Metric Measure Spaces and Upper Regularity of Dirichlet Forms.

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Università degli Studi di Pisa

Facolta di Scienze Matematiche, Fisiche e Naturali

Corso di Laurea Magistrale in Matematica

Tesi di Laurea Magistrale

Metric Measure Spaces and Upper

Regularity of Dirichlet Forms

22 September 2017

CANDIDATO

Lorenzo Portinale

lorenzo.portinale@sns.it

RELATORE

Prof. Luigi Ambrosio

Scuola Normale Superiore, Pisa

CONTRORELATORE

Prof. Dario Trevisan

Università di Pisa

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Notations

x = (x1, ..., xn) point in Rn,

|x| _{euclidean norm of x ∈ R}n_, < x, y >= x · y euclidean scalar product,

Ω _{open set in R}n,

I interval in R,

(M, g) Riemannian manifold with g positive definite metric on M ,

∆f Laplacian of f ,

Ricg Ricci tensor related to the metric g, volg volume form on (M, g),

Br ball centered at the origin with radius r, Br(x) ball centered at x ∈ Rn with radius r,

supp(f ) support of f , ∇f (x) gradient of f in x, |∇f |(x) slope of f in x,

|∇−_{f |(x)} _{descending slope of f in x,}

|∇+_{f |(x)} _{ascending slope of f in x,}

Lip_a(f, x) asymptotic Lipschitz constant of f in x, |Df | distributional gradient of f ,

|Df |∗ minimal relaxed slope of f ,

|Df |w minimal weak upper gradient of f ,

Lip(f ) Lipschitz constant of f ,

Chd Cheeger energy induced by the distance d,

Ln _{Lebesgue measure in R}n_,

1A characteristic function of a measurable set A,

Lp_{(X, m)} _{Lebesgue space of functions defined on a measure space (X, m),}

Cl(Ω) space of l−time continuously differentiable functions on Ω, C∞(Ω) space of infinitely time differentiable functions on Ω, C_b(Ω) space of continuous and bounded functions on Ω,

C_c(Ω) space of continuous and with bounded support functions on Ω, C([a, b], X) space of continuous curves defined in [a, b] with values in X,

AC([a, b], X) space of absolutely continuous curves defined in [a, b] with values in X,

W1,p(Ω) Sobolev spaces on Ω,

P(X) space of probability measures on X,

P2(X) space of probability measures on X with finite second moment,

W2 2−Wasserstein distance onP2(X),

Entm m−relative entropy functional,

Sym(R, n) set of symmetrix matrices of dimension n × n with real coeffi-cients.

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Chapter 1

Introduction

Metric measure spaces have become a crucial point of investigation through the last thirty years; born as the natural generalization of the concept of weighted Riemannian manifold, these spaces have been deeply studied under several points of view, starting from the classical abstraction of differential tools in the metric setting to the theory of abstract Sobolev spaces.

A metric measure space is a triple (X, d, m) where (X, d) is a complete and separable metric space, while (X, m) is a measure space with m a nonnegative σ−finite Borel measure on X (where we consider the topology induced by the distance d).

The model we have to keep in mind is the one of a complete oriented Riemannian Manifold (M, g): here the distance one has to take into account is the geodesic one, whereas the reference measure is usually the volume measure induced by the metric on the space, that is:

volg=

q

det(g) dx1∧ ... ∧ dxk

written in the language of differential forms (the measure is the one induced by the integration of the form on the manifold).

In this setting, a powerful result was proven by Myers around the forties to link hy-pothesis on the curvature and topological/metric characteristics of the space. In particular, the statement says that if (M, g) is a n−complete manifold with the Ricci tensor bounded from below, namely:

Ric_g _{≥ (n − 1)kg , k ∈ R}+ (1.0.1)

then M has to be compact, its fondamental group is finite and it has an explicit estimate on the diameter given by:

diam(M ) ≤ √π

k.

This result is based on the double structure we have on the manifold: from a request concerning differential properties of the space (the hypothesis on the Ricci tensor, due to the fact that it is defined using the covariant derivative) we can deduce metric (and topological as well) properties of M .

