An asymptotic variant of the Fubini theorem for maps into CAT(0)-spaces

AN ASYMPTOTIC V ARIANT OF THE FUBINI THEOREM F OR MAPS INTO CA T(0)-SP A CES KEI FUNANO Abstract. The classical F ubini theorem as serts that t he m ultiple in tegral is equal to the repeated one for any in tegra ble function on a product measure space. In this pap er , we derive an asymptotic v ariant of t he F ubini theorem for maps into CA T(0)-spa c es from the L 1 and L 2 -concentration of the maps. 1. Intro duction and st a te ment of the main resul t The classical F ubini theorem ass erts that the m ultiple in tegral is eq ual to the repeated one for an y in tegrable function on a pro duct measure space. In this pap er, w e pro v e a n asymptotic v ariant of the F ubini theorem for maps into CA T(0)-spaces. F or this purpose, let us defin e the ex p ectation (in tegral) for a map from a probabilit y space in to a CA T(0)-space. Throughout this section, let N b e a CA T(0)-space. Let (Ω , A , P ) b e a Probability space. F or an N -v alued random v ariable Z : Ω → N such that the push-forw ard measure Z ∗ P has finite momen t of o rder 1, w e define its exp e ctation E ( Z ) as the cente r of mass of the measure Z ∗ P (see Subsection 2.2 for the definition o f the cen ter of mass). This definition o f exp ectations is based on the classical p oin t of view of [4]. In [4], C. F. Gauss defined the expectations of ra ndom v ariables with v alues in Euclidean spaces as the a b o v e w ay . In the conte xt of metric spaces, this p oint of view w as succe ssfully used b y [1], [8], [16], and man y o t hers. Let ( X , d X , µ X ) and ( Y , d Y , µ Y ) b e t w o mm-spaces. Here, an mm -sp ac e is a triple ( X , d X , µ X ) of a set X , a complete sep arable distance function d X on X , and a Borel probabilit y measure µ X on ( X, d X ) with full-supp ort. The pr o d uct mm-sp ac e o f X and Y is the mm-space X × Y equipp ed with the ℓ 2 -distance function and the pro duct probabilit y measure. F or a Bo r el measurable map f : X × Y → N suc h that the push-forw ard measure f ∗ ( µ X × µ Y ) has finite momen t of order 1 and y ∈ Y , w e shall cons ider the map f y : X → N defined b y f y ( x ) := f ( x, y ). Note that the push-forward me asure ( f y ) ∗ ( µ X ) has finite momen t o f order 1 fo r µ Y -a.e. y ∈ Y . F or defining the rep eated in tegral fo r the map f , we assume the f ollo wing: (1) T he map g f : Y → N defined by g f ( y ) := E ( f y ) is Borel measurable. Date : Nov ember 20, 2018. 2000 Mathematics Subje ct Classific ation. 53C21, 5 3C23. Key wor ds and phr ases. CA T(0)-space, the F ubini theorem, L 2 -concentration of maps, mm-space. This work was partially supp orted b y Research F ello wships o f the Japan So ciety for the Promotio n o f Science for Y oung Scientists. 