Gamma-convergence of Cheeger energies with respect to increasing distances
aa r X i v : . [ m a t h . M G ] F e b GAMMA-CONVERGENCE OF CHEEGER ENERGIES WITH RESPECTTO INCREASING DISTANCES
DANKA LU ˇCI ´C AND ENRICO PASQUALETTO
Abstract.
We prove a Γ-convergence result for Cheeger energies along sequences of metricmeasure spaces, where the measure space is kept fixed, while distances are monotonicallyconverging from below to the limit one. As a consequence, we show that the infinitesimalHilbertianity condition is stable under this kind of convergence of metric measure spaces. Introduction
In the successful theory of weakly differentiable functions over metric measure spaces, aleading role is played by the so-called Cheeger p -energy, which was introduced in [2] andgeneralises the classical Dirichlet p -energy functional. The purpose of this paper is to studythe convergence of Cheeger p -energies along a sequence of metric measure spaces, where theunderlying set and the measure are fixed, while distances monotonically converge from below.More precisely, given a metric measure space (X , d , m ) and a sequence ( d i ) i ∈ N of distanceson X inducing the same topology as d and satisfying d i ր d , we prove (in Theorem 4.1) thatfor any p ∈ (1 , ∞ ) the Cheeger p -energies E d i Ch ,p : L p ( m ) → [0 , + ∞ ] associated with (X , d i , m )converge to E d Ch ,p in the sense of Mosco. As shown in Example 4.4, this kind of statementmight totally fail in the case where d i ց d . Since the family of quadratic forms is closedunder Mosco-convergence, an interesting consequence of Theorem 4.1 is the stability of theinfinitesimal Hilbertianity condition (that was introduced in [5] and states the quadraticity ofthe Cheeger 2-energy functional) with respect to increasing limits of the involved distances.Sub-Riemannian manifolds constitute a significant example of metric structures where theabove results apply, as the induced length distances can be monotonically approximated frombelow by Riemannian ones; cf. the discussion in Remark 4.3.A previous result on the Mosco-convergence of Cheeger energies was obtained in [6, The-orem 6.8] for sequences of CD ( K, ∞ ) spaces that converge with respect to (a variant of) thepointed measured Gromov–Hausdorff topology. However, since measured Gromov–Hausdorffconvergence is a zeroth-order concept, while the Cheeger energy is a first-order one, we cannotexpect such Mosco-convergence result to hold on arbitrary metric measure spaces. Indeed,given an arbitrary metric measure space (X , d , m ), one can easily construct a sequence of dis-crete measures ( m i ) i ∈ N that weakly converge to m ; consequently, since the Cheeger energies Date : February 15, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Cheeger energy, Mosco-convergence, infinitesimal Hilbertianity. associated with the spaces (X , d , m i ) are identically zero, the Mosco-convergence result willgenerally fail. In the case of CD ( K, ∞ ) spaces, the convergence of the Cheeger energies isboosted by the uniform lower bound on the Ricci curvature (encoded in the CD condition),which is a second-order notion. Conversely, in our main Theorem 4.1 we do not require anyregularity at the level of the involved metric measure spaces, but instead we consider a notionof convergence that is much stronger than the pointed measured Gromov–Hausdorff one.We conclude the introduction by briefly describing an approximation result for Lipschitzfunctions (Proposition 3.3) that will have an essential role in the proof of Theorem 4.1. Underthe same assumptions as in the Mosco-convergence result for Cheeger energies, we prove thatevery d -Lipschitz function f can be approximated (in the integral sense) by a d i -Lipschitzfunction g , for some index i ∈ N sufficiently large, such that the integral of the p -power ofthe asymptotic slope of g is close to that of f . This goal is achieved by appealing to theasymptotic-slope-preserving extension result for Lipschitz functions obtained in [4]. Acknowledgements.
The authors wish to thank Tapio Rajala for the careful reading ofa preliminary version of this paper. The first named author was supported by the project2017TEXA3H “Gradient flows, Optimal Transport and Metric Measure Structures”, fundedby the Italian Ministry of Research and University. The second named author was supportedby the Balzan project led by Luigi Ambrosio.2.