A resulting natural thought of many mathematicians has been to try to extend such a result in a more abstract setting, but to do this, firstly, one needs to introduce something that extends the classical differential tools, for example the Ricci tensor (or better, in the Myers problem, the lower bound on the Ricci tensor).

The decisive observation is to notice the relationship between two different but linked objects we have on a weighted Riemannian manifold. On the one hand, there is the distance, induced by geodesics on the manifold; on the other hand, we have the classical Dirichlet

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energy on the space, given by:

E(f ) = Z

M

|∇f |2_dm

where ∇ is the Levi-Civita covariant derivative on M .

As we are going to see inchapter 3, there is a strong correspondence between the energy and the distance; in the smooth framework of a Riemannian manifold, it is possible to recover the metric structure from the energy one and viceversa. This means that, in some sense, they represent the same thing.

Following these ideas, two really different approaches were born: the one the BE(K, N ) spaces and the one of CD(K, N ) spaces.

The most important reference for the first one is given by the seminal work of Bakry-Émery ([7], 1984) and the abstract setting is the one of the Dirichlet spaces. The main tool and the starting point of the theory is a local and symmetric Dirichlet form on a Polish measure space (X, m) whose domain is a dense subset of L2(X, m) (namely a Dirichlet space); details will be presented throughoutchapter 3 but the idea of the theory (or more precisely of a Dirichlet form) is to try to define a general concept of "energy" that plays the same role of the classical Dirichlet energy in the smooth setting. In the work of Bakry-Émery, a particular class of Dirichlet spaces were introduced, the so-called

BE(K, N ) spaces, which are named after the two authors: here K ∈ R+, N ∈ N indicates a "lower bound on the Ricci curvature" and a "dimension" on the space respectively. The definition of these two concepts in the abstract setting of Dirichlet spaces passes through "Bochner’s formula", which is true on Riemannian manifold, and it is given by:

∆ ₁ 2|∇f | 2 = ||Hess_gf ||2_g+ Ric_g(∇f, ∇f ) (1.0.2) and maybe the more important Bochner’s inequality:

∆ ₁ 2|∇f | 2_{≥ K|∇f |}2₊ 1 N (∆gf ) 2 (1.0.3) which holds for any N ≥ n (dimension of M ) if and only if K is a lower bound for the Ricci curvature. As a result, it is natural to take (1.0.3) as a definition both for the dimension and the curvature lower bound in Dirichlet spaces, after giving a suitable meaning to the objects one can find in the formula.

The BE(K, N ) spaces were the first general approach to the theory of spaces with Ricci curvature bounded from below, but they were definitely not the only one. At the beginning of the 20th century, Lott-Villani ([17]) and Sturm ([20]) dealt with the problem in the framework of optimal transportation; they were in particular interested in looking for a suitable class of metric measure spaces which was stable under the so-called

Gromov-Hausdorff convergence. This time the starting point is the distance and not the energy:

a metric measure space (X, d, m) is said to be a CD(K, ∞) space if the relative entropy functional is K−convex along the geodesics in the Wasserstein space (P₂(X), W₂), where the entropy is the functional Entm:P2(X) → R+ given by:

Ent_m(ρµ) = Z

X

ρ log ρ dm (1.0.4)

and +∞ if the measure is not absolutely continuous with respect to m.

Again, in the classical Riemannian setting, this condition turns out to be equivalent to the condition of having Ricci curvature bound from below; therefore it is taken as definition

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9 in the general setting of metric measure spaces. Nevertheless, given a metric/measure structure, one can build an "energy" called Cheeger energy; as we explain inchapter 3, this energy needs not to be a Dirichlet form (in particular, it needs not to be quadratic), but in some sense it gives a way starting from the metric measure structure to define a "form" as well as in the Dirichlet spaces is possible to induce a distance.

These two different notions of Ricci curvature bound from below worked indipendently for some years without having too much to deal with each other, until the work of Ambro-sio, Gigli and Savaré with [6] and [4], where RCD(K, ∞) has been introduced: they are just CD spaces where the Cheeger energy is quadratic and then in particular a Dirichlet form. Under this assumption, we have at the same time a metric measure space and a Dirichlet one, that means RCD spaces are the meeting point of the theory of BE and

CD spaces. Moreover, they turned out to be a closed subset of CD with respect to the

Gromov-Hausdorff convergence.