1 2 KEI FUNAN O (2) T he push-forw ard measure ( g f ) ∗ ( µ Y ) ha v e finite momen t of o rder 1. If the map f is unifo rmly con tinuous, then the map g f satisfies the ab o v e (1 ) and (2) (see Lemmas 3.1 and 3.2). It seems that the ab ov e (1) and (2) hold for an a r bitrary Borel measurable map f , but the author do es not kno w ho w to prov e it as of no w. F or a Borel measurable map f satisfying the ab ov e (1) and (2), w e define its r ep e ate d inte gr al E y ( E ( f y )) by the exp ectatio n E ( g f ). W e will see t ha t the F ubini theorem E ( f ) = E y ( E ( f y )) do es not hold in general for a nonlinear CA T(0)-space N (see Example 3.3). Ho w ev er, w e succee d to estimate their difference d N ( E ( f ) , E y ( E ( f y ))) b y the term of the L 1 and L 2 -v ariation of the map f : Let X b e an mm-space and p ≥ 1. Giv en a Borel measurable map f : X → N , w e define its L p -variation b y V p ( f ) :=  Z Z X × X d N ( f ( x ) , f ( x ′ )) p dµ X ( x ) dµ X ( x ′ )  1 /p . A main theorem of this paper is the follow ing: Theorem 1.1. L et X and Y b e two mm-sp ac es. Then, for any uniform ly c ontinuous map f : X × Y → N such that the push- f orwar d me asur e f ∗ ( µ X × µ Y ) has the finite mom ent of or der 1 , we have d N ( E ( f ) , E y ( E ( f y ))) ≤ V 1 ( f ) (1.1) and d N ( E ( f ) , E y ( E ( f y ))) ≤ 1 √ 3 V 2 ( f ) (1.2) Jensen’s inequalit y easily leads to the inequalit y (1 .1 ). I n the pro of of the inequalit y (1.2), w e iterate some K-T. Sturm’s inequalit y ab out the cen ter of mass of a probability measure on a CA T(0)-space (see Prop osition 2.8). W e emphasize that the co efficien t 1 / √ 3 of the inequalit y (1.2) cannot b e obtained only from the inequalities (1.1) and V 1 ( f ) ≤ V 2 ( f ). Let { X n } ∞ n =1 b e a sequence of mm-spaces and { N n } ∞ n =1 a sequen ce o f CA T(0)-spaces. F or p ≥ 1, w e say that a sequence { f n : X n → N n } ∞ n =1 of Borel measurable maps L p - c onc en tr ates if V p ( f n ) → 0 a s n → ∞ . F rom the inequalit y ( 1 .1), the L p -concen tration of uniformly con tin uous maps implies that the F ubini theorem for the maps “almostly” holds. The L 2 -concen tration theory of maps in to CA T(0)-spaces w as first studied by M . Gromov in [5]. In [2], the author a lso studied relationships b et w een the L´ evy-Milman concen tration theory of 1-Lipsc hitz maps and the L p -concen tration theory of 1-Lipsc hitz maps (see [10], [11 ], [12], [13], and [15] fo r further inf o rmation a b out the L´ evy-Milman concen tration theory). Motiv ated b y Gromov ’s w orks in [5], [6], and [7], the author studied the L p -concen tration theory o f 1-Lipsc hitz maps into Hadamard manifolds and R -trees in [2] and [3]. Com bining Theorem 1.1 with author’s works and Gro mov’s w orks, w e obta in the follow ing corollary: W e shall consider eac h compact connected Riemannian manifold M a s an mm-space equipp ed with the v olume measure normalized to ha v e the total v olume 1. W e denote b y λ 1 ( M ) the non-zero first eigenv alue of the Laplacian on M . AN ASYMPTOTIC V ARI ANT OF THE FUBINI THEOREM FOR MAPS IN TO CA T(0)-SP ACES 3 Corollary 1.2. L et M b e a c omp act R iema n nian ma n ifold. Then, for any n -dimensional Hadamar d manifold N ′ and 1 -Lipsch i tz map f : M × M → N ′ , we have d N ′ ( E ( f ) , E y ( E ( f y ))) ≤ 2 s 2 n 3 λ 1 ( M ) . F or an R -tr e e T a nd a 1 -Lipschitz map f : M × M → T , we also have d T ( E ( f ) , E y ( E ( f y ))) 2 ≤ 8(38 + 16 √ 2) 3 λ 1 ( M ) . 2. Preliminaries 2.1. The W asserstein distance function of order 1. Let ( X , d X ) b e a complete metric space. F or p ≥ 1 , w e indicate b y P p ( X ) the set of all probability measures ν suc h that ν has the separable support and R X d X ( x, y ) p dν ( y ) < + ∞ for some (hence all) x ∈ X . F or µ, ν ∈ P 1 ( X ), we define the Wasserstein distanc e d W 1 ( µ, ν ) of or der 1 b et w een µ and ν as the infim um of R X × X d X ( x, y ) dπ ( x, y ), where π ∈ P 1 ( X × X ) runs o v er all c ouplings of µ and ν , that is, the probability measures π with the prop ert y that π ( A × X ) = µ ( A ) and π ( X × A ) = ν ( A ) for an y Borel subset A ⊆ X . Theorem 2.1 (L. V. K a n torovic h, cf. [17, Theorem 5.1, R emark 6.5]) . F or any µ, ν ∈ P 1 ( X ) , we ha v e d W 1 ( µ, ν ) = sup n Z X ψ ( x ) dµ ( x ) − Z X ψ ( x ) dν ( x ) o , wher e the supr emum is taken over al l 1 -Lipsch i tz function ψ : X → R . 2.2. Basics of the cen ter of mass of a measure on CA T(0)-spaces. In this sub- section, we review Sturm’s w orks ab o ut probability measures on a CA T(0)-spaces , whic h is n eeded for the pro of of the main theorem. Refer [9] and [16] for details. W e shall recall some standar d terminologies in metric geometry . Let ( X , d X ) b e a metric space. A rectifiable curv e γ : [0 , 1] → X is called a ge o desic if its arclength coincides with the distance d X ( γ (0) , γ (1)) and it has a constan t speed, i.e., parameterized prop ortionally to the arc length. W e sa y that ( X , d X ) is a ge o desic metric sp ac e if an y t w o p o in ts in X are joined b y a geo desic b etw ee n them. A geo desic metric space N is called a CA T(0 ) -sp ac e if we hav e d N ( x, γ (1 / 2) ) 2 ≤ 1 2 d N ( x, y ) 2 + 1 2 d N ( x, z ) 2 − 1 4 d N ( y , z ) 2 for any x, y , z ∈ N and an y minimizing geo desic γ : [0 , 1] → N from y to z . F or example, Hadamard manifolds, Hilb ert spaces, and R -trees are all CA T(0)-spaces . F or any ν ∈ P 1 ( X ) and z ∈ X , we consider the f unction h z ,ν : X → R de fined b y h z ,ν ( x ) := Z X { d X ( x, y ) 2 − d X ( z , y ) 2 } dν ( y ) . 