Preliminaries
Let (X , d ) be a given metric space. We denote by τ ( d ) the topology on X induced by thedistance d . The open ball and the closed ball of center x ∈ X and radius r > B d r ( x ) := (cid:8) y ∈ X (cid:12)(cid:12) d ( x, y ) < r (cid:9) , ¯ B d r ( x ) := (cid:8) y ∈ X (cid:12)(cid:12) d ( x, y ) ≤ r (cid:9) , respectively. The space of d -Lipschitz functions f : X → R will be denoted by LIP d (X). Givenany f ∈ LIP d (X) and E ⊆ X, we denote by Lip d ( f ; E ) ∈ [0 , + ∞ ) and lip d a ( f ) : X → [0 , + ∞ )the Lipschitz constant of f | E and the asymptotic slope of f , respectively. Videlicet, we setLip d ( f ; E ) := sup (cid:26) (cid:12)(cid:12) f ( x ) − f ( y ) (cid:12)(cid:12) d ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) x, y ∈ E, x = y (cid:27) , lip d a ( f )( x ) := inf r> Lip d (cid:0) f ; B d r ( x ) (cid:1) , for every x ∈ X , where we adopt the convention that Lip d ( f ; ∅ ) = Lip d (cid:0) f ; { x } (cid:1) := 0. For the sake of brevity,we will use the shorthand notation Lip d ( f ) := Lip d ( f ; X). Observe that lip d a ( f ) ≤ Lip d ( f ). Remark 2.1.
Let X be a non-empty set. Let d and d ′ be distances on X such that d ≤ d ′ .Then for any x, x ′ , y, y ′ ∈ X it holds that (cid:12)(cid:12) d ( x, y ) − d ( x ′ , y ′ ) (cid:12)(cid:12) ≤ d ( x, x ′ ) + d ( y, y ′ ) ≤ d ′ ( x, x ′ ) + d ′ ( y, y ′ ) ≤ √ d ′ × d ′ ) (cid:0) ( x, y ) , ( x ′ , y ′ ) (cid:1) , thus d : X × X → [0 , + ∞ ) is ( d ′ × d ′ )-continuous, where d ′ × d ′ stands for the product distance( d ′ × d ′ ) (cid:0) ( x, y ) , ( x ′ , y ′ ) (cid:1) := p d ′ ( x, x ′ ) + d ′ ( y, y ′ ) , for every ( x, y ) , ( x ′ , y ′ ) ∈ X × X . AMMA-CONVERGENCE OF CHEEGER ENERGIES WITH RESPECT TO INCREASING DISTANCES 3
Moreover, given f ∈ LIP d (X) and E ⊆ X, we can estimate (cid:12)(cid:12) f ( x ) − f ( y ) (cid:12)(cid:12) ≤ Lip d ( f ; E ) d ′ ( x, y )for every x, y ∈ E . This shows that LIP d (X) ⊆ LIP d ′ (X) and thatLip d ′ ( f ; E ) ≤ Lip d ( f ; E ) , for every f ∈ LIP d (X) and E ⊆ X . In particular, we obtain that lip d ′ a ( f ) ≤ lip d a ( f ) for every f ∈ LIP d (X). (cid:4) By a metric measure space (X , d , m ) we mean a complete and separable metric space (X , d ),which is endowed with a boundedly-finite Borel measure m ≥
0. One of the possible waysto introduce Sobolev spaces on (X , d , m ) is via relaxation of upper gradients. Instead of theoriginal approach that was introduced by Cheeger [2], we present its equivalent reformulation(via relaxation of the asymptotic slope) that was studied by Ambrosio–Gigli–Savar´e in [1].Given a metric measure space (X , d , m ) and an exponent p ∈ (1 , ∞ ), let us define theasymptotic p -energy functional E d a,p : L p ( m ) → [0 , + ∞ ] as E d a,p ( f ) := ( p R lip d a ( f ) p d m , + ∞ , if f ∈ LIP d (X) is boundedly-supported,otherwise.Then the Cheeger p -energy functional E d Ch ,p : L p ( m ) → [0 , + ∞ ] is defined as the L p ( m )-lowersemicontinuous envelope of E d a,p . Videlicet, for any function f ∈ L p ( m ) we define E d Ch ,p ( f ) := inf (cid:26) lim n →∞ E d a,p ( f n ) (cid:12)(cid:12)(cid:12)(cid:12) ( f n ) n ⊆ L p ( m ) , f n → f strongly in L p ( m ) (cid:27) . It turns out that E d Ch ,p is weakly lower semicontinuous, meaning that E d Ch ,p ( f ) ≤ lim n E d Ch ,p ( f n )whenever f ∈ L p ( m ) and ( f n ) n ⊆ L p ( m ) satisfy f n ⇀ f weakly in L p ( m ). The p -Sobolev spaceon (X , d , m ) is then defined as the finiteness domain of E d Ch ,p , videlicet W ,p (X) := (cid:8) f ∈ L p ( m ) (cid:12)(cid:12) E d Ch ,p ( f ) < + ∞ (cid:9) . It holds that W ,p (X) is a Banach space if endowed with the following norm: k f k W ,p (X) := (cid:0) k f k pL p ( m ) + p E d Ch ,p ( f ) (cid:1) /p , for every f ∈ W ,p (X) . In general, the 2-Sobolev space is not Hilbert. A metric measure space (X , d , m ) is said to beinfinitesimally Hilbertian [5] provided the associated 2-Sobolev space W , (X) is Hilbert, orequivalently provided E d Ch , is a quadratic form. Remark 2.2.