Aim of this work is to deeply investigate the interaction between "energy" and distance and to study differences and links between the two different perspectives: on the one side the metric measure spaces framework, on the other side the theory of the Dirichlet forms.

In chapter 2 we start with metric measure spaces theory; the main definitions are given, as the notion of slope and asymptotic Lipschitz constant of a measurable function. Then starting from the metric (and measure) structure (X, d, m), we give the definition of Cheeger energy, that is:

Chd(f ) = inf lim inf n→+∞ Z X Lip2_a(fn, x) dm(x) : {fn}n⊂ Lipb(X, d) , fn→ f in L2(X, m) ,

where Lip_a_{(f, ·) is the asymptotic Lipschitz constant of a function f : X → R, namely:} Lip_a(f, x) = inf

>0_y,z∈Bsup (x) , y6=z

|f (y) − f (z)|

d(y, z) .

The theory of relaxed slopes and weak upper gradients are presented so as to show two different though equivalent ways to represent the Cheeger energy; in the final part of chapter 2the Hopf-Lax semigroup is introduced with its main properties, with particular attention to the duality formulaTheorem 2.3.6.

Dirichlet forms will be introduced inchapter 3in order to present the other point of view, that is to start directly from the "energy" (the form) and then trying to find the most natural way to induce a metric structure from the energy one. In this work we will mainly treat Dirichlet forms E which admit an integral representation with respect to a reference measure m, namely

E(f ) = Z

X

Γ(f ) dm

where Γ(·) is usually named carré du champs of E . We present some examples of Dirichlet spaces, with particular attention to the case of forms induced by elliptic diffusion operators in Rn_{, namely:}

EA(f ) =

Z

Ω

< A∇f, ∇f > dx, ∀f ∈ H1(Ω) (1.0.5) where Ω ⊂ Rn is an open set and A : Ω → Sym(R, n) is a measurable map such that

λI ≤ A ≤ ΛI; in this framework we focus in particular on the interaction between energy,

distance and (Riemannian) metric (if A(·) is smooth).

Given a Dirichlet form, we introduce the notion of intrinsic distance:

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so as to have a canonical way to move from the "energy" to the metric structure. In the last part ofchapter 3we will study the energy measure spaces, which means to work with the induced metric measure spaces (X, dE, m) where E is a Dirichlet form on (X, m).

The central topic of chapter 4 is the natural question that arises from the previous constructions: what is the relationship between the induced Cheeger of an energy measure space and the original Dirichlet form? In some sense we are looking for a Riemannian structure in an energy measure space.

First of all we see that the "projection" that maps a Dirichlet form E into the induced Cheeger energy ChE is always decreasing, in the sense that:

ChE(f ) ≥ E (f )

for all f ∈ V. In order to prove the so-called regularity of a form E, namely

ChdE = E (1.0.6)

we need to prove the other inequality; this turns out to be equivalent to the so-called upper

regularity of E , a notion introduced for the first time in the works of Ambrosio, Gigli and

Savaré ([4]) and then in the setting of extended metric spaces by Ambrosio, Erbar and Savaré ([2]). A Dirichlet form E is said to be upper regular if for any f in the domain of E, it is possible to find a sequence {fn}n ⊂ Lipb(X, dE) and a sequence of bounded and

upper semicontinuous functions {gn}n in such a way that fn→ f in L2(X, m), gn≥

q

Γ(fn), m − a.e. in X and lim sup n→+∞

Z

X

g2_ndm ≤ E (f ). This condition turns out to be equivalent to (1.0.6); the proof, following the original idea of Ambrosio, Gigli and Savaré, is presented throughout the wholechapter 4.

Finally, we come back to the elliptic diffusion Dirichlet forms in Rn in the last chapter; the aim of chapter 5 is to deepen the analysis of the upper regularity of this particular class of forms.

The first important result in this setting dates back to 1997 and is due to Sturm [19].

