aa r X i v : . [ m a t h . M G ] M a y Cheeger’s energyon the harmonic Sierpinski gasket
Ugo Bessi*
Abstract
Koskela and Zhou have proven that, on the harmonic Sierpinski gasket with Kusuoka’s measure, the”natural” Dirichlet form coincides with Cheeger’s energy. We give a different proof of this result, which usesthe properties of the Lyapounov exponent of the gasket.
Introduction
This paper deals with the harmonic Sierpinski gasket S endowed with Kusuoka’s measure κ ; we referthe reader to sections 1 and 2 below for the definitions and properties of these objects.Let P v denote the orthogonal projection in R along the vector v : P v w = ( w, v ) || v || · v. As we recall in section 2 below, there is a Borel vector field v : S → R such that, if one defines E : C ( R , R ) × C ( R , R ) → R E ( φ, ψ ) = Z S ( ∇ φ, P v ( x ) ∇ ψ )d κ ( x ) (1)then E extends to a local, self-similar Dirichlet form densely defined on L ( S, κ ). Heuristically, this meansthat v x is the ”tangent space at x ” seen by the Brownian motion on S . As a remark on notation, we shalldenote the vector field at x ∈ S both by v ( x ) and by v x .There is another quadratic form densely defined on L ( S, κ ), and it is Cheeger’s energy ([2], [4]); in [12](see also [8]), the following theorem is proven.
Theorem 1.
The Dirichlet form E coincides with the double of Cheeger’s energy; in symbols, Ch ( u ) = 12 E ( u, u ) (2) * Dipartimento di Matematica, Universit`a Roma Tre, Largo S. Leonardo Murialdo, 00146 Roma, Italy. email: [email protected]
Work partially supported by the PRIN2009 grant ”CriticalPoint Theory and Perturbative Methods for Nonlinear Differential Equations1 or all u ∈ D ( Ch ) = D ( E ) . Morally, this means that v x is also the ”tangent space at x ” seen by large families of absolutely continuouscurves, the so-called test plans with bounded deformation.The proof of [12] is rather delicate; the aim of this paper is to provide a different proof which linksCheeger’s energy to the Lyapounov exponent of the dynamical system underlying S . Indeed, a third wayof looking at the ”tangent space” of S is the following: there is an expansive map F : R → R whichsends S into itself and whose derivative is defined κ -a. e. on S . Heuristically, the vector field v ( x ) of (1)coincides with the ”less expansive” direction of F and it determines the only viable velocity for curves in S ; more precisely, we shall prove that for ”almost” all absolutely continuous curves γ : [0 , → S which aredifferentiable at t ∈ (0 , || v ( x ) || ≡ γ t = ±|| ˙ γ t || · v ( γ t ) . If φ ∈ C ( R , R ), the formula above implies, practically by definition of Cheeger’s derivative | Dφ | w , that | Dφ | w ( x ) ≤ | ( ∇ φ, v ( x )) | for κ -a. e. x ∈ S which is half of the proof of formula (2).For the opposite inequality, we use two facts. First, we prove that in Cheeger’s definition of the energy(formula (1.13) below) we can restrict to C functions. Second, using again the Lyapounov exponent weshow that, if f ∈ C , then its local Lipschitz constant on S , usually denoted by Lip a ( f, x ), is larger than | ( ∇ f ( x ) , v ( x )) | .The paper is organised as follows. In section 1 below we recall the main definitions and properties of theHarmonic Sierpinski gasket and Cheeger’s energy. In section 2 we introduce Kusuoka’s measure, followingthe approach of [3]. In section 3, we study the shape of smaller and smaller cells of the gasket: heuristically,they are ”skinny” triangles with the long side aligned with v ( x ). We shall also see how Kusuoka’s measuredistributes inside these triangles. Theorem 1 is proven in section 4. Acknowledgements.
The author would like to thank the referee for the useful comments. § We recall the definition of the harmonic Sierpinski gasket; we follow [8]. We consider the threelinear contractions of R T = (cid:18) (cid:19) , T = √ √
310 12 ! , T = − √ − √
310 12 ! A ¯ B ¯ C :¯ A = (0 , , ¯ B = (cid:18) , √ (cid:19) , ¯ C = (cid:18) , − √ (cid:19) . We define the three affine contractions of R ψ ( x ) = ¯ A + T ( x − ¯ A ) , ψ ( x ) = ¯ B + T ( x − ¯ B ) , ψ ( x ) = ¯ C + T ( x − ¯ C ) . (1 . S ⊂ R , which is called the harmonicSierpinski gasket, such that S = [ i =1 ψ i ( S ) . (1 . S is contained in the equilateral triangle ¯ A ¯ B ¯ C ; since the three vertices ¯ A , ¯ B and ¯ C are fixed pointsof ψ , ψ and ψ respectively, they are contained in S ; they are often called the boundary of S ; we refer thereader to [15] for this terminology and other interesting facts on the gasket.The three maps ψ , ψ and ψ are the three branches of the inverse of an expansive map F : S → S .More precisely, let us consider the three triangles ψ i ( ¯ A ¯ B ¯ C ) for i = 1 , ,
