h ◦ ℑ f ) dµ + i Z G ( h ◦ ℜ g + h ◦ ℑ f ) dµ + Z H ( h ◦ ℜ g + h ◦ ℑ g ) dµ + i Z H ( h ◦ ℜ g + h ◦ ℑ g ) dµ. Note that E ∪ F ∪ G ∪ H = X and that E, F, G, H are pairwise disjoint. Thus,Φ( f ∨ g ) + Φ( f ∧ g ) = Φ( f ) + Φ( g ) . Hence Φ is a valuation.It remains to show that Φ is continuous. Let f ∈ L p ( C , µ ) and let { f k } bea sequence in L p ( C , µ ) with f k → f in L p ( C , µ ). Next, we will show that Φ( f k )converges to Φ( f ) by showing that every subsequence, Φ( f k l ), of Φ( f k ) has a sub-sequence, Φ( f k lm ), converges to Φ( f ). Set f = α + iβ and f k = α k + iβ k with α, β, α k , β k ∈ L p ( R , µ ) such that α k → α and β k → β in L p ( R , µ ). Let { f k l } be asubsequence of { f k } , then { f k l } converges to f in L p ( C , µ ). Then there exists a sub-sequence { f k lm } of { f k l } which converges to f in L p ( C , µ ), where f k lm = α k lm + iβ k lm with α k lm , β k lm ∈ L p ( R , µ ) such that α k lm → α and β k lm → β in L p ( R , µ ). Since h is continuous, we have ( h ◦ α k lm )( x ) → ( h ◦ α )( x ) , a.e. [ µ ] . | h ( a ) | ≤ γ | a | p + δ for all a ∈ R , we get | ( h ◦ α k lm )( x ) | ≤ γ | α k lm | p + δ , a.e. [ µ ] . If µ ( X ) < ∞ , apply α k lm → α in L p ( R , µ ) to getlim m →∞ Z X γ | α k lm | p + δ dµ = Z X γ | α | p dµ + δ µ ( X ) . And we take δ = 0 in the above equation if µ ( X ) = ∞ . By a modification ofLebesgue’s Dominated Convergence Theorem (see [46, Proposition 2.2.]), we have h ◦ ( α ) ∈ L ( R , µ ) andlim m →∞ Z X ( h ◦ α k lm ) dµ = Z X ( h ◦ α ) dµ. (3 . m →∞ Z X ( h ◦ α k lm ) dµ = Z X ( h ◦ α ) dµ, (3 . m →∞ Z X ( h ◦ β k lm ) dµ = Z X ( h ◦ β ) dµ, (3 . m →∞ Z X ( h ◦ β k lm ) dµ = Z X ( h ◦ β ) dµ. (3 . | Φ( f k lm ) − Φ( f ) | = | Z X ( h ◦ α k lm − h ◦ α ) dµ + Z X ( h ◦ β k lm − h ◦ β ) dµ + i (cid:16) Z X ( h ◦ α k lm − h ◦ α ) dµ + Z X ( h ◦ β k lm − h ◦ β ) dµ (cid:17) |≤ | Z X ( h ◦ α k lm − h ◦ α ) dµ | + | Z X ( h ◦ β k lm − h ◦ β ) dµ | + | Z X ( h ◦ α k lm − h ◦ α ) dµ | + | Z X ( h ◦ β k lm − h ◦ β ) dµ | . From (3.2)-(3.5), we conclude Φ( f k lm ) → Φ( f ). Hence, Φ is continuous. (cid:3) Lemma 3.2
If the function Φ : L p ( C , µ ) → C is a valuation, thenΦ( f ) = Φ( ℜ f ) + Φ( i ℑ f ) , for all f ∈ L p ( C , µ ). Proof. If ℜ f, ℑ f ≥ ℜ f, ℑ f ≤
0, then, by (1.2) and (1.3), we haveΦ( ℜ f ) + Φ( i ℑ f ) = Φ( f ) + Φ(0) . (3 . ℜ f ≥ , ℑ f ≤ ℜ f ≤ , ℑ f ≥
0, then, by (1.2) and (1.3), we haveΦ( f ) + Φ(0) = Φ( ℜ f ) + Φ( i ℑ f ) . (3 . f ) = Φ( ℜ f ) + Φ( i ℑ f )for all f ∈ L p ( C , µ ). (cid:3) If we restrict f to belong to L p ( R , µ ), then it is obvious that Φ is a valuation on L p ( R , µ ). Also, we can construct another valuation on L p ( R , µ ) which is related toΦ. Lemma 3.3
Let Φ : L p ( C , µ ) → C be a valuation. If the functions Φ ′ : L p ( R , µ ) → C is defined by Φ ′ ( f ) = Φ( if )for all f ∈ L p ( R , µ ), then Φ ′ is a valuation on L p ( R , µ ). Proof.