Theorem 1.0.1 (Sturm). Let Ω ⊂ Rn be an open set and consider the class of uniform elliptic matrices-valued operators M (λ, Λ). Suppose that n ≥ 2; then, for any A ∈ M (λ, Λ) it is always possible to find ¯A ∈ M (¯λ, ¯Λ) in such a way that:

dA= d_A¯

whereas ¯A < A on a set Y ⊂ Ω of positive m−measure, in the sense that: vtA(x)v < v¯ tA(x)v, ∀v ∈ Rn, ∀x ∈ Y.

The previous result shows that the map

A → dA

is not one-to-one, differently from what happens concerning the duality between the operator A and the induced form E_A(see the classical work of Spagnolo [18]). Moreover, the same result can be used to prove that in dimensione n ≥ 2 it is always possible to find examples of elliptic diffusion forms which are not regular (in the sense ofEquation 1.0.6) and then not upper regular too, invoking the equivalenceTheorem 4.0.3.

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11 The regularity of a form remains a not trivial condition to ask, even if we are working in the simpler context of the elliptic diffusion forms in Rn; in [14] Koskela and Zhou present an interesting example of a Dirichlet forms in R2 such that Cheeger energy induced by d_A is not quadratic (and then, in particular, there is no regularity). This and the previous remarks show that some more conditions must be asked to A if we want to get the upper regularity of E_A.

Precisely, in the last part of the work we investigate the additional condition needed on the coefficients of the diffusion operator A(·) so that the induced form E_A turns out to be upper regular. First of all we prove that in dimension n = 1 no condition is needed to get the upper regularity (Theorem 5.1.1).

Theorem 1.0.2. Let us suppose n = 1, Ω = (a, b), a, b ∈ R and take any a(·) ∈ M (λ, Λ); let Ea be the induced Dirichlet form, which is:

E_a(f ) = Z b

a

a(t) ˙f2(t) dt, ∀f ∈ H1([a, b]). (1.0.7)

Then Ea is upper regular.

On the other hand, things work differently in bigger dimension; as we have already pointed out, Sturm’s theorem ([19]) can be used to prove that it is possible to find examples of forms E_A which are not upper regular, for any dimension n ≥ 2.

Nevertheless, we prove inTheorem 5.2.2that it is sufficient to ask a bit more regularity on the coefficients of A(·); we have to suppose A(·) to be directionally upper semicontinuous which means that for all v ∈ Rn, the map

x 7→< A(x)v, v >

is upper semicontinuous in Ω.

Theorem 1.0.3. Let Ω ⊂ Rn be a bounded open set, n ∈ N and take any A ∈ M (λ, Λ). Suppose that A is directionally upper semicontinuous. Then EA is upper regular.

Some open questions are left in this framework; in particular, there is not a pointwise equivalent characterization of the upper regularity of elliptic diffusion forms in dimension bigger than one.

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Chapter 2

Metric Measure Spaces

This chapter is devoted to the introduction of metric measure spaces theory.

Definition 2.0.1. A metric measure space is given by (X, d, m) where:

(a) (X, d) is a complete and separable metric space, with d a finite distance on X. (b) (X, m) is a measure space, with m a nonnegative σ−finite Borel measure, with respect

to the metric topology on X induced by d.

In the first part of the chapter the basic tools are introduced, such as the concept of slope and asymptotic Lipschitz constant of a measurable function f : X → R; therefore the Cheeger energy is defined and its main property are treated. We will see two different ways to represent the Cheeger energy with respect to the reference measure m, using first the theory of relaxed slopes and then the one of weak upper gradients. In the last part of the chapter Hopf-Lax semigroup will be introduced, because of its central role in the equivalence theorem (Theorem 4.0.3) in chapter 4.

The classical model of metric measure space that we have to keep in mind throughout the entire work is given by the Riemmanian setting.

Example 2.0.2. The most simple example of metric measure space is given by (X, d, m)

where:

• X = (M(N )_{, g) is a N −dimensional Riemannian manifold, with g a strictly positive}

definite metric on M(N ).

• d = dgeo is the (complete) geodesic distance on X induced by g.