3; they are depicted on the right offigure 1 below, where ψ ( ¯ A ¯ B ¯ C ) = ¯ Abc , ψ ( ¯ A ¯ B ¯ C ) = ¯ Bca and ψ ( ¯ A ¯ B ¯ C ) = ¯ Cba . Note that the sets ψ i ( ¯ A ¯ B ¯ C )intersect only at the vertices denoted by lowercase letters.We can find three disjoint open sets O , O and O such that O contains the closed triangle Abc savethe two points { b, c } , O contains Bca save { c, a } and O contains Cba save { b, a } . We can also require that O j ⊂ ψ − ( O j ). We define F : [ i =1 O i → ( ¯ A ¯ B ¯ C )by F ( x ) = ψ − i ( x ) if x ∈ O i . This does not define F on the three points { a, b, c } ; we set arbitrarily F ( a ) = C , F ( b ) = A and F ( c ) = B .We can afford this arbitrariness since it is standard ([13], see [3] for an alternative proof) that { a, b, c } is anull set for Kusuoka’s measure.A basic fact is that the dynamics of F on S can be coded. Namely, let us consider the space of sequencesΣ: = {{ x i } i ≥ : x i ∈ (1 , , ∀ i ≥ } with the product topology. Let η ∈ (0 ,
1) be the common Lipschitz constant of ψ , ψ and ψ (by (1.1)we gather that η = , but the precise value is immaterial). If x = ( x x . . . ) ∈ Σ, using the fact thatdiam( ¯ A ¯ B ¯ C ) = √ we get that diam( ψ x ◦ . . . ◦ ψ x l ( ¯ A ¯ B ¯ C )) ≤ √ η l +1 . (1 . A ¯ B ¯ C is a closed triangle, the sets ψ x ◦ . . . ◦ ψ x l ( ¯ A ¯ B ¯ C ) are compact; formula (1.2) implies that ψ i ( S ) ⊂ S which in turn implies that for all l ≥ ψ x ◦ . . . ◦ ψ x l ◦ ψ x l +1 ( ¯ A ¯ B ¯ C ) ⊂ ψ x ◦ . . . ◦ ψ x l ( ¯ A ¯ B ¯ C ) . (1 . \ l ≥ ψ x ◦ . . . ◦ ψ x l ( ¯ A ¯ B ¯ C )is a single point, which we call Φ( x ) = Φ( x x . . . ). Formulas (1.3) and (1.4) imply easily that the mapΦ: Σ → S is continuous.Recall that the triangles ψ i ( ¯ A ¯ B ¯ C ) intersect only at the vertices, which have the lowercase letters inthe figure above; this easily implies that, if x ∈ S is coded by w w . . . and w ′ w ′ . . . with w = w ′ , then x ∈ { a, b, c } . Iterating, we get that the points in S which have multiple codings are of the type ψ x ◦ . . . ◦ ψ x l ( P ) with P ∈ ( ¯ A, ¯ B, ¯ C ) . Note that the points defined by the formula above are a countable set. It is easy to see that these pointshave at most two pre-images, i. e. that Φ is at most two to one.Next we note that, if x ∈ S \ { B, C } , then ψ ( x ) ∈ O and thus F ◦ ψ ( x ) = x ; the same argumentholds for ψ and ψ yielding (1.5) below, while (1.6) is immediate. F ◦ ψ ( x ) = x ∀ x ∈ G \ { B, C } ,F ◦ ψ ( x ) = x ∀ x ∈ G \ { A, C } ,F ◦ ψ ( x ) = x ∀ x ∈ G \ { A, B } . (1 . ψ i ◦ F ( y ) = y ∀ y ∈ O i . (1 . Dψ j is constant, we haveomitted its argument. DF | ψ j ( x ) = ( Dψ j ) − ∀ x ∈ G \ { A, B, C } , ∀ j ∈ (1 , ,
3) (1 . Dψ j = ( DF ( y )) − ∀ y ∈ O j . (1 . σ : Σ → Σ σ : { x i } i ≥ → { x i +1 } i ≥ . From (1.5) it follows easily that Φ conjugates F and σ :Φ ◦ σ ( { x i } i ≥ ) = F ◦ Φ( { x i } i ≥ ) (1 . x ) ∈ S where F ◦ Φ( x ) is defined arbitrarily, i. e. { a, b, c } .4et x = ( x x . . . ) ∈ Σ; we shall often use the notation ψ x,l = ψ x ...x l = ψ x ◦ . . . ◦ ψ x l . (1 . ψ x ◦ . . . ◦ ψ x l looks redundant, but we shall use both of them.We define the cell (or cylinder) [ x . . . x l ] as[ x . . . x l ] = ψ x,l ( S ) = ψ x ...x l ( S ) . (1 . Absolutely continuous curves.
It is a standard fact ([8]) that S is arcwise connected; this prompts us torecall a definition of [1]. Definition.
We say that γ ∈ AC ([0 , , S ) if1) γ is absolutely continuous from [0 ,
1] to R ,2) Z || ˙ γ s || d s < + ∞ γ s ∈ S for all s ∈ [0 , x, y ∈ S we defined geod ( x, y ) = min (cid:26)Z || ˙ γ s || d s : γ ∈ AC ([0 , , S ) , γ = x, γ = y (cid:27) . It turns out ([8], see also [19] and [10]) that the minimum is attained and that the minimal curves are ofclass C ; they are geodesics of constant speed, i. e. for all 0 ≤ s ≤ t ≤ geod ( γ s , γ t ) = ( t − s )d geod ( x, y ) . By [8], the geodesic of constant speed connecting x and y is unique; we call it γ x,y .Another fact (again [8]) is that the union M : = γ ¯ A ¯ B ([0 , ∪ γ ¯ B ¯ C ([0 , ∪ γ ¯ A ¯ C ([0 , . S is contained in theclosure of the bounded component. Test plans and Cheeger’s energy.
A Borel probability measure on C ([0 , , S ) is called a test plan if thetwo points below hold.1) π concentrates on AC ([0 , , S ).2) π satisfies Z C ([0 , ,S ) d π ( γ ) Z || ˙ γ s || d s < + ∞ . m on S ; from section 3 onwards, m will be Kusuoka’s measure κ , but the definition we give below holds for a completely arbitrary m . For t ∈ [0 ,
1] let us denote by e t theevaluation map e t : C ([0 , , S ) → Se t : γ → γ t . If π is a test plan, let us define by ( e t ) ♯ π the push-forward of π by e t ; in other words, if f : S → R is abounded Borel function, then Z S f ( x )d[( e t ) ♯ π ]( x ) = Z C ([0 , ,S ) f ( γ t )d π ( γ ) . We say that the test plan π has bounded deformation with respect to m if for all t ∈ [0 ,
1] we have that( e t ) ♯ π = ρ t m and if there is C > || ρ t || L ∞ ( m ) ≤ C ∀ t ∈ [0 , . We recall the definition of weak gradient from [2].
Definition.