For f, g ∈ L p ( R , µ ), by (1.2) and (1.3), we have i ( f ∨ g ) = if ∨ ig and i ( f ∧ g ) = if ∧ ig. (3 . ′ ( f ∨ g ) + Φ ′ ( f ∧ g ) = Φ( if ∨ ig ) + Φ( if ∧ ig ) = Φ( if ) + Φ( ig ) = Φ ′ ( f ) + Φ ′ ( g )for all f, g ∈ L p ( R , µ ). Thus, Φ ′ is a valuation on L p ( R , µ ). (cid:3) Lemma 3.4
Let Φ : L p ( R , µ ) → C be a valuation. If the functions Φ , Φ : L p ( R , µ ) → R are defined by Φ( f ) = Φ ( f ) + i Φ ( f )for all f ∈ L p ( R , µ ), then Φ , Φ both are a real-valued valuation on L p ( R , µ ). Proof.
Since Φ is a valuation, we haveΦ( f ∨ g ) + Φ( f ∧ g ) = Φ ( f ∨ g ) + i Φ ( f ∨ g ) + Φ ( f ∧ g ) + i Φ ( f ∧ g )= Φ( f ) + Φ( g ) = Φ ( f ) + i Φ ( f ) + Φ ( g ) + i Φ ( g )for all f, g ∈ L p ( R , µ ). Thus,Φ ( f ∨ g ) + Φ ( f ∧ g ) = Φ ( f ) + Φ ( g ) , and Φ ( f ∨ g ) + Φ ( f ∧ g ) = Φ ( f ) + Φ ( g ) , for all f, g ∈ L p ( R , µ ). Therefore, Φ , Φ both are a real-valued valuation on L p ( R , µ ). (cid:3)
7n order to establish a representation theorem for continuous complex-valuedvaluations on L p ( C , µ ), we will use the corresponding representation theorem forreal case which was obtained by Tsang [46]. Theorem 3.5 ([46]) Let ( X, F , µ ) be a non-atomic measure space and let Φ : L p ( R , µ ) → R be a continuous translation invariant valuation. If there exists acontinuous function h : R → R with h (0) = 0 such that Φ( bχ E ) = h ( b ) µ ( E ) for all b ∈ R and all E ∈ F with µ ( E ) < ∞ , then there exist constants γ, δ ≥ | h ( a ) | ≤ γ | a | p + δ for all a ∈ R , andΦ( f ) = Z X ( h ◦ f ) dµ for all f ∈ L p ( R , µ ). In addition, δ = 0 if µ ( X ) = ∞ . Proof of Theorem 1.1
Let Φ : L p ( C , µ ) → C be a continuous valuation. For f ∈ L p ( C , µ ), by Lemma3.2, Lemma 3.3 and Lemma 3.4, we haveΦ( f ) = Φ( ℜ f ) + Φ( i ℑ f ) = Φ ( ℜ f ) + i Φ ( ℜ f ) + Φ ( i ℑ f ) + i Φ ( i ℑ f )= Φ ( ℜ f ) + i Φ ( ℜ f ) + Φ ′ ( ℑ f ) + i Φ ′ ( ℑ f ) , where Φ( ℜ f ) = Φ ( ℜ f ) + i Φ ( ℜ f ) , Φ ′ ( ℑ f ) = Φ ( i ℑ f ) , and Φ ′ ( ℑ f ) = i Φ ( i ℑ f ).Since ℜ ( f ∨ g ) = ℜ f ∨ ℜ g, ℜ ( f ∧ g ) = ℜ f ∧ ℜ g , ℑ ( f ∨ g ) = ℑ f ∨ ℑ g , and ℑ ( f ∧ g ) = ℑ f ∧ ℑ g . Moreover, Lemma 3.3 and Lemma 3.4 imply that Φ , Φ , Φ ′ , Φ ′ all are areal-valued valuation on L p ( R , µ ).If we restrict to f ∈ L p ( R , µ ), then the continuity of Φ implies that Φ , Φ arecontinuous on L p ( R , µ ). If we consider f ∈ L p ( C , µ ) with ℜ f = 0, then the continuityof Φ implies that Φ ′ , Φ ′ are continuous on L p ( R , µ ). Thus, Φ , Φ , Φ ′ , Φ ′ all are acontinuous real-valued valuation on L p ( R , µ ). It follows from Theorem 3.5 that thereexist continuous functions h k : R → R with the properties that h k (0) = 0 and thereexist constants γ k , δ k ≥ | h k ( a ) | ≤ γ k | a | p + δ k for all a ∈ R ( k = 1 , , , f ) = Z X ( h ◦ ℜ f + h ◦ ℑ f ) dµ + i Z X ( h ◦ ℜ f + h ◦ ℑ f ) dµ for all f ∈ L p ( C , µ ). In addition, δ k = 0 ( k = 1 , , ,
4) if µ ( X ) = ∞ . (cid:3) If µ is Lebesgue measure, then L p ( C , µ ) becomes the space of Lebesgue measur-able complex-valued functions. We usually write as L p ( C , R n ). Lemma 3.6
Let h k : R → R be continuous functions with the property that thereexist γ k ≥ | h k ( a ) | ≤ γ k | a | p for all a ∈ R ( k = 1 , . . . , L p ( C , R n ) → C is defined byΦ( f ) = Z R n ( h ◦ ℜ f + h ◦ ℑ f ) dx + i Z X ( h ◦ ℜ f + h ◦ ℑ f ) dx, Proof.