• m = e−Vvolg, where volg is the standard volume measure of (M(N ), g) and V a

smooth potential, namely V ∈ C∞(M ).

This represents the "smooth setting", the reference example of metric measure space which is built using the differentiable structure of a manifold.

Concerning this part of the work, the most important references we have been inspired by are mainly [3] and [1]; see also [13], [10] for the general theory of metric measure spaces.

2.1 "Derivatives" in metric setting

Let’s start our "differential" description with a lagrangian point of view, working with curves on a general metric space (X, d). Some natural questions arise:

What "to differentiate a curve" does mean? Which curves can be differentiated?

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As usual, the idea is trying to analyze the classical setting in a general perspective; in particular we’d like to find equivalent formulations of smooth concepts using only the metric structure of the space. Therefore one can give new coherent definitions in the abstract setting using those equivalences; following this idea, we recall that given a Rn-valued curve γ defined on [0, 1], the absolute continuity of γ can be equivalently described by the following conditions:

• The curve is absolutely continuous in the classical sense, namely:

∀ > 0 , ∃δ = δ() > 0 : ∀ partition of [0, 1] {0 ≤ t₁ < ... < tn= 1} n X j=1 |tj+1− tj| ≤ δ =⇒ n X j=1 |f (tj+1) − f (tj)| ≤ .

• There exists a function g : [0, 1] → R+ _in_L1_{([0, 1]) such that:}

|γ(t) − γ(s)| ≤ Z t

s

g(r) dr

for all s, t ∈ [0, 1], s ≤ t.

It is well-known from the theory on the real line that the absolute continuity of a curve is the right concept to get an almost-everywhere differentiability on its interval of definition.

We shall try to repeat the same theory in the abstract setting starting from the second version/definition of absolute continuity and try to find something similar to the classical differentiability for a curve on a general metric space.

Hence, let (X, d) be a metric space and a curve γ : [a, b] → X.

Definition 2.1.1. We say that γ is absolutely continuous if there exists a positive function g ∈ L1_{([a, b]) such that:}

d(γ(x), γ(y)) ≤

Z y

x

g(t) dt. (2.1.1)

We denote by AC([a, b], X) the space of absolutely continuous curves in X.

Among all the functions g that satisfy (2.1.1) for a given absolutely continuous function

f , there is a minimal one that coincides with the modulus of ˙γ(t) in the smooth setting

(seeTheorem 2.1.2 in this special case).

In the following theorem we characterize this minimal element in the general framework of a metric space.

Theorem 2.1.2. Let γ ∈ AC([a, b], X); then the following limit exists:

| ˙γ|(t) := lim

h→0

d(γ(t + h), γ(t))

|h| .

In addition, | ˙γ| is the smallest admissible g (up to L1−negligible sets) in (2.1.1).

Definition 2.1.3. | ˙γ| is called metric derivative of γ, while we called length of γ the value:

l(γ) :=

Z b

a

| ˙γ|(t) dt.

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2.1. "DERIVATIVES" IN METRIC SETTING 15 In the class of absolutely continuous curves is then possible to make a similar theory as in the smooth setting; for example, one can define the geodesics on the space using a purely metric point of view.

Definition 2.1.4. A curve γ ∈ AC([a, b], X) is said to be a geodesic if:

d(γ(t), γ(s)) = |s − t|d(γ(b), γ(a)) for any s, t ∈ [a, b]. (2.1.2) Moreover, one defines the geodesic distance on the space X as the distance given by:

dg(x, y) := sup (tj)    N X j=1 d(γ(tj+1), γ(tj)) : a = t1≤ ... ≤ tN = b    . (2.1.3)

Finally a metric space (X, d) is said to be a length space if the geodesic distance coincides with the starting one, that is d_g = d.

We have just seen how to "differentiate" a curve on a metric space; suppose now to have a scalar function on the space, that is:

f : X → R.

We want to find the correct way to generalize the usual gradient in Rn in this general framework.