Let f ∈ L ( S, m ); we say that g ∈ L ( S, m ) is a weak gradient of f if, for all test plans π withbounded deformation with respect to m and for π -a. e. curve γ ∈ C ([0 , , S ) we have | f ( γ ) − f ( γ ) | ≤ Z g ( γ s ) || ˙ γ s || d s. If f admits a weak gradient g ∈ L ( S, m ), then it admits a weak gradient | Df | w minimal in the followingtwo senses: if g ∈ L ( S, m ) is a weak gradient, then ||| Df | w || L ( S,m ) ≤ || g || L ( S,m ) and | Df | w ( x ) ≤ g ( x ) m -a. e..Cheeger’s energy is the function Ch : L ( S, m ) → [0 , + ∞ ]defined by Ch ( f ) = 12 Z S | Df | w d m if f has a weak gradient in L ( S, m ), and Ch ( f ) = + ∞ otherwise.Though there is a square in its definition, Ch in general is not a quadratic form (see [2] for an example).However, if m = κ , then theorem 1 of the introduction implies that Ch is quadratic.6e shall need an equivalent definition ([2], [4]) of | Df | w . Namely, if f : S → R is a function, we define | Df | ( x ) = lim r → sup y ∈ [ B ( x,r ) \{ x } ] ∩ S | f ( y ) − f ( x ) ||| y − x || where || · || is the euclidean norm on R .It turns out that, if f, | Df | w ∈ L ( S, m ), then | Df | w is the L function g of minimal norm for whichthere is a sequence of Lipschitz functions f n : S → R such that1) f n → f in L ( S, m ) and2) | Df n | → g weakly in L ( S, m ).Points 1) and 2) above are Cheeger’s characterisation of | Df | w ; this characterisation implies that Ch ( f ) = inf lim inf n → + ∞ Z S | Df n | d m where the inf is over all sequences { f n } n ≥ of Lipschitz functions such that f n → f in L ( S, m ).In section 4 we shall use a different formula, which we found in chapter 3 of [5]. We define the localLipschitz constant of f by Lip a ( f, x ) = lim r → sup (cid:26) | f ( y ) − f ( z ) ||| y − z || : z = y, z, y ∈ B ( x, r ) (cid:27) . Then, Ch ( f ) = inf lim inf n → + ∞ Z S | Lip a ( f n ) | d m (1 . { f n } n ≥ of Lipschitz functions which converge to f in L ( S, m ). § In this section, we recall the definition and some properties of Kusuoka’s measure ([13]); we follow theapproach of [3].Let M denote the space of 2 × M is a Hilbert space for the Hilbert-Schmidtinner product ( A, B ) HS : = tr ( t AB ) = tr ( AB )where t A denotes the transpose of A ; for the second equality, recall that A is symmetric.We denote by C ( S, M ) the space of continuous functions from S to M ; its dual is the space M ( S, M )of the M -valued Borel measures on S .We want to define the integral against µ ∈ M ( S, M ); in order to do this, we recall that the totalvariation of µ is a finite measure || µ || on the Borel sets of S . The polar decomposition of µ is given by µ = M x || µ || M : S → M is a Borel field of symmetric matrices which satisfies || M x || HS = 1 for || µ || -a. e. x ∈ S . (2 . A : S → M is Borel and || A || HS ∈ L ( S, || µ || ), then by (2.1) and Cauchy-Schwarz we have that( A x , M x ) HS ∈ L ( S, || µ || ). Consequently, we can define the scalar Z S ( A x , d µ ( x )) HS : = Z S ( A x , M x ) HS d || µ || ( x ) . The duality coupling between C ( S, M ) and M ( S, M ) is given by h· , ·i : C ( S, M ) × M ( S, M ) → R h A, µ i : = Z S ( A x , d µ ( x )) HS . If Q ∈ C ( S, M ) and µ ∈ M ( S, M ), we define the scalar measure ( Q, µ ) HS by Z S f ( x )d( Q, µ ) HS ( x ): = Z S ( f Q, d µ ) HS (2 . f ∈ C ( S, R ). In other words, ( Q, µ ) HS = ( Q x , M x ) HS · || µ || .If v, w : S → R are Borel functions such that || v || · || w || ∈ L ( S, || µ || ) then, again by (2.1) and Cauchy-Schwarz, ( v x , M x w x ) ∈ L ( S, || µ || ) and we can define Z S ( v x , d µ ( x ) w x ): = Z S ( v x , M x w x )d || µ || ( x ) . (2 . µ ∈ M ( S, M ) is semipositive definite if µ ( E ) is a semipositive definite matrix for all Borelsets E ⊂ S . Lusin’s theorem easily implies ([3]) that µ is semipositive definite if and only if h A, µ i ≥ . A ∈ C ( S, M ) such that A x ≥ x ∈ S .We denote by M + ( S, M ) the set of all semipositive definite measures of M ( S, M ); by the character-isation of (2.4), M + ( S, M ) is a convex set of M ( S, M ), closed for the weak ∗ topology.Let now Q ∈ C ( S, M ) be such that Q x is positive-definite for all x ∈ S ; since S is compact there is ǫ > Q x ≥ ǫId ∀ x ∈ S. (2 . Q we define P Q ( S, M ) as the set of all µ ∈ M + ( S, M ) such that Z S ( Q x , d µ ( x )) HS = 1 . As shown in [3], if Q satisfies (2.5) there is D ( ǫ ) > µ ∈ M + ( S, M ), || µ || ≤ D ( ǫ )( Q, µ ) HS . (2 . || µ || with respect to ( Q, µ ) HS is bounded by D ( ǫ ). As aconsequence, if µ ∈ P Q ( S, M ), then || µ || ( S ) ≤ D ( ǫ ) . By the characterisation (2.4), P Q ( S, M ) is a convex subset of M ( S, M ), closed for the weak ∗ topology; bythe formula above, it is compact.Let ψ , ψ , ψ be the affine maps of section 1; we define the Ruelle operator as L : C ( S, M ) → C ( S, M )( L A )( x ): = X i =1 t Dψ i ( x ) · A ψ i ( x ) · Dψ i ( x ) . The adjoint of
L L ∗ : M ( S, M ) → M ( S, M )is defined by hL A, µ i = h A, L ∗ µ i for all A ∈ C ( S, M ) and µ ∈ M ( S, M ).The following proposition is proven as in the scalar case, for which we refer to [16] and [20]; the detailsfor the matrix case are in [3]. Proposition 2.1.
1) The operator L has a simple, positive eigenvalue β > . Let ˜ Q ∈ C ( S, M ) be aneigenfunction of β ; then, up to multiplying ˜ Q by a scalar, we have ˜ Q x = Id ∀ x ∈ S. In other words, the eigenspace of β is generated by a matrix field constantly equal to the identity.2) There is a unique τ ∈ P Id ( S, M ) such that L ∗ τ = βτ.
3) Let us define the scalar measure κ : = ( Id, τ ) HS as in (2.2); then, κ is a probability measure ergodic forthe map F . Moreover, κ is non-atomic.4) For f, g ∈ C ( R , R ) we define E ( f, g ) = Z S ( ∇ f ( x ) , d τ ( x ) ∇ g ( x )) where the integral has been defined in (2.3). Then, E is self-similar; in other words, for the maps ψ i ofsection 1 and all f, g ∈ C ( R , R ) we have that E ( f, g ) = 1 β X i =1 E ( f ◦ ψ i , g ◦ ψ i ) . ) The measure τ has the Gibbs property; in other words, with the notation of (1.10) and (1.11) we havethat τ [ x . . . x l − ] = 1 β l ( Dψ x ...x l − ) · τ ( S ) · t ( Dψ x ...x l − ) . This formula further simplifies since (see for instance [3] or [8]) τ ( S ) = Id . Since ψ x ...x l is affine, itsderivative is constant; therefore, we haven’t specified the point where we calculate it. Lemma 2.2 below gives an expression for the push-forward ( ψ j ...j n − ) ♯ τ ; the notation is that of (1.10). Lemma 2.2.
Let j , . . . , j n − ∈ (1 , , ; then, ( ψ j ...j n − ) ♯ τ = β n ( Dψ j ...j n − ) − · τ | ψ j ...jn − ( S ) · t ( Dψ j ...j n − ) − . (2 . By τ | ψ j ...jn − ( S ) we have denoted the restriction of τ to ψ j ...j n − ( S ) ; note that ( ψ j ...j n − ) ♯ τ is supportedon ψ j ...j n − ( S ) by definition. Proof.