Let M denote the collection of Lebesgue measurable sets in R n . Take X = R n , F = M and µ Lebesgue measure in Lemma 3.1 to conclude that Φ is acontinuous valuation on L p ( C , R n ).For every t ∈ R n and every f ∈ L p ( C , R n ), we haveΦ( f ( x − t )) = Z R n ( h ◦ℜ f + h ◦ℑ f )( x − t ) dx + i Z X ( h ◦ℜ f + h ◦ℑ f )( x − t ) dx = Φ( f ) , which means that Φ is translation invariant. (cid:3) Proof of Theorem 1.2
It follows from Lemma 3.6 that (1.4) determines a continuous translation invari-ant valuation on L p ( C , R n ).Conversely, let Φ : L p ( C , R n ) → C be a continuous translation invariant valua-tion. Taking X = R n , F = M and µ Lebesgue measure in the proof of Theorem1.1, we obtain Φ( f ) = Φ ( ℜ f ) + i Φ ( ℜ f ) + Φ ′ ( ℑ f ) + i Φ ′ ( ℑ f ) , where Φ , Φ , Φ ′ , Φ ′ all are a real-valued valuation on L p ( R , µ ). Theorem 2.1 impliesthat there exist continuous functions h k : R → R with the property that there existconstants γ k ≥ | h k ( a ) | ≤ γ k | a | p for all a ∈ R ( k = 1 , , , f ) = Z R n ( h ◦ ℜ f + h ◦ ℑ f ) dx + i Z R n ( h ◦ ℜ f + h ◦ ℑ f ) dx for all f ∈ L p ( C , R n ). (cid:3) Let W denote the σ -algebra defined as W = { E : E ⊆ S n − , { λx : x ∈ E, ≤ λ ≤ } ∈ M} . Also denote by σ the spherical Lebesgue measure. If µ is the spherical Lebesguemeasure, then L p ( C , σ ) denotes the space of spherical Lebesgue measurable complex-valued functions. We usually write as L p ( C , S n − ). Lemma 3.7
Let h k : R → R be continuous functions with the properties that h k (0) = 0 and there exist γ k , δ k ≥ | h k ( a ) | ≤ γ k | a | p + δ k for all a ∈ R ( k =1 , . . . , L p ( C , S n − ) → C is defined byΦ( f ) = Z S n − ( h ◦ ℜ f + h ◦ ℑ f ) du + i Z S n − ( h ◦ ℜ f + h ◦ ℑ f ) du, then Φ is a continuous rotation invariant valuation. Proof.
Take X = S n − , F = W and µ = σ in Lemma 3.1 to conclude that Φ is acontinuous valuation on L p ( C , S n − ). 9ote that θu ∈ S n − for every θ ∈ O( n ) and every u ∈ S n − . Since the sphericalLebesgue measure is rotation invariant, we haveΦ( f ◦ θ − ) = Z S n − ( h ◦ℜ f + h ◦ℑ f )( θu ) du + i Z S n − ( h ◦ℜ f + h ◦ℑ f )( θu ) du = Φ( f )for all f ∈ L p ( C , S n − ), which finishes the proof. (cid:3) Proof of Theorem 1.3
It follows from Lemma 3.7 that (1.5) determines a continuous rotation invariantvaluation on L p ( C , S n − ).Conversely, let Φ : L p ( C , S n − ) → C be a continuous rotation invariant valuation.Taking X = S n − , F = W and µ = σ in the proof of Theorem 1.1, we obtainΦ( f ) = Φ ( ℜ f ) + i Φ ( ℜ f ) + Φ ′ ( ℑ f ) + i Φ ′ ( ℑ f ) , where Φ , Φ , Φ ′ , Φ ′ all are a real-valued valuation on L p ( R , S n − ). Theorem 2.2implies that there exist continuous functions h k : R → R with the properties that h k (0) = 0 and there exist constants γ k , δ k ≥ | h k ( a ) | ≤ γ k | a | p + δ k for all a ∈ R ( k = 1 , , , f ) = Z S n − ( h ◦ ℜ f + h ◦ ℑ f ) du + i Z S n − ( h ◦ ℜ f + h ◦ ℑ f ) du for all f ∈ L p ( C , S n − ). (cid:3) Acknowledgment
The work was supported in part by the Natural Science Foun-dation of Hunan Province (2019JJ50172).
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