Definition 2.1.5. _{Let f : X → R be a scalar function defined on X. Given any x ∈ X,}

the slope of f is defined by:

|∇f |(x) := lim sup

y→x

|f (y) − f (x)|

d(y, x) . (2.1.4)

In a similar way, the descending slope is defined by: |∇−f |(x) := lim sup

y→x

(f (y) − f (x))₋

d(y, x) (2.1.5)

while the ascending slope by:

|∇+_{f |(x) := lim sup}

y→x

(f (y) − f (x))₊

d(y, x) (2.1.6)

where (·)− and (·)+ are respectively the negative and positive part of a function.

Remark 2.1.6. Suppose (X, d) = (Rm, | · − · |); let f : Rn→ R and let x ∈ Rn_{be a point of}

differentiability for f. Then:

|∇f |(x) = |∇f (x)|. Indeed, since f is differentiable in x we have:

f (y) = f (x) + ∇f (x) · (y − x) + o(|y − x|) as y → x;

then, if we compute the slope of f in x we find: lim sup

y→x

|f (y) − f (x)|

|y − x| = lim supy→x

∇f (x) · (y − x) |y − x| = inf >0_0<|y−x|<sup ∇f (x) · (y − x) |y − x| = |∇f (x)|.

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This is why this new definition represents the natural extension of the (modulus of the) "gradient" in the metric setting.

For example, if f ∈ Lip(Rn), then Theorem 6.1.1implies: |∇f |(x) = |∇f (x)| for a.e. x ∈ Rn.

Nevertheless, if one works with non differentiable functions, it is possible to find examples where the slopes defined inDefinition 2.1.5don’t coincide.

Example 2.1.7. Let’s take X = R with the euclidean distance. Consider the function:

f (t) =        1 + t if t ∈ (−1, 0] 1 − t if t ∈ [0, 1) 0 otherwise

which is a C∞_{−function on R \ {−1, 0, 1}. It can be easily seen by definition that:}

|∇f |(t) = (

1 if t ∈ [−1, 1] 0 otherwise

but the ascending and the descending slope are two more different functions. Indeed: |∇−f |(t) = ( 1 if t ∈ (−1, 1) 0 otherwise while |∇+f |(t) = ( 1 if t ∈ (−1, 1) \ {0} 0 otherwise.

This is consequence of the fact that the discending slope (as the name suggests) on a point t sees only directions where the function goes down (with respect to the point t itself) and evaluate the "slope" through those directions only. The same happens with the ascending slope with respect to the directions where the functions goes up.

This is the reason why the slopes of the function f are different on its maximum and minimum values.

Obviously the slope of a function is not the only notion one can introduce in a metric space so as to generalize the concept of gradient: another important tool is given by the so-called asymptotic Lipschitz constant.

Definition 2.1.8. We denote by asymptotic Lipschitz constant of a function f the following

map:

Lip_a(f, x) := lim

δ→0Lip(f, Bδ(x)) = infδ>0Lip(f, Bδ(x))

where Lip(f, B_δ(x)) denotes the Lipschitz constant of f on the set B_δ(x), namely: Lip(f, B_δ(x)) = sup

z,w∈Bδ(x),z6=w

|f (z) − f (w)|

d(z, w) .

Note that just from definition we have:

Lip_a(f, x) ≥ |∇f |(x), for all x ∈ X.

The next proposition shows some of the basic properties of the asymptotic Lipschitz constant of a function; note that the same properties are satisfied by the slope too.

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2.1. "DERIVATIVES" IN METRIC SETTING 17

Proposition 2.1.9. _{Let f, g : X → R be two continuous functions on X; then the following} inequalities hold:

Lip_a(f + g, x) ≤ Lip_a(f, x) + Lip_a(g, x), ∀x ∈ X,

Lip_a(f g, x) ≤ |f (x)| Lip_a(f, x) + |g(x)| Lip_a(g, x), ∀x ∈ X. (2.1.7)

Proof. The first inequality in (2.1.7) easily comes from the trivial inequalities: |a + b| ≤ |a| + |b| and sup x {f (x) + g(x)} ≤ sup x {f (x)} + sup x {g(x)}. On the other hand, we have:

|f (z)g(z) − f (w)g(w)| ≤ |f (z)(g(z) − g(w))| + |g(w)(f (z) − f (w))|. This and the other trivial property of