Since the sets [ x . . . x l − ] generate the Borel sets of S , it suffices to show that, for all l ≥ n ≥ { j i } i ≥ , { x i } i ≥ ∈ Σ,[( ψ j ...j n − ) ♯ τ ][ x . . . x l − ] = β n ( Dψ j ...j n − ) − · τ ([ x . . . x l − ] ∩ ψ j ...j n − ( S )) · t ( Dψ j ...j n − ) − . (2 . ψ j ...j n − ) ♯ τ is supported on the cell [ j . . . j n − ], we can as well suppose that x x . . . x n − = j j . . . j n − : otherwise, we have zero on both sides of (2.8). Using this fact together with (1.10) and(1.11) we get the equality below.[ x n x n +1 . . . x l − ] = ψ − j ...j n − ([ x x . . . x l − ]) . (2 . τ [ ψ x ...x l − ( S )] = 1 β l ( Dψ x ...x l − ) · τ ( S ) · t ( Dψ x ...x l − ) τ [ ψ x n ...x l − ( S )] = 1 β l − n ( Dψ x n ...x l − ) · τ ( S ) · t ( Dψ x n ...x l − ) . By the chain rule, the last two formulas imply that τ [ ψ x n ...x l − ( S )] = β n ( Dψ x ...x n − ) − · τ [ ψ x ...x l − ( S )] · t ( Dψ x ...x n − ) − . (2 . x x . . . x n − = j j . . . j n − .[( ψ j ...j n − ) ♯ τ ][ x . . . x l − ] = τ ( ψ − j ...j n − ([ x . . . x l − ])) = τ [ x n . . . x l − ] = β n ( Dψ j ...j n − ) − · τ [ x . . . x l − ] · t ( Dψ j ...j n − ) − . x x . . . x n − = j j . . . j n − and we are done. \\\ We begin to note that (2.2) with Q = Id easily implies that κ = ( Id, τ ) HS ≤ || τ || . (2 . || τ || and κ are mutually absolutely continuous; since the polar decom-position of τ is τ = T x || τ || with || T x || HS = 1 for || τ || -a. e. x ∈ S , we get that τ = λ x T x κ where λ : S → (0 , + ∞ ) is the Radon-Nikodym derivative d || τ || d κ ; it is bounded away from zero and + ∞ by(2.6) and (2.11).By [13] we have that there is a Borel vector field v : S → R of unitary vectors such that T x = P v ( x ) for || τ || -a. e. x ∈ S where P v is the projection we defined in the introduction. By the last two formulas, τ = λ x P v ( x ) κ. By Rokhlin’s theorem (see [17], [18] or theorem 0.5.1 of [14]) the first equality below holds for κ -a. e. x = ( x x . . . ) ∈ S ; the second equality follows from point 5) of proposition 2.1 and the definition of κ . λ x P v ( x ) = lim l → + ∞ τ [ x . . . x l − ] κ [ x . . . x l − ] =lim l → + ∞ ( Dψ x ...x l − ) · τ ( S ) · t ( Dψ x ...x l − ) tr [( Dψ x ...x l − ) · τ ( S ) · t ( Dψ x ...x l − )] . (2 . τ ( S ) = Id ; thus, the last formula reduces to λ x P v ( x ) = lim l → + ∞ ( Dψ x ...x l − ) · t ( Dψ x ...x l − ) || ( Dψ x ...x l − ) || HS . (2 . λ x ≡ H : S \ { a, b, c } → L ( R d ) H ( x ) = Dψ i ( x ) if x ∈ ψ i ( S ) \ { a, b, c } . This function H is well defined, since all the points which belong to more than one ψ i ( S ) are in { a, b, c } ; itis also defined κ -a. e. since we saw above that κ ( { a, b, c } ) = 0.11e set H l ( x ) = H ( x ) ◦ H ( F ( x )) ◦ . . . ◦ H ( F l − ( x )) (2 . H l is defined save on the countable set [ l ≥ [ x ...x l − ψ x ...x l − ( { a, b, c } )i. e. save on the points where the coding Φ is not injective.If x = Φ( w w . . . ), using (1.9) we see that, for x in the full measure set where H l is defined, H l ( x ) = Dψ w ...w l − . (2 . v, w, a ∈ R d and || v || = 1, then( v ⊗ w ) a = ( a, v ) · w. (2 . Lemma 2.3.
Let the vector field v ( x ) be as in (2.13). Then, for each l ≥ there is a vector field of unitvectors v l ( x ) such that, for κ -a. e. x ∈ S , || H l ( x ) || H l ( x ) || HS − v l ( x ) ⊗ v ( x ) || HS → . (2 . Equivalently, if x = Φ( w w . . . ) , we get by (2.15) that || Dψ w ...w l || Dψ w ...w l || HS − v l ( x ) ⊗ v ( x ) || HS → . (2 . We are not asserting that the convergence is uniform.
Proof.
Let x = Φ( w w . . . ); the first equality below is the definition of v l , the second one follows from(2.15). v l ( x ): = 1 || t H l ( x ) || HS t H l ( x ) v ( x ) = t ( Dψ w ...w l − ) v ( x ) || t ( Dψ w ...w l − ) || HS . By (2.13) and the fact that λ x ≡ κ -a. e. x ∈ G ,( v l ( x ) , v l ( x )) → ( P v ( x ) v ( x ) , v ( x )) = 1 . (2 . w ⊥ v ( x ) we obtain the equality below, while the limit follows from (2.13) and (2.15). (cid:18) t H l ( x ) w || t H l ( x ) || HS , t H l ( x ) w || t H l ( x ) || HS (cid:19) → ( P v ( x ) w, w ) = 0 . In other words, for l large t H l ( x ) || t H l ( x ) || HS brings the unit vector v ( x ) into the unit vector v l ( x ), and annihilates v ( x ) ⊥ ; this is tantamount to saying that || t H l ( x ) || t H l ( x ) || HS − v ( x ) ⊗ v l ( x ) || HS → κ -a. e. x ∈ S . Transposing, we get (2.17). \\\§ Let x = Φ( w w . . . ) ∈ S and let the cell [ w . . . w l ] be defined as in (1.11). Let ¯ A , ¯ B and ¯ C be the threepoints of the ”boundary” of S as in section 1; we shall say that¯ A w,l = ψ w,l ( ¯ A ) , ¯ B w,l = ψ w,l ( ¯ B ) , ¯ B w,l = ψ w,l ( ¯ B ) (3 . w . . . w l ]. Let us denote by ˜ S the set of the points of S where (2.13)holds; we saw before formula (2.12) that ˜ S is a Borel set and κ ( ˜ S ) = 1. Let ˆ S denote ˜ S minus the countableset of points of (3.1); again ˆ S is Borel; since κ is non-atomic, κ ( ˆ S ) = 1.In section 1 we saw that S is contained inside the closed set M of (1.12); we define by d ( x, A ) theEuclidean distance of x ∈ R from a set A ⊂ R and we set S θ = { x ∈ S : d ( x, M ) ≥ θ } . (3 . S θ is S minus a thin strip around its three ”sides”, i . e. the curve M .The definition of S θ in (3.2) easily implies that [ n ≥ S n = S \ M. By [8], κ ( M ) = 0; together with the formula above this implies that there is θ > θ ∈ (0 , θ ], S θ is not empty.Again since κ ( M ) = 0, (2.6) implies that || τ || ( M ) = 0; as a consequence, τ ( S θ ) → τ ( S ) = Id as θ → . (3 . P ⊥ v the projection on the orthogonal subspace to v : P ⊥ v w = w − P v ( w ) . Lemma 3.1.