Let (X , d , m ) be a metric measure space. Let d ′ be a distance on X with d ≤ d ′ and τ ( d ) = τ ( d ′ ), thus (X , d ′ , m ) is a metric measure space as well. Then Remark 2.1 yields E d ′ a,p ≤ E d a,p , E d ′ Ch ,p ≤ E d Ch ,p , (2.1)for any given exponent p ∈ (1 , ∞ ). (cid:4) DANKA LUˇCI´C AND ENRICO PASQUALETTO An approximation result
Aim of this section is to achieve an approximation result for Lipschitz functions ( i.e. ,Proposition 3.3), which will be a key tool in order to prove our main Theorem 4.1.
Remark 3.1.
Let X be a non-empty set and ( d i ) i ∈ ¯ N a sequence of distances on X satisfying d i ( x, y ) ր d ∞ ( x, y ) , for every x, y ∈ X . Then d i → d ∞ uniformly on each subset of X × X that is compact with respect to τ ( d ∞ × d ∞ ).Indeed, Remark 2.1 grants that d i : X × X → R is ( d ∞ × d ∞ )-continuous for all i ∈ ¯ N . (cid:4) We begin with a preliminary approximation result, where the given Lipschitz function isuniformly approximated on a compact set and just the global Lipschitz constant is controlled.
Lemma 3.2.
Let (X , d ) be a metric space. Suppose there exists a sequence ( d i ) i ∈ N of distanceson X such that d i ( x, y ) ր d ( x, y ) as i → ∞ for every x, y ∈ X . Let f ∈ LIP d (X) be given.Then for any K ⊆ X compact and ε > there exist i ∈ N and g ∈ LIP d i (X) such that max K | g − f | ≤ ε, (3.1a)Lip d i ( g ) ≤ Lip d ( f ) . (3.1b) Proof.
Call L := Lip d ( f ) and fix a dense sequence ( x j ) j ∈ N in K . Given any n ∈ N , we define˜ g n ( x ) := (cid:0) − L d ( x, x ) + f ( x ) (cid:1) ∨ · · · ∨ (cid:0) − L d ( x, x n ) + f ( x n ) (cid:1) − n , for every x ∈ X . Note that (˜ g n ) n ∈ N ⊆ LIP d (X) and ˜ g n ≤ ˜ g n +1 ≤ f for all n ∈ N . We claim that ˜ g n ( x ) → f ( x )as n → ∞ for every x ∈ K . To prove it, fix x ∈ K and δ >
0. Pick ¯ n ∈ N such that 1 / ¯ n ≤ δ and d ( x, x ¯ n ) ≤ δ . Then for any n ≥ ¯ n it holds that˜ g n ( x ) ≥ − L d ( x, x ¯ n ) + f ( x ¯ n ) − n ≥ f ( x ) − L d ( x, x ¯ n ) − n ≥ f ( x ) − (2 L + 1) δ, which grants that ˜ g n ( x ) ր f ( x ) by arbitrariness of δ . Therefore, we have that ˜ g n → f uniformly on K , so that there exists n ∈ N for which the function ˜ g := ˜ g n satisfies | ˜ g − f | ≤ ε/ K . Given any i ∈ N , let us define the function g i ∈ LIP d i (X) as g i ( x ) := (cid:0) − L d i ( x, x ) + f ( x ) (cid:1) ∨ · · · ∨ (cid:0) − L d i ( x, x n ) + f ( x n ) (cid:1) − n , for every x ∈ X . Note that g i ց ˜ g pointwise on K , as a consequence of the assumption d i ր d . Since each g i is continuous with respect to d , we deduce that g i → ˜ g uniformly on K , thus for some i ∈ N the function g := g i satisfies | g − ˜ g | ≤ ε/ K . Hence, it holds that | g − f | ≤ ε on K ,yielding (3.1a). Finally, we have that Lip d i ( g ) ≤ L = Lip d ( f ) by construction, whence (3.1b)and accordingly the statement follow. (cid:3) By combining Lemma 3.2 with a partition of unity argument and the extension result in[4], we show that also the asymptotic slope can be kept under control (in an integral sense).
AMMA-CONVERGENCE OF CHEEGER ENERGIES WITH RESPECT TO INCREASING DISTANCES 5
Proposition 3.3.
Let (X , d , m ) be a metric measure space. Suppose to have a sequence ( d i ) i ∈ N of distances on X such that d i ( x, y ) ր d ( x, y ) as i → ∞ for every x, y ∈ X and τ ( d i ) = τ ( d ) for every i ∈ N . Fix an exponent p ∈ (1 , ∞ ) and a boundedly-supported function f ∈ LIP d (X) .Then for any ε > there exist i ∈ N and g ∈ LIP d i (X) boundedly-supported such that Z | g − f | p d m ≤ ε, (3.2a) Z lip d i a ( g ) p d m ≤ Z lip d a ( f ) p d m + ε. (3.2b) Proof.