4 KEI FUNAN O Note that Z X | d X ( x, y ) 2 − d X ( z , y ) 2 | dν ( y ) ≤ d X ( x, z ) Z X { d X ( x, y ) + d X ( z , y ) } dν ( y ) < + ∞ . A p oin t z 0 ∈ X is called t he c enter of mass of the measure ν ∈ B 1 ( X ) if for any z ∈ X , z 0 is a unique minimizing p oin t o f the function h z ,ν . W e denote the p oin t z 0 b y c ( ν ) . Note that if the measure ν moreo v er satisfies that ν ∈ P 2 ( X ), then we ha v e Z X d X ( c ( ν ) , y ) 2 dν ( y ) = inf x ∈ X Z X d X ( x, y ) 2 dν ( y ) . A me tric space X is said to be c entric if ev ery ν ∈ P 1 ( X ) has the cen ter of mass. Prop osition 2.2 (cf. [16, Prop osition 4 . 3]) . A CA T( 0 )-sp ac e is c entric. A simple v a riational argumen t implie s the follo wing lemma: Lemma 2.3 (cf. [16, Prop osition 5.4]) . L et H b e a Hilb ert sp ac e. T hen, for e ac h ν ∈ P 1 ( H ) , we have c ( ν ) = Z H y dν ( y ) . Lemma 2.4 (cf. [16, Prop o sition 5.10 ]) . L et N b e a Hadamar d manifold a n d ν ∈ P 1 ( N ) . Then, x = c ( ν ) if and only if Z N exp − 1 x ( y ) dν ( y ) = 0 . In p articular, identifying the tangent sp ac e o f N at c ( ν ) with the Euclide an sp ac e of the same dimension, we have c ((exp − 1 c ( ν ) ) ∗ ( ν )) = 0 . Let (Ω , A , P ) b e a probabilit y space and N a cen tric metric space. F or an N -v alued random v ariable Z : Ω → N satisfying Z ∗ P ∈ P 1 ( N ), w e de fine its exp e ctation E ( Z ) ∈ N b y the p oint c ( Z ∗ P ). Let X be a geo desic metric space. A function ϕ : X → R is called c on v ex if the function ϕ ◦ γ : [0 , 1] → R is con v ex for eac h geo desic γ : [0 , 1] → X . Prop osition 2.5 ( Conv exit y of a distance function, cf. [16, Corollary 2.5]) . L et N b e a CA T(0)-sp ac e and γ , η : [0 , 1] → N b e two ge o desics. Then, for any t ∈ [0 , 1] , we have d N ( γ ( t ) , η ( t )) ≤ (1 − t ) d N ( γ (0) , η (0)) + t d N ( γ (1) , η (1)) . Theorem 2.6 (Jensen’s inequalit y , cf. [16, Theorem 6.2]) . L et N b e a CA T(0)-s p ac e. Then, for any lower semic ontinuous c onvex function ϕ : N → R and ν ∈ P 1 ( N ) , we have ϕ ( c ( ν )) ≤ Z N ϕ ( x ) dν ( x ) , pr ovide d the rig ht-hand side is wel l- d efine d. Applying Prop osition 2.5 to Theorem 2.6 , w e obtain the follo wing corollary: AN ASYMPTOTIC V ARI ANT OF THE FUBINI THEOREM FOR MAPS IN TO CA T(0)-SP ACES 5 Corollary 2.7. L et N b e a CA T(0)-sp ac e. Then, for an y p 0 ∈ N and ν ∈ P 1 ( N ) , we have d N ( p 0 , c ( ν )) ≤ Z N d N ( p 0 , p ) dν ( p ) . Prop osition 2.8 (V a r iance inequalit y , [16, Prop osition 4.4]) . L et N b e a CA T(0 ) -sp ac e and ν ∈ P 1 ( N ) . Then, for any z ∈ N , we have Z N { d N ( z , x ) 2 − d N ( c ( ν ) , x ) 2 } dν ( x ) ≥ d N ( z , c ( ν )) 2 . (2.1) Note that if N is a Hilb ert spac e, then we hav e the equalit y in (2.1) . Prop osition 2.9 (cf. [16, Theorem 2 .5]) . L et N b e a CA T(0)-sp ac e. Then, for any µ, ν ∈ P 1 ( N ) , we have d N ( c ( µ ) , c ( ν )) ≤ d W 1 ( µ, ν ) . 3. Proof of t he main theorem Let X a nd Y b e a t w o mm-spaces and N a CA T(0)-space. Giv en a uniformly con tinuous map f : X × Y → N with f ∗ ( µ X × µ Y ) ∈ P 1 ( N ), w e easily see that ( f y ) ∗ ( µ X ) ∈ P 1 ( N ) for µ Y -a.e. y ∈ Y . Since Y has t he f ull-supp ort and the map f is uniformly con tin uous, w e see that ( f y ) ∗ ( µ X ) ∈ P 1 ( N ) for any y ∈ N . W e shall consider the map g f : Y → N defined b y g f ( y ) := E ( f y ). Lemma 3.1. The map g f : Y → N is uniformly c ontinuous. In p articular, the map is Bor el me as ur able. Pr o of. F rom Theorem 2.1 and Prop osition 2.6, for an y y , y ′ ∈ Y , w e hav e d N ( g f ( y ) , g f ( y ′ )) ≤ d W 1 (( f y ) ∗ ( µ X ) , ( f y ′ ) ∗ ( µ X )) = sup n Z N ψ ( z ) d ( f y ) ∗ ( µ X )( z ) − Z N ψ ( z ) d ( f y ′ ) ∗ ( µ X )( z ) o = sup n Z X ψ ( f ( x, y )) dµ X ( x ) − Z X ψ ( f ( x, y ′ )) dµ X ( x ) o ≤ Z X d N ( f ( x, y ) , f ( x, y ′ )) dµ X ( x ) , where eac h suprem um is taken o v er all 1- L ipschitz function ψ : N → R . Observ e t ha t the righ t-hand side of the ab ov e inequality con v erges to zero as d Y ( y , y ′ ) → 0. This completes the proof.  Lemma 3.2. We have ( g f ) ∗ ( µ Y ) ∈ P 1 ( N ) . Pr o of. T aking an y p oin t p 0 ∈ N , from Corollar y 2.7 , w e obtain Z Y d N ( E ( f y ) , p 0 ) dµ Y ( y ) ≤ Z X × Y d N ( f ( x, y ) , p 0 ) d ( µ X × µ Y )( x, y ) < + ∞ . This comple tes the proof.  6 KEI FUNAN O The following example asserts that the equality E ( f ) = E y ( E ( f y )) do es not hold for non-linear CA T(0)- spaces in general: Example 3.3. F or i = 1 , 2 , 3, let T i := { ( i, r ) | r ∈ [0 , + ∞ ) } b e a cop y of [0 , + ∞ ) equipped with the usual Euclidean distance function. The trip o d T is the metric space obtained by gluing together all these spaces T i , i = 1 , 2 , 3, at their origins with the in trinsic distance function. Let { a, b } b e an arbitrary t w o-p oin t mm-space equipp ed with the uniform probability measure. Let us conside r the map f : { a, b } 2 → T defined by f ( a, a ) := (1 , 1) ∈ T 1 , f ( b, a ) := ( 2 , 1) ∈ T 2 , and f ( a, b ) = f ( b, b ) := (3 , 1) ∈ T 3 . In this case, we easily see that E ( f ) = (0 , 0), E ( f a ) = (0 , 0), E ( f b ) = (3 , 1), and therefore E y ( E ( f y )) = (3 , 1 / 2). Pr o of of The or em 1.1. Iterating Corollary 2.7, w e ha v e d N ( E ( f ) , E y ( E ( f y ))) ≤ Z X × Y d N ( f ( x, y ′ ) , E y ( E ( f y ))) d ( µ X × µ Y )( x, y ′ ) ≤ Z X × Y × Y d N ( f ( x, y ′ ) , E ( f y ′′ )) d ( µ X × µ Y × µ Y )( x, y ′ , y ′′ ) ≤ V 1 ( f ) . Thereb y , w e obta in the ineq uality (1.1 ). T o prov e the inequalit y (1.2), w e are going to iterate Prop osition 2.8. Since f ∗ ( µ X × µ Y ) 6∈ P 2 ( N ) implies V 2 ( f ) = + ∞ , w e assume that f ∗ ( µ X × µ Y ) ∈ P 2 ( N ). F rom Propo- sition 2.8, w e ha v e Z X × Y d N ( f ( x, y ′ ) , E ( f )) 2 d ( µ X × µ Y )( x, y ′ ) (3.1) = Z Y dµ Y ( y ′ ) Z X d N ( f y ′ ( x ) , E ( f )) 2 dµ X ( x ) ≥ Z Y dµ Y ( y ′ ) n Z X d N ( f y ′ ( x ) , E ( f y ′ )) 2 dµ X ( x ) + d N ( E ( f y ′ ) , E ( f )) 2 o = Z X × Y d N ( f y ′ ( x ) , E ( f y ′ )) 2 d ( µ X × µ Y )( x, y ′ ) + Z Y d N ( E ( f y ′ ) , E ( f )) 2 dµ Y ( y ′ ) ≥ Z X × Y d N ( f y ′ ( x ) , E ( f y ′ )) 2 d ( µ X × µ Y )( x, y ′ ) + Z Y d N ( E ( f y ′ ) , E y ( E ( f y ))) 2 dµ Y ( y ′ ) + d N ( E ( f ) , E y ( E ( f y ))) 2 . Since d N ( f y ′ ( x ) , E ( f y ′ )) 2 + d N ( E ( f y ′ ) , E y ( E ( f y ))) 2 ≥ 1 2 d N ( f y ′ ( x ) , E y ( E ( f y ))) 2 , AN ASYMPTOTIC V ARI ANT OF THE FUBINI THEOREM FOR MAPS IN TO CA T(0)-SP ACES 7 substituting this in to the ine quality (3.1), w e get Z X × Y d N ( f ( x, y ′ ) , E ( f )) 2 d ( µ X × µ Y )( x, y ′ ) ≥ 1 2 Z X × Y d N ( f ( x, y ′ ) , E y ( E ( f y ))) 2 d ( µ X × µ Y )( x, y ′ ) + d N ( E ( f ) , E y ( E ( f y ))) 2 Since Z X × Y d N ( f ( x, y ′ ) , E ( f )) 2 d ( µ X × µ Y )( x, y ′ ) ≤ Z X × Y d N ( f ( x, y ′ ) , E y ( E ( f y ))) 2 d ( µ X × µ Y )( x, y ′ ) , w e therefore obtain d N ( E ( f ) , E y ( E ( f y ))) 2 ≤ 1 2 Z X × Y d N ( f ( x, y ′ ) , E y ( E ( f y ))) 2 d ( µ X × µ Y )( x, y ′ ) . (3.2) By v irtue of Prop osition 2.8, w e also get Z X × Y d N ( f ( x, y ′ ) , E y ( E ( f y ))) 2 d ( µ X × µ Y )( x, y ′ ) (3.3) ≤ Z X × Y d ( µ X × µ Y )( x, y ′ ) n Z Y  d N ( f y ′ ( x ) , E ( f y ′′ )) 2 − d N ( E ( f y ′ ) , E y ( E ( f y ))) 2  dµ Y ( y ′′ ) o = Z X × Y × Y d N ( f y ′ ( x ) , E ( f y ′′ )) 2 d ( µ X × µ Y × µ Y )( x, y ′ , y ′′ ) − Z Y d N ( E ( f y ′ ) , E y ( E ( f y ))) 2 dµ Y ( y ′ ) . Since d N ( f y ′ ( x ) , E ( f y ′′ )) 2 ≤ Z X  d N ( f y ′′ ( x ′ ) , f y ′ ( x )) 2 − d N ( f y ′′ ( x ′ ) , E ( f y ′′ )) 2  dµ X ( x ′ ) from Prop osition 2.8, substituting this in to (3.3), w e ha v e Z X × Y d N ( f ( x, y ′ ) , E y ( E ( f y ))) 2 d ( µ X × µ Y )( x, y ′ ) ≤ V 2 ( f ) 2 − Z X × Y d N ( f y ′ ( x ) , E ( f y ′ )) 2 d ( µ X × µ Y )( x, y ′ ) − Z Y d N ( E ( f y ′ ) , E y ( E ( f y ))) 2 dµ Y ( y ′ ) ≤ V 2 ( f ) 2 − 1 2 Z X × Y d N ( f y ′ ( x ) , E y ( E ( f y ))) 2 d ( µ X × µ Y )( x, y ′ ) . W e therefore obtain Z X × Y d N ( f ( x, y ′ ) , E y ( E ( f y ′ ))) 2 d ( µ X × µ Y )( x, y ′ ) ≤ 2 3 V 2 ( f ) 2 . 8 KEI FUNAN O Com bining this with the inequalit y (3 .2), we finally obtain the inequalit y ( 1 .2). This completes the pro of.  4. Applica tions 4.1. Pro duct inequalities. Prop osition 4.1 (Y. G. Reshetn y ak, cf. [16, Prop osition 2.4 ]) . F or a ny four p oints x 1 , x 2 , x 3 , x 4 in a CA T ( 0 )-sp ac e N , we hav e d N ( x 1 , x 3 ) 2 + d N ( x 2 , x 4 ) 2 ≤ d N ( x 1 , x 2 ) 2 + d N ( x 2 , x 3 ) 2 + d N ( x 3 , x 4 ) 2 + d N ( x 4 , x 1 ) 2 . Giv en an mm-space X and a metric s pace Y w e define Obs L p -V ar Y ( X ) := sup { V p ( f ) | f : X → Y is a 1-Lipsc hitz map } , and call it the ob servable L p -variation of X . The idea of the observ able L p -v ariation comes from the quan tum and statistical mec hanics, that is, we think of µ X as a state on a configura t ion space X and f is interprete d as an observ able. Corollary 4.2. L et X and Y b e two mm-sp ac es a n d N a CA T(0)-sp ac e. Then, we hav e Obs L 2 - V ar N ( X × Y ) 2 ≤ Obs L 2 - V ar N ( X ) + Obs L 2 - V ar N ( Y ) . (4.1) Pr o of. Let f : X × Y → N b e a n arbitrary 1-Lipsc hitz map. Then, putting Z := X × Y , from Prop osition 4.1, w e obtain V 2 ( f ) 2 = 1 2 Z Z × Z { d N ( f ( x, y ) , f ( x ′ , y ′ )) 2 + d N ( f ( x, y ′ ) , f ( x ′ , y )) 2 } d ( µ Z × µ Z )( x, y , x ′ , y ′ ) ≤ 1 2 Z Z × Z { d N ( f ( x, y ) , f ( x ′ , y )) 2 + d N ( f ( x ′ , y ) , f ( x ′ , y ′ )) 2 + d N ( f ( x ′ , y ′ ) , f ( x, y ′ )) 2 + d N ( f ( x, y ′ ) , f ( x, y ) ) 2 } d ( µ Z × µ Z )( x, y , x ′ , y ′ ) = Z X V 2 ( f x ) 2 dµ X ( x ) + Z Y V 2 ( f y ) 2 dµ Y ( y ) ≤ Obs L 2 -V ar N ( X ) 2 + Obs L 2 -V ar N ( Y ) 2 . This comple tes the proof.  Lemma 4.3. L et X and Y b e two mm - s p ac es and Z a metric sp ac e. Then, for any p ≥ 1 , we have Obs L p - V ar Z ( X × Y ) p ≤ 2 p − 1 Obs L p - V ar Z ( X ) p + 2 p − 1 Obs L p - V ar Z ( Y ) p . (4.2) Pr o of. Giv en an y 1-L ipschitz map f : X × Y → Z , putting W := X × Y , w e ha v e V p ( f ) p ≤ Z W × W 2 p − 1 { d Z ( f ( x, y ) , f ( x, y ′ )) p + d Z ( f ( x, y ′ ) , f ( x ′ , y ′ )) p } d ( µ W × µ W )( x, y , x ′ , y ′ ) = 2 p − 1 Z X V p ( f x ) p dµ X ( x ) + 2 p − 1 Z Y V p ( f y ) p dµ Y ( y ) ≤ 2 p − 1 Obs L p -V ar Z ( X ) p + 2 p − 1 Obs L p -V ar Z ( Y ) p . AN ASYMPTOTIC V ARI ANT OF THE FUBINI THEOREM FOR MAPS IN TO CA T(0)-SP ACES 9 This comple tes the proof.  Note that the inequ ality (4.1) is sharp er t han the inequalit y (4.2) in the case where p = 2 and Z is a CA T(0)-space. Com bining Theorem 1.1 and Lemma 4.3 w e obtain the follo wing corolla ry: Corollary 4.4. L et { X n } ∞ n =1 and { Y n } ∞ n =1 b e a se quenc es of m m-sp ac es and { N n } ∞ n =1 b e a se quenc es o f CA T(0)-sp ac es. T hen, assuming that Obs L 1 - V ar N n ( X n ) → 0 as n → ∞ and Obs L 1 - V ar N n ( Y n ) → 0 as n → ∞ , we have d N n ( E ( f ) , E y n ( E ( f y n ))) → 0 as n → ∞ for any se quenc e { f n : X n × Y n → N n } ∞ n =1 of 1 -Lipschitz maps. 4.2. The non-zero first eigen v alue of Laplacian and the observ able L 2 -v ariation. Although the same metho d in [5 ] and [7] implies the f ollo wing prop osition, w e prov e it for the completeness. Prop osition 4.5 (cf. [5, Section 13], [7, Section 3 1 2 . 41]) . L et M b e a c omp ac t c onne cte d Riemannian manifold a nd N ′ an n -dimensio nal Hadamar d ma nifold. Then, we ha v e Obs L 2 - V ar N ′ ( M ) ≤ 2 r n λ 1 ( M ) . Pr o of. Let f : M → N ′ b e an arbitrary 1-Lipsc hitz map. W e shall pro v e tha t Z M d N ′ ( f ( x ) , E ( f )) 2 dµ M ( x ) ≤ n λ 1 ( M ) . (4.3) If the inequalit y (4.3) ho lds, t hen w e finish the pr o of sinc e V 2 ( f ) ≤ 2  Z M d N ′ ( f ( x ) , E ( f )) 2 dµ M ( x )  1 / 2 ≤ 2 r n λ 1 ( M ) . Supp ose that Z M d N ′ ( f ( x ) , E ( f )) 2 dµ M ( x ) > n λ 1 ( M ) . (4.4) W e iden tify the tangent space of N ′ at the p oin t E ( f ) with the Euclidean space R n and consider the map f 0 := exp − 1 E ( f ) ◦ f : M → R n . Ac cording to the hing e theorem (see [14, Chapter IV, R emark 2.6 ]) , the map f 0 is a 1-Lipsc hitz map. Note that | f 0 ( x ) | = d N ′ ( f ( x ) , E ( f )) for an y x ∈ X b ecause the map exp − 1 E ( f ) is isometric on rays is suing from the point E ( f ). Hence, from the inequalit y (4.4), w e ha v e Z M | f 0 ( x ) | 2 dµ M ( x ) > n λ 1 ( M ) . 10 KEI FUNAN O Denoting b y ( f 0 ( x )) i the i - th comp onen t of f 0 ( x ), we therefore see that there ex ists i 0 suc h that Z M | ( f 0 ( x )) i 0 | 2 dµ M ( x ) > 1 λ 1 ( M ) . (4.5) Note the f unction ( f 0 ) i 0 has t he mean zero from Lemmas 2.3 and 2.4. Com bining this with the inequalit y (4.5), w e therefore obtain λ 1 ( M ) = inf R M | grad x g | 2 dµ M ( x ) R M | g ( x ) | 2 dµ M ( x ) < λ 1 ( M ) , where the infim um is tak en o v er all Lipsc hitz function g : M → R with the mean zero. This is a c ontradiction. This completes the pro of.  One c an obtain a similar r esult to Proposition 4.5 for a finite connected graph. Theorem 4.6 (cf. [2, P rop osition 5.7]) . L et X b e an mm- s p ac e and T a n R -tr e e. Th e n, we have Obs L 2 - V ar T ( X ) 2 ≤ (38 + 16 √ 2) Obs L 2 - V ar R ( X ) 2 . Com bining Prop osition 4 .5 with Theorem 4.6, we obtain the following corolla ry: Corollary 4.7. L et M b e a c omp a ct c onne ct e d Riemannian manifold and T an R -tr e e. Then, we have Obs L 2 - V ar T ( M ) 2 ≤ 4(38 + 16 √ 2) λ 1 ( M ) . Pr o of of Cor ol lary 1.2. The corollary follo ws from Theorem 1.1 together with Corollary 4.2, P rop osition 4.5, and Corollary 4 .7. This comple tes the pr o of.  Ac kno wledgemen ts. 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