Let w = ( w w . . . ) ∈ Σ be such x = Φ( w ) ∈ ˆ S ; let v : ˆ S → R be the vector field of (2.13).Then, we can label the boundary points A w,n , B w,n and C w,n of (3.1) in such a way that the two assertionsbelow hold.1) For n → + ∞ we have that max[ || P ⊥ v ( x ) ( A w,n − B w,n ) || , || P ⊥ v ( x ) ( B w,n − C w,n ) || , || P ⊥ v ( x ) ( C w,n − A w,n ) || ] || P v ( x ) ( A w,n − B w,n ) || → . (3 . euristically, the cell ψ w,n ( S ) is ”long” in the direction v ( x ) and ”short” in the orthogonal direction; thevertices are re-labeled in such a way that the side A w,n B w,n has the longest projection on v ( x ) .2) Let θ ∈ (0 , θ ] , let S θ be as in (3.2) and let us suppose that x ∈ ψ w,n k ( S θ ) for a sequence n k ր + ∞ ; then, max || P ⊥ v ( x ) ( x − A w,n k ) |||| P v ( x ) ( x − A w,n k ) || , || P ⊥ v ( x ) ( x − B w,n k ) |||| P v ( x ) ( x − B w,n k ) || ! → . (3 . Again heuristically, x sees both boundary points on the ”long” side of the cell much farther away in the v ( x ) direction than in the orthogonal one. Note that neither (3.4) nor (3.5) are uniform in x ∈ ˆ S . Proof.
We begin with (3.4). Let ¯ A ¯ B ¯ C be the same equilateral triangle of section 1; its sides are √ long,its height 1. Elementary geometry tells us that, given any v ∈ R \ { } ,max( || P v ( ¯ B − ¯ A ) || , || P v ( ¯ B − ¯ C ) || , || P v ( ¯ C − ¯ A ) || ) = diam( P v ( ¯ A ¯ B ¯ C )) ≥ . Clearly, the diameter of the projection is minimal when v is parallel to one of the heights of ¯ A ¯ B ¯ C .Let x = Φ( w ) ∈ ˆ S and let v n ( x ) and v ( x ) be as in lemma 2.3; we can relabel ¯ A , ¯ B , ¯ C to A , B , C insuch a way that || P v n ( x ) ( B − A ) || is maximal; using (2.16), and the fact that || v ( x ) || = 1 we get the equalitybelow; the first inequality comes from the formula above; the second on comes, for some δ n ց
0, from (2.19). || v n ( x ) ⊗ v ( x )( B − A ) || = ( v n ( x ) , B − A ) ≥|| v n ( x ) || ≥ − δ n . (3 . A w,n = ψ w,n ( A ) , B w,n = ψ w,n ( B ) , C w,n = ψ w,n ( C ) . Together with the fact that ψ w,n is affine, this definition implies the equality below. Since x ∈ ˆ S , (2.18)holds; this implies that, for n ≥ n ( x ), the first inequality follows for a matrix F n with || F n || HS ≤ ǫ . Thisbound on the norm implies the second inequality, while the last one comes from (3.6). || P v ( x ) ( A w,n − B w,n ) |||| Dψ w,n || HS = || P v ( x ) ◦ Dψ w,n ( A − B ) |||| Dψ w,n || HS ≥|| v n ( x ) ⊗ v ( x )( A − B ) || − || P v ( x ) ◦ F n ( A − B ) || ≥|| v n ( x ) ⊗ v ( x )( A − B ) || − ǫ || A − B || ≥ − δ n − √ ǫ. (3 . n large, the formula below holds. || P ⊥ v ( x ) ( A w,n − B w,n ) |||| Dψ w,n || HS = || P ⊥ v ( x ) ◦ Dψ w,n ( A − B ) |||| Dψ w,n || HS = || P ⊥ v ( x ) F n ( A − B ) || ≤ ǫ || A − B || = ǫ √ . (3 . δ n → a ( θ ) > x ∈ S θ , || P v n ( x ) (˜ x − A ) || ≥ a ( θ ) · || ˜ x − A || and || P v n ( x ) (˜ x − B ) || ≥ a ( θ ) · || ˜ x − B || . (3 . A and B in (3.6). Indeed, since S θ is a compact set contained in the open,equilateral triangle ABC , we see that there is b ( θ ) > AB and A ˜ x belongs to[ b ( θ ) , π − b ( θ )]. Since P v n ( x ) = P − v n ( x ) , we can as well suppose that ( v n ( x ) , B − A ) >
0; since AB is theside with the longest projection on v n ( x ), the angle between AB and v n ( x ) lies in [ − π , π ]. Summing up, weget that that the angle between v n ( x ) and A ˜ x lies in [ − π + b ( θ ) , π − b ( θ )]. The first formula of (3.9) nowfollows with a ( θ ) = sin( b ( θ )); the second one follows analogously for the same a ( θ ).Since x ∈ ψ w,n ( S θ ), there is ˜ x ∈ S θ such that x = ψ w,n (˜ x ). The first inequality below follows as in (3.7)for a matrix F n with || F n || HS ≤ ǫ , this latter fact implying the second inequality; the last one follows, forsome δ n ց
0, from (3.9) and (2.19). || P v ( x ) ( x − A w,n ) |||| Dψ w,n || HS ≥ || v n ( x ) ⊗ v ( x )(˜ x − A ) || − || P v ( x ) ◦ F n (˜ x − A ) || ≥ ( v n ( x ) , ˜ x − A ) − ǫ || ˜ x − A || ≥ [ a ( θ )(1 − δ n ) − ǫ ] · || ˜ x − A || . Analogously, || P v ( x ) ( x − B w,n ) |||| Dψ w,n || HS ≥ [ a ( θ )(1 − δ n ) − ǫ ] · || ˜ x − B || . Given ǫ >
0, we see as in (3.8) that, for n large enough (depending on w and thus on x ) we have || P ⊥ v ( x ) ( x − A w,n ) |||| Dψ w,n || HS ≤ ǫ || ˜ x − A || , || P ⊥ v ( x ) ( x − B w,n ) |||| Dψ w,n || HS ≤ ǫ || ˜ x − B || . Now (3.5) follows from the last three formulas. \\\
Lemma 3.2.