First of all, fix a point ¯ x ∈ X and a radius
R > f ) ⊆ B d R (¯ x ). Denoteby B the ball ¯ B d R +2 (¯ x ). Moreover, fix any ε ′ ∈ (0 , /
4) such that (cid:20)(cid:16) p Lip d ( f ) p − + 1 (cid:17) m ( B ) + (cid:16)
15 Lip d ( f ) + sup X | f | + 7 (cid:17) p (cid:21) ε ′ ≤ ε. (3.3) Step 1: Construction of the auxiliary function ˜ h . Since X ∋ x Lip d (cid:0) f ; B d /n ( x ) (cid:1) is a Borel function for any n ∈ N and lip d a ( f )( x ) = lim n →∞ Lip d (cid:0) f ; B d /n ( x ) (cid:1) for every x ∈ X,by virtue of Egorov’s theorem there exist K ⊆ B compact and r > m ( B \ K ) ≤ ε ′ andLip d (cid:0) f ; B d r ( x ) (cid:1) ≤ lip d a ( f )( x ) + ε ′ , for every x ∈ K. (3.4)Choose some points x , . . . , x k ∈ K for which K ⊆ S kj =1 B d r ( x j ). Fix a d -Lipschitz partitionof unity { ψ , . . . , ψ k } of K subordinated to (cid:8) K ∩ B d r ( x ) , . . . , K ∩ B d r ( x k ) (cid:9) . Videlicet, eachfunction ψ j : K → [0 ,
1] is d -Lipschitz, satisfies spt( ψ j ) ⊆ K ∩ B d r ( x j ), and P kj =1 ψ j ( x ) = 1for every x ∈ K . Since d i → d uniformly on K × K (by Remark 3.1), there exists i ∈ N suchthat d ( x, y ) ≤ d i ( x, y ) + ε ′ r for every x, y ∈ K and i ≥ i . Given any j = 1 , . . . , k , pick somefunction f j ∈ LIP d (X) such that f j | B d r ( x j ) = f | B d r ( x j ) and Lip d ( f j ) = Lip d (cid:0) f ; B d r ( x j ) (cid:1) , thuswe can find (by Lemma 3.2) an index i ( j ) ≥ i and a function h j ∈ LIP d i ( j ) (X) such that (cid:12)(cid:12) h j ( x ) − f j ( x ) (cid:12)(cid:12) ≤ ε ′ (cid:0) k Lip d ( ψ j ) (cid:1) ∨ , for every x ∈ K, (3.5a)Lip d i ( j ) ( h j ) ≤ Lip d ( f j ) = Lip d (cid:0) f ; B d r ( x j ) (cid:1) . (3.5b)Let us denote i := max (cid:8) i (1) , . . . , i ( k ) (cid:9) and ˜ d := d i | K × K . Moreover, we define ˜ h : K → R as˜ h ( x ) := k X j =1 ψ j ( x ) h j ( x ) , for every x ∈ K. Step 2: Estimates for the Lipschitz constant of ˜ h . We claim that ˜ h ∈ LIP ˜ d ( K ) andthat Lip ˜ d (˜ h ) ≤ ε ′ + 5 Lip d ( f ). In order to prove it, fix any y, z ∈ K . Then we have that (cid:12)(cid:12) ˜ h ( y ) − ˜ h ( z ) (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 ψ j ( y ) (cid:0) h j ( y ) − h j ( z ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 (cid:0) ψ j ( y ) − ψ j ( z ) (cid:1)(cid:0) h j ( z ) − f ( z ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k X j =1 ψ j ( y ) (cid:12)(cid:12) h j ( y ) − h j ( z ) (cid:12)(cid:12) + k X j =1 (cid:12)(cid:12) ψ j ( y ) − ψ j ( z ) (cid:12)(cid:12)(cid:12)(cid:12) h j ( z ) − f ( z ) (cid:12)(cid:12) . (3.6) DANKA LUˇCI´C AND ENRICO PASQUALETTO
Observe that the first term in the second line of the above formula can be estimated as k X j =1 ψ j ( y ) (cid:12)(cid:12) h j ( y ) − h j ( z ) (cid:12)(cid:12) ≤ k X j =1 ψ j ( y ) Lip d i ( j ) ( h j ) d i ( j ) ( y, z ) (3.5b) ≤ Lip d ( f ) d i ( y, z ) . (3.7)In order to estimate the second term in (3.6), fix j = 1 , . . . , k . We consider three cases:i) If z ∈ B d r ( x j ), then f ( z ) = f j ( z ) and accordingly (cid:12)(cid:12) ψ j ( y ) − ψ j ( z ) (cid:12)(cid:12)(cid:12)(cid:12) h j ( z ) − f ( z ) (cid:12)(cid:12) (3.5a) ≤ Lip d i ( ψ j ) d i ( y, z ) ε ′ k Lip d ( ψ j ) ≤ ε ′ k d i ( y, z ) . ii) If z / ∈ B d r ( x j ) and y ∈ B d r ( x j ), then f j ( y ) = f ( y ) and d ( y, z ) > r . In particular, d ( y, z ) d i ( y, z ) ≤ d i ( y, z ) + ε ′ r d i ( y, z ) ≤ ε ′ r d ( y, z ) − ε ′ r < ε ′ − ε ′ < , (3.8)whence it follows that (cid:12)(cid:12) ψ j ( y ) − ψ j ( z ) (cid:12)(cid:12)(cid:12)(cid:12) h j ( z ) − f ( z ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) ψ j ( y ) − ψ j ( z ) (cid:12)(cid:12)h(cid:12)(cid:12) h j ( z ) − f j ( z ) (cid:12)(cid:12) + (cid:12)(cid:12) f j ( z ) − f j ( y ) (cid:12)(cid:12) + (cid:12)(cid:12) f j ( y ) − f ( z ) (cid:12)(cid:12)i ≤ Lip d i ( ψ j ) d i ( y, z ) (cid:12)(cid:12) h j ( z ) − f j ( z ) (cid:12)(cid:12) + ψ j ( y ) (cid:12)(cid:12) f j ( z ) − f j ( y ) (cid:12)(cid:12) + ψ j ( y ) (cid:12)(cid:12) f ( y ) − f ( z ) (cid:12)(cid:12) (3.5b) ≤ Lip d i ( ψ j ) d i ( y, z ) ε ′ k Lip d ( ψ j ) + ψ j ( y ) Lip d ( f j ) d ( y, z ) + ψ j ( y ) Lip d ( f ) d ( y, z ) ≤ ε ′ k d i ( y, z ) + 2 ψ j ( y ) Lip d ( f ) d ( y, z ) (3.8) ≤ (cid:18) ε ′ k + 4 ψ j ( y ) Lip d ( f ) (cid:19) d i ( y, z ) . iii) If z / ∈ B d r ( x j ) and y / ∈ B d r ( x j ), then trivially (cid:12)(cid:12) ψ j ( y ) − ψ j ( z ) (cid:12)(cid:12)(cid:12)(cid:12) h j ( z ) − f ( z ) (cid:12)(cid:12) = 0.By combining the estimates we obtained in i), ii), iii) with (3.7) and (3.6), we deduce that (cid:12)(cid:12) ˜ h ( y ) − ˜ h ( z ) (cid:12)(cid:12) ≤ (cid:0) ε ′ + 5 Lip d ( f ) (cid:1) d i ( y, z ) , for every y, z ∈ K. This proves that ˜ h ∈ LIP ˜ d ( K ) and Lip ˜ d (˜ h ) ≤ ε ′ + 5 Lip d ( f ), yielding the sought conclusion. Step 3: Estimates for the asymptotic slope of ˜ h . Next we claim thatlip ˜ d a (˜ h )( x ) ≤ lip d a ( f )( x ) + 2 ε ′ , for every x ∈ K. (3.9)To prove it, fix any δ < ε ′ r and y, z ∈ B ˜ d δ ( x ). Define F := (cid:8) j = 1 , . . . , k : d ( x, x j ) < r/ (cid:9) .If j / ∈ F , then y, z / ∈ B d r ( x j ) and thus ψ j ( y ) = ψ j ( z ) = 0, as it is granted by the estimates d ( y, x j ) ≥ d ( x, x j ) − d ( x, y ) ≥ r − d i ( x, y ) − ε ′ r > (cid:18) − ε ′ (cid:19) r − δ > (cid:18) − ε ′ (cid:19) r > r, and similarly for d ( z, x j ). If j ∈ F , then B d r ( x j ) ⊆ B d r ( x ) and f j ( z ) = f ( z ). The latter claimfollows from the fact that z ∈ B d r ( x j ), which is granted by the estimates d ( z, x j ) ≤ d ( z, x ) + d ( x, x j ) < d i ( z, x ) + ε ′ r + 3 r < δ + (cid:18) ε ′ + 32 (cid:19) r < (cid:18) ε ′ + 32 (cid:19) r < r. AMMA-CONVERGENCE OF CHEEGER ENERGIES WITH RESPECT TO INCREASING DISTANCES 7
Therefore, by using (3.6) and the above considerations, we obtain that (cid:12)(cid:12) ˜ h ( y ) − ˜ h ( z ) (cid:12)(cid:12) (3.5a) ≤ X j ∈ F ψ j ( y ) Lip d i ( j ) ( h j ) d i ( j ) ( y, z ) + X j ∈ F Lip d i ( ψ j ) d i ( y, z ) ε ′ k Lip d ( ψ j ) (3.5b) ≤ (cid:20) X j ∈ F ψ j ( y ) Lip d (cid:0) f ; B d r ( x j ) (cid:1) + ε ′ (cid:21) d i ( y, z ) ≤ h Lip d (cid:0) f ; B d r ( x ) (cid:1) + ε ′ i d i ( y, z ) (3.4) ≤ (cid:2) lip d a ( f )( x ) + 2 ε ′ (cid:3) d i ( y, z ) . Thanks to the arbitrariness of y, z ∈ B ˜ d δ ( x ), we deduce that Lip ˜ d (cid:0) ˜ h ; B ˜ d δ ( x ) (cid:1) ≤ lip d a ( f )( x ) + 2 ε ′ ,whence by letting δ ց Step 4: Construction of the function g . Given any point x ∈ K , it holds that (cid:12)(cid:12) ˜ h ( x ) − f ( x ) (cid:12)(cid:12) ≤ k X j =1 ψ j ( x ) (cid:12)(cid:12) h j ( x ) − f ( x ) (cid:12)(cid:12) = k X j =1 ψ j ( x ) (cid:12)(cid:12) h j ( x ) − f j ( x ) (cid:12)(cid:12) (3.5a) ≤ ε ′ . (3.10)In particular, we have that sup K | ˜ h | ≤ sup X | f | +1. Recall also that Lip ˜ d (˜ h ) ≤ d ( f )+ ε ′ , asproven in Step 2 . Therefore, by applying [4, Theorem 1.1] we can find a function h ∈ LIP d i (X)with h | K = ˜ h such that lip d i a ( h )( x ) = lip ˜ d a (˜ h )( x ) for every x ∈ K andLip d i ( h ) ≤ Lip ˜ d (˜ h ) + ε ′ ≤ d ( f ) + 2 ε ′ =: C. (3.11)Define G := (cid:8) x ∈ X : d i ( x, spt( f ) ∩ K ) ≤ (cid:9) and observe that sup G | h | ≤ C + sup X | f | + 1.Indeed, given any point x ∈ G , one has that (cid:12)(cid:12) h ( x ) (cid:12)(cid:12) ≤ inf y ∈ spt( f ) ∩ K h(cid:12)(cid:12) h ( x ) − h ( y ) (cid:12)(cid:12) + (cid:12)(cid:12) h ( y ) (cid:12)(cid:12)i ≤ Lip d i ( h ) inf y ∈ spt( f ) ∩ K d i ( x, y ) + sup K | h | (3.11) ≤ C d i ( x, spt( f ) ∩ K ) + sup K | ˜ h | ≤ C + sup X | f | + 1 . Moreover, we have that G ⊆ B = ¯ B d R +2 (¯ x ). Indeed, by using that spt( f ) ⊆ B d R (¯ x ), we get d ( x, ¯ x ) ≤ inf y ∈ spt( f ) ∩ K (cid:2) d ( x, y ) + d ( y, ¯ x ) (cid:3) ≤ inf y ∈ spt( f ) ∩ K d i ( x, y ) + R ≤ R + 2 , for every x ∈ G . Let us now define the d i -Lipschitz cut-off function η : X → [0 ,
1] as η ( x ) := (cid:16)(cid:0) − d i ( x, spt( f ) ∩ K ) (cid:1) ∧ (cid:17) ∨ , for every x ∈ X . It holds that η = 1 on a neighbourhood of spt( f ) ∩ K and that Lip d i ( η ) ≤
1. Given that η = 0in X \ G , it also holds that spt( η ) ⊆ G . We then define the function g : X → R as g := ηh . Step 5: Conclusion.