There is δ ( θ ) → as θ → such that, for all x = Φ( w ) ∈ ˆ S , lim n → + ∞ κ ( ψ w,n ( S θ )) κ ( ψ w,n ( S )) ≥ − δ ( θ ) . (3 . As in lemma 3.1, we are not asserting that the limit is uniform in w ∈ Σ . Proof.
The first equality below comes from lemma 2.2. The second equality comes from the definition ofthe push-forward. β n · || Dψ w,n || HS · τ ( ψ w,n ( S θ )) = Dψ w,n || Dψ w,n || HS · [( ψ w,n ) ♯ τ ]( ψ w,n ( S θ )) · t Dψ w,n || Dψ w,n || HS =15 ψ w,n || Dψ w,n || HS · τ ( S θ ) · t Dψ w,n || Dψ w,n || HS . If A lies in a bounded set of M d , the family of maps α A : B → BA t B is equicontinuous on the set || B || HS ≤ α θ : B → Bτ ( S θ ) t B are equicontinuous as θ varies in [0 , θ ∈ [0 , || β n · || Dψ w,n || HS · τ ( ψ w,n ( S θ )) − v n ( x ) ⊗ v ( x ) · τ ( S θ ) · t ( v n ( x ) ⊗ v ( x )) || HS =sup θ ∈ [0 , || Dψ w,n || Dψ w,n || HS · τ ( S θ ) · t Dψ w,n || Dψ w,n || HS − v n ( x ) ⊗ v ( x ) · τ ( S θ ) · t ( v n ( x ) ⊗ v ( x )) || HS → . n → + ∞ .We recall that the map ρ B : A → t BAB is Lipschitz with a Lipschitz constant which only depends on || B || HS ; next we note that || v n ( x ) ⊗ v ( x ) || is bounded, since v n ( x ) is bounded by definition; by this and (3.3)there is δ ′′ ( θ ) → θ →
0, independent of n , such that || v n ( x ) ⊗ v ( x ) · τ ( S θ ) · v ( x ) ⊗ v n ( x ) − v n ( x ) ⊗ v ( x ) · τ ( S ) · v ( x ) ⊗ v n ( x ) || HS ≤ δ ′′ ( θ ) . Since τ ( S ) = Id by proposition 2.1, this is tantamount to || v n ( x ) ⊗ v ( x ) · τ ( S θ ) · v ( x ) ⊗ v n ( x ) − P v n ( x ) || HS ≤ δ ′′ ( θ ) . By the last formula and (3.11) we get that, for n large enough (depending on x ∈ ˆ S ), || β n · || Dψ w,n || HS · τ ( ψ w,n ( S θ )) − P v n ( x ) || HS ≤ δ ′′ ( θ ) . Taking the Hilbert-Schmidt product with the identity we get that (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) β n · || Dψ w,n || HS · τ ( ψ w,n ( S θ )) , Id (cid:19) HS − (cid:12)(cid:12)(cid:12)(cid:12) < δ ′′ ( θ ) . Analogously, (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) β n · || Dψ w,n || HS · τ ( ψ w,n ( S )) , Id (cid:19) HS − (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ ′′ ( θ ) . The first equality below comes from the definition of Kusuoka’s measure κ ; the second one comes since weare integrating the constant matrix field Id ; the inequality comes from the last two formulas; it holds for n large enough, depending on x ∈ ˆ S . (cid:12)(cid:12)(cid:12)(cid:12) κ ( ψ w,n ( S θ )) κ ( ψ w,n ( S )) − (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R ψ w,n ( S θ ) ( Id, d τ ( x )) HS R ψ w,n ( S ) ( Id, d τ ( x )) HS − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ( Id, τ ( ψ w,n ( S θ ))) HS ( Id, τ ( ψ w,n ( S ))) HS − (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ ′′ ( θ ) . If we set δ ( θ ): = 4 δ ′′ ( θ ), (3.10) follows from the last formula. \\\ efinition. Let ˆ S be defined as at the beginning of this section and let δ ( θ ) > F θ as the set of all the x = Φ( w ) ∈ ˆ S such that for infinitely many n ’s we have x ∈ ψ w,n ( S θ ) and κ ( ψ w,n ( S θ )) κ ( ψ w,n ( S )) ≥ − δ ( θ ) . (3 . S θ in (3.2), the first formula above says that x is ”well in the interior” of the cell ψ w,n ( S ). The second inequality holds for all x = Φ( w ) ∈ ˆ S and n ≥ n ( w ) by lemma 3.2. Lemma 3.3.
Let θ be defined as after (3.2) and let the function δ ( θ ) be as in lemma 3.2; then, κ ( F θ ) ≥ − δ ( θ ) for all θ ∈ (0 , θ ] . (3 . Proof.
Let us consider the set of the x = Φ( w ) ∈ ˆ S which satisfy (3.12) for some fixed n ∈ N ; clearly, it is F n,θ : = [ w ...w n ψ w ...w n ( S θ ) ∩ ˆ S where the union is over all words w . . . w n for which the second formula of (3.12) holds. Note that the unionis disjoint: indeed, the two cells ψ w ...w n ( S ) and ψ w ′ ...w ′ n ( S ) either coincide (and then w . . . w n = w ′ . . . w ′ n )or are disjoint or intersect at the boundary points; but ψ w ′ ...w ′ n ( S θ ) does not contain the boundary points.Recall that, by definition, F θ = lim sup n → + ∞ F n,θ . Thus, it suffices to show that κ ( F n,θ ) ≥ − δ ( θ )from a certain n onwards; equivalently, we are going to show that κ ( F cn,θ ) ≤ δ ( θ ) . (3 . ψ w ...w n ( S ) ∩ ˆ S are disjoint. In other words, if x ∈ ˆ S , x lies in only one ψ w ′ ...w ′ n ( S ); thus, x F n,θ only in two cases:1) x ∈ ψ w ...w n ( S ) with the word ( w . . . w n ) which does not satisfy the second formula of (3.12), or2) ( w , . . . , w n ) satisfies the second formula of (3.12), but x ∈ ψ w ...w n ( S \ S θ ).By (3.10) and Rokhlin’s theorem ([17], [18]) we see easily that the measure of the union of the cells 1)tends to zero as n → + ∞ . Thus, it suffices to prove that the measure of the union of ψ w ...w n ( S \ S θ ) forthe words satisfying 2) is smaller than 2 δ ( θ ). Since these cells are disjoint (recall that the boundary pointswhere they intersect are not in ˆ S ), we get the equality below, where the sum is over the words w . . . w n which satisfy the second formula of (3.12); the first inequality follows by (3.12), the second one from the factthat the union is disjoint and κ is probability. κ (cid:16)[ ψ w ...w n ( S \ S θ ) (cid:17) = X κ ( ψ w ...w n ( S \ S θ )) ≤ δ ( θ ) X κ ( ψ w ...w n ( S )) ≤ δ ( θ ) . This is (3.14), which we have seen to imply the thesis. \\\§
We begin the proof of theorem 1 showing that, if f ∈ C ( R , R ), then Ch ( f ) ≤ E ( f ). The core of theproof is in lemma 4.3 below, which says the following: if π is a test plan with bounded deformation then,for π -a. e. curve γ and L -a. e. t ∈ [0 , γ is differentiable at t and ˙ γ t is parallel to the unit vector v ( γ t )of (2.13). Thus, if φ ∈ C ( R , R ), we get that | ( ∇ φ ( x ) , v ( x )) | is a weak gradient of φ ; by the definition ofCheeger’s derivative, this implies that | Dφ | w ( x ) ≤ | ( ∇ φ ( x ) , v ( x )) | .We set ˆ F = [ n ≥ F n (4 . F θ is defined as in formula (3.12). By lemma 3.3 we have that κ ( ˆ F ) = 1; since F n ⊂ ˆ S by definition,we have that ˆ F ⊂ ˆ S ; recall that ˆ S has been defined at the beginning of section 3. Lemma 4.1.
Let the set ˆ F be defined as in (4.1); let x = Φ( w ) ∈ ˆ F and let γ ∈ AC ([0 , , S ) , t ∈ (0 , be such that γ t = x and γ is differentiable at t . Let the vector field v ( x ) be as in (2.13). Then, ˙ γ t = ±|| ˙ γ t || · v ( γ ( t )) . (4 . Proof.
By the definition of ˆ F in (4.1) we have that x ∈ F l for some l ∈ N . By the definition of F l in(3.12) there is a sequence n k ր + ∞ such that x ∈ ψ w,n k ( S l ) ∀ k ≥ . (4 . x ∈ ˆ S , we can label the boundary of the cell ψ w,n k ( S ) as in lemma 3.1; we get that (3.5) holds forthe point x = Φ( w ). Since γ t = x and γ can escape out of a cell only hitting its boundary, three cases arepossible.1) There is h k → γ ( t + h k ) ∈ { A w,n k , B w,n k } . By (4.3), (3.5) holds and implies that γ ( t + h k ) − γ ( t ) h k → λv ( x )for some λ ∈ R ; since we are supposing that γ is differentiable at t , (4.2) follows.18) There are h + k > h − k < h + k , h − k → γ ( t + h + k ) = γ ( t + h − k ) = C w,k . Since γ is differentiable at t , we get the first equality below, while the second one follows by the formulaabove. ˙ γ t = lim k → + ∞ γ ( t + h + k ) − γ ( t + h − k ) h + k − h − k = 0 . This implies (4.2) for the speed || ˙ γ t || = 0.3) The only remaining alternative is that, for some h >
0, either γ ([ t, t + h ]) ⊂ ψ w,n k ( S ) or γ ([ t − h , t ]) ⊂ ψ w,n k ( S )for infinitely many k ’s.Let us suppose that the first case happens and let us fix h ∈ (0 , h ]. Since the diameter of ψ w,n k ( S )tends to zero by (1.3), the limit below holds, while the inequality holds for h ∈ (0 , h ]. || γ ( t + h ) − γ ( t ) || h ≤ diam( ψ w,n k ( S )) h → . Thus, || γ ( t + h ) − γ ( t ) || h = 0for all h ∈ (0 , h ]. Letting h → γ is differentiable at t , we get that ˙ γ t = 0, proving (4.2)also in this case. \\\ Lemma 4.2.
Let π be a test plan with bounded deformation and let ˆ F ⊂ S be defined as (4.1). Then,denoting by L the Lebesgue measure on R , we have that L ( γ − ( ˆ F c )) = 0 for π -a. e. γ ∈ C ([0 , , S ) . (4 . Proof.
We consider [0 , × C ([0 , , S ) with the measure L ⊗ π ; we define the map g : [0 , × C ([0 , , S ) → [0 , × Sg : ( t, γ ) → ( t, γ t ) . Since π has bounded deformation, it is immediate that there is C > g ♯ ( L ⊗ π ) ≤ C ( L ⊗ κ ) . We saw after (4.1) that κ ( ˆ F c ) = 0, which implies by Fubini that ( L ⊗ κ )([0 , × ˆ F c ) = 0. Together withthe last formula this implies that g ♯ ( L ⊗ π )([0 , × ˆ F c ) = 0, i. e., by the definition of push-forward,( L ⊗ π )( { ( t, γ ) : γ t ˆ F } ) = 0 .
19y Fubini, this implies the thesis. \\\
Lemma 4.3.
Let φ ∈ C ( R , R ) . Then, | Dφ | w ( x ) ≤ | ( P v ( x ) , ∇ φ ( x )) | for κ -a. e. x ∈ S . (4 . Proof.
By the definition of | Dφ | w ( x ) it suffices to show that | ( P v ( x ) , ∇ φ ( x )) | is a weak gradient of φ ; bythe definition of weak gradient in section 1, this follows if we show that, for all test plans π with boundeddeformation, for π -a.e. γ ∈ C ([0 , , S ) and for L -a. e. t ∈ [0 ,
1] which is a point of differentiability of γ wehave ˙ γ t = ±|| ˙ γ t || · v ( γ ( t )) . (4 . π be a test plan and let us call B ⊂ C ([ − , , S ) the full measure set of curves for which (4.4)holds; since π is a test plan, excluding a zero-measure set we can suppose that all curves in B are absolutelycontinuous. If we set A γ = γ − ( ˆ F ), we see by (4.4) that L ( A γ ) = 1 for all γ ∈ B . Since γ is absolutelycontinuous, if we set ˆ A γ = { t ∈ A γ : γ is differentiable at t } we see that L ( ˆ A γ ) = 1 too. The upshot of all this is that it suffices to show (4.6) for all γ ∈ B and all t ∈ ˆ A γ . To say that t ∈ ˆ A γ is equivalent to say that γ t ∈ ˆ F and γ is differentiable at t ; now (4.6) followsfrom lemma 4.1. \\\ We need to prove the opposite inequality to (4.5); for this we shall use lemma 3.1 and Cheeger’scharacterisation of the energy (1.13). We begin with two lemmas; before stating them we recall from the endof section 1 that
Lip a ( f, x ) is the local Lipschitz constant calculated on S ; the distance on S is that inducedby the immersion in R . Lemma 4.4.
Let f ∈ Lip ( S ) . Then, Ch ( f ) = inf lim inf n → + ∞ Z S Lip a ( f n )d κ where the inf is over all sequences { f n } n ≥ such that1) f n ∈ C ( R , R ) and2) f n → f in L ( S, κ ) . Proof.
Since C functions are Lipschitz, the inequality Ch ( f ) ≤ inf lim inf n → + ∞ Z S Lip a ( f n )d κ Step 1.
We begin to assert that in (1.13) it suffices to consider sequences { g n } n ≥ of Lipschitz functionswhich are C on each cell ψ w ...w n ( ¯ A ¯ B ¯ C ).We prove the assertion. Let h : S → R be Lipschitz and let us consider the cell ψ w ...w n ( S ); its threeboundary points A w ...w n , B w ...w n and C w ...w n define a triangle. It is standard (Kirszbraun’s theorem, seefor instance [6]) that we can extend h restricted to the three points A w ...w n , B w ...w n , C w ...w n to a function˜ h defined on the whole triangle A w ...w n , B w ...w n , C w ...w n which has the same Lipschitz constant h has onthe three points. By definition, h and ˜ h coincide on the three boundary points.We use another standard approximation: there exists ǫ n → g n such that g n is C inthe interior of each triangle ψ w ...w n ( S ) and satisfies i ) g n ( A w ...w n ) = h ( A w ...w n ), g n ( B w ...w n ) = h ( B w ...w n ) and g n ( C w ...w n ) = h ( C w ...w n ). ii ) | g n ( x ) − ˜ h ( x ) | ≤ ǫ n for all x in the triangle. iii ) |∇ g n ( x ) | ≤ L w ...w n + ǫ n for all x in the triangle, where L w ...w n is the Lipschitz constant of ˜ h in thetriangle (or of h on the three points A w ...w n , B w ...w n , C w ...w n , which is the same.) Here and in the followingwe denote by ∇ f the standard gradient of f in R .For a fixed n ∈ N , the triangles ψ w ...w n ( ABC ) intersect only at the vertices; this implies by i ) that, ifwe define ˜ g n to be equal to g n on each triangle, the definition is well-posed and the function ˜ g n is continuous.Moreover, it is C in the interior of each triangle.Since h is Lipschitz, ii ) and the definition of ˜ h imply that g n → h uniformly on S , and thus in L ( S, κ ).On the other side, by iii ) and the definition of the
Lip a in section 1 we have thatlim sup n → + ∞ |∇ g n | ( x ) ≤ Lip a ( h, x )for all x ∈ S , save for the points on the boundary of a cell, where the gradient is not defined; but these area countable set and have zero measure.Since h is Lipschitz, we get that L w ...w n is bounded; now iii ), the formula above and dominatedconvergence imply that lim sup n → + ∞ Z S |∇ g n | ( x )d κ ( x ) ≤ Z S Lip a ( h, x )d κ ( x ) . (4 . { h n } n ≥ be a sequence of Lipschitz functions such that12 Z S Lip a ( h n , x )d κ ( x ) → Ch ( f )and || h n − f || L ( S,κ ) → . By the aforesaid we can approximate each h n with a function g n which is • ) continuous on S , • ) C on the k -th generation cells for some large k = k ( n )21 ) and such that, by (4.7), 12 Z S |∇ g n | ( x )d κ ( x ) ≤ Z S Lip a ( h n , x )d κ ( x ) + 1 n and || h n − g n || L ( S,κ ) → . From the last four formulas we get that g n → f in L ( S, κ ) and that12 Z S |∇ g n | ( x )d κ ( x ) → Ch ( f ) . Clearly, this implies step 1.
Step 2.
Next, we assert that in (1.13) we can take f n ∈ C ( R , R ).In order to show this, let us fix n and let us gather all the boundary points of the n -th generation cells[ w . . . w n ] in a finite set { P j } l n j =1 . For r > O ( r ) = S ∩ l n [ j =1 B ( P j , r n ) . Since κ is non-atomic, given ǫ n > r n > κ ( O ( r n )) < ǫ n . (4 . g n be the function of the previous step, which is C on all the n -th generation triangles; it is easy tosee that, whatever is ǫ n > r n >
0, which depends on ǫ n ), we can find f n ∈ C ( R , R ) which coincideswith g n on S \ O ( r n ) and such that, for all x ∈ S , |∇ f n | ( x ) ≤ Lip ( g n ). Choosing ǫ n small enough in (4.8),this implies first that f n → f in L ( S, κ ), second that Z S ||∇ f n | − |∇ g n | | d κ → n → + ∞ . As at the end of step 1 this implies that12 Z S |∇ f n | ( x )d κ ( x ) → Ch ( u )which in turn implies (1.13) for { f n } n ≥ . \\\ Lemma 4.5.
Let f ∈ C ( R , R ) and let the local Lipschitz constant Lip a ( f, x ) be defined as above; let v ( x ) be as in (2.13). We assert that Lip a ( f, x ) ≥ | P v ( x ) ∇ f ( x ) | ∀ x ∈ ˆ S. (4 . roof. Let ψ w,n ( S ) be the n -th generation cell which contains x ; we relabel its boundary A w,n , B w,n and C w,n in such a way that (3.4) holds. Since A w,n and B w,n belong to S and converge to x we have that Lip a ( f, x ) ≥ lim n → + ∞ | f ( B w,n ) − f ( A w,n ) ||| B w,n − A w,n || . Since f ∈ C ( R , R ), (3.4) implies (4.9). \\\ End of the proof of theorem 1. Step 1.
We assert that it suffices to show that E ( f ) = 2 Ch ( f ) when f ∈ C ( R ).We prove the assertion using theorem 3.8 of [8] (see section 4 of [11] for the original treatment and theproof). This theorem says that D ( E ), the domain of E , is a Hilbert space for the inner product( u, v ) D ( E ) : = ( u, v ) L ( S,κ ) + E ( u, v )and that C ( R ) is dense in D ( E ).Since E and 2 Ch coincide on C ( R ), we get that D ( E ) with the inner product ( · , · ) D ( E ) is also thecompletion of C ( R ) for the norm || u || D ( Ch ) : = || u || L ( S,κ ) + 2 Ch ( u ) . This implies that D ( Ch ) ⊃ D ( E ); the opposite inclusion follows by (1.13) if we show that D ( E ) contains theLipschitz functions.This follows from two facts: first, on D ( E ), E is defined by relaxation E ( u ) = inf lim inf n → + ∞ E ( u n ) . (4 . { u n } n ≥ ⊂ C ( R , R ) such that u n → u in L ( S, κ ).The second fact is that, if u ∈ Lip ( S ), then u can be approximated, in the uniform topology, by asequence { u n } n ≥ ⊂ C ( R ) such that ||∇ u n || sup is bounded. This implies the second inequality below; thefirst one comes from the fact that E is lower semicontinuous; the equality comes from the definition of E inproposition 2.1. E ( u ) ≤ lim inf n → + ∞ E ( u n ) = lim inf n → + ∞ Z S ( ∇ u n ( x ) , P v ( x ) ∇ u n ( x ))d κ ( x ) < + ∞ . Step 2.
We show that E ( f ) = 2 Ch ( f ) when f ∈ C ( R , R ).First of all, lemma 4.3 implies that 2 Ch ( f ) ≤ E ( f ) . (4 . f ∈ C ( R , R ) , let us define its pre-Cheeger energy pCh ( f ) as pCh ( f ) = 12 Z S Lip a ( f, x ) d κ ( x ) .
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