Note that g ∈ LIP d i (X), spt( g ) ⊆ G , and sup X | g | ≤ C + sup X | f | + 1.Let us estimate Lip d i ( g ). Since (cid:12)(cid:12) g ( x ) − g ( y ) (cid:12)(cid:12) ≤ η ( x ) (cid:12)(cid:12) h ( x ) − h ( y ) (cid:12)(cid:12) + (cid:12)(cid:12) η ( x ) − η ( y ) (cid:12)(cid:12)(cid:12)(cid:12) h ( y ) (cid:12)(cid:12) holdsfor every x, y ∈ X, we obtain that (cid:12)(cid:12) g ( x ) − g ( y ) (cid:12)(cid:12) ≤ (cid:0) C + sup G | h | (cid:1) d i ( x, y ) whenever y ∈ G ,whence it follows that Lip d i ( g ) ≤ C + sup X | f | + 1. The same computations giveLip d i ( g ; E ) ≤ Lip d i ( h ; E ) + sup E | h | , for every E ⊆ X . (3.12) DANKA LUˇCI´C AND ENRICO PASQUALETTO
On the one hand, since g and h agree on a neighbourhood of spt( f ) ∩ K , for any x ∈ spt( f ) ∩ K we have that (cid:12)(cid:12) g ( x ) − f ( x ) (cid:12)(cid:12) ≤ ε ′ by (3.10) and lip d i a ( g )( x ) ≤ lip d a ( f )( x ) + 2 ε ′ by (3.9). On theother hand, if x ∈ K \ spt( f ), then f ( x ) = lip d a ( f )( x ) = 0, thus accordingly we can deducefrom (3.10) that (cid:12)(cid:12) g ( x ) − f ( x ) (cid:12)(cid:12) = η ( x ) (cid:12)(cid:12) h ( x ) (cid:12)(cid:12) ≤ ε ′ , while (3.9), (3.10), and (3.12) ensure thatlip d i a ( g )( x ) = lim δ ց Lip d i (cid:0) g ; B d i δ ( x ) (cid:1) ≤ lim δ ց Lip d i (cid:0) h ; B d i δ ( x ) (cid:1) + lim δ ց sup B d iδ ( x ) | h | = lip d i a ( h )( x ) + (cid:12)(cid:12) h ( x ) (cid:12)(cid:12) ≤ ε ′ . All in all, we have shown that (cid:12)(cid:12) g ( x ) − f ( x ) (cid:12)(cid:12) ≤ ( ε ′ , C + sup X | f | + 1 , if x ∈ K, if x ∈ X \ K, (3.13a)lip d i a ( g )( x ) ≤ ( lip d a ( f )( x ) + 3 ε ′ , C + sup X | f | + 1 , if x ∈ K, if x ∈ X \ K. (3.13b)It remains to check that g satisfies (3.2a) and (3.2b). Recall that spt( f ) , spt( g ) ⊆ B . Then Z | g − f | p d m = Z K | g − f | p d m + Z B \ K | g − f | p d m (3.13a) ≤ m ( K ) ( ε ′ ) p + m ( B \ K ) (cid:16) C + sup X | f | + 1 (cid:17) p ≤ (cid:20) m ( B ) + (cid:16) C + sup X | f | + 1 (cid:17) p (cid:21) ε ′ . Moreover, it holds that Z lip d i a ( g ) p d m = Z K lip d i a ( g ) p d m + Z B \ K lip d i a ( g ) p d m (3.13b) ≤ Z K (cid:0) lip d a ( f ) + 3 ε ′ (cid:1) p d m + m ( B \ K ) (cid:16) C + sup X | f | + 1 (cid:17) p ≤ Z lip d a ( f ) p d m + 3 pε ′ Z B lip d a ( f ) p − d m + (cid:16) C + sup X | f | + 1 (cid:17) p ε ′ ≤ Z lip d a ( f ) p d m + (cid:20) p Lip d ( f ) p − m ( B ) + (cid:16) C + sup X | f | + 1 (cid:17) p (cid:21) ε ′ . By taking (3.3) into account, we can finally conclude that (3.2a) and (3.2b) are verified. (cid:3) Mosco-convergence of Cheeger energies
By applying Proposition 3.3, we can easily obtain our main Γ-convergence result.
Theorem 4.1.
Let (X , d , m ) be a metric measure space. Let ( d i ) i ∈ N be a sequence of completedistances on X such that d i ր d as i → ∞ . Suppose τ ( d i ) = τ ( d ) for all i ∈ N . Fix p ∈ (1 , ∞ ) .Then E d i Ch ,p Mosco-converges to E d Ch ,p as i → ∞ . Videlicet, the following properties hold: AMMA-CONVERGENCE OF CHEEGER ENERGIES WITH RESPECT TO INCREASING DISTANCES 9 i) Weak Γ -lim inf. If ( f i ) i ∈ N ⊆ L p ( m ) weakly converges to f ∈ L p ( m ) , then it holds E d Ch ,p ( f ) ≤ lim i →∞ E d i Ch ,p ( f i ) . ii) Strong Γ -lim sup. Given any f ∈ L p ( m ) , there exists a sequence ( f i ) i ∈ N ⊆ L p ( m ) that strongly converges to f and satisfies E d Ch ,p ( f ) ≥ lim i →∞ E d i Ch ,p ( f i ) . Proof.
Item i) can be easily proven: given any f ∈ L p ( m ) and ( f i ) i ∈ N ⊆ L p ( m ) with f i ⇀ f weakly in L p ( m ), the weak lower semicontinuity of E d Ch ,p : L p ( m ) → [0 , + ∞ ] grants that E d Ch ,p ( f ) ≤ lim i →∞ E d Ch ,p ( f i ) (2.1) ≤ lim i →∞ E d i Ch ,p ( f i ) . Let us then pass to the verification of item ii). Let f ∈ L p ( m ) be given. If f / ∈ W ,p (X),then E d Ch ,p ( f ) = + ∞ and accordingly the Γ-lim sup inequality is trivially verified (by taking,for instance, f i := f for every i ∈ N ). Now suppose f ∈ W ,p (X). By definition of E d Ch ,p ,we can find a sequence ( ˜ f n ) n ⊆ LIP d (X) of boundedly-supported functions such that ˜ f n → f strongly in L p ( m ) and E d Ch ,p ( f ) = lim n E d a,p ( ˜ f n ). By Proposition 3.3, we can find ι : N → N increasing and a sequence ( g n ) n of boundedly-supported functions g n ∈ LIP d ι ( n ) (X) such that Z | g n − ˜ f n | p d m ≤ n , E d ι ( n ) a,p ( g n ) ≤ E d a,p ( ˜ f n ) + 1 n . In particular, g n → f strongly in L p ( m ) and E d Ch ,p ( f ) ≥ lim n E d ι ( n ) a,p ( g n ) ≥ lim n E d ι ( n ) Ch ,p ( g n ).Finally, we define the recovery sequence ( f i ) i ⊆ L p ( m ) in the following way: f i := g n , for every n ∈ N and i ∈ (cid:8) ι ( n ) , . . . , ι ( n + 1) − (cid:9) . Notice that f i → f strongly in L p ( m ). Moreover, Remark 2.2 grants that E d i Ch ,p ( f i ) ≤ E d ι ( n ) Ch ,p ( g n )whenever ι ( n ) ≤ i < ι ( n + 1), which implies that lim i E d i Ch ,p ( f i ) = lim n E d ι ( n ) Ch ,p ( g n ) ≤ E d Ch ,p ( f ).This gives the Γ-lim sup inequality, thus accordingly the statement is achieved. (cid:3) It readily follows from Theorem 4.1 that the infinitesimal Hilbertianity condition is stableunder taking increasing limits of the distances (while keeping the measure fixed). Videlicet:
Corollary 4.2.
Let (X , d , m ) be a metric measure space. Let ( d i ) i ∈ N be a sequence of completedistances on X such that d i ր d as i → ∞ and τ ( d i ) = τ ( d ) for every i ∈ N . Suppose (X , d i , m ) is infinitesimally Hilbertian for every i ∈ N . Then (X , d , m ) is infinitesimally Hilbertian.Proof. Theorem 4.1 implies that E d i Ch , → E d Ch , with respect to the strong topology of L ( m ),thus [3, Theorem 11.10] grants that E d Ch , is a quadratic form, which gives the statement. (cid:3) Remark 4.3.
Let (M , d ) be (the metric space associated with) a generalised sub-Riemannianmanifold, in the sense of [7, Definition 4.1]. Then there exists a sequence ( d i ) i ∈ N of distanceson M, induced by Riemannian metrics, such that d i ր d ; cf. [7, Corollary 5.2]. Suppose d and each d i are complete distances. Fix a Radon measure m on M. Then [8, Theorem 4.11] ensures that each (M , d i , m ) is infinitesimally Hilbertian. Therefore, by applying Corollary 4.2we can conclude that (M , d , m ) is infinitesimally Hilbertian as well. This argument providesan alternative proof of [7, Corollary 5.6]. (cid:4) We conclude the paper by illustrating an example which shows that the results of thissection cannot hold if the assumption of monotone convergence from below of the distancesis replaced by a monotone convergence from above.
Example 4.4.
Let (X , d , m ) be any metric measure space such that d ≤
1. Given any i ∈ N ,we define the ‘snowflake’ distance d i on X as d i ( x, y ) := d ( x, y ) − i for every x, y ∈ X. Thenwe have d i ( x, y ) ց d ( x, y ) as i → ∞ for all x, y ∈ X and τ ( d i ) = τ ( d ) for all i ∈ N . Sinceabsolutely continuous curves in (X , d i ) are constant, it follows from the results in [1] that E d i Ch ,p ( f ) = 0 , for every p ∈ (1 , ∞ ) and f ∈ L p ( m ) . In particular, each space (X , d i , m ) is infinitesimally Hilbertian. This shows that Theorem 4.1and Corollary 4.2 might fail if we replace the assumption d i ր d with d i ց d . (cid:4) References [1]
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Universit`a di Pisa, Dipartimento di Matematica, Largo Bruno Pontecorvo 5,56127 Pisa, Italy
Email address : [email protected] (Enrico Pasqualetto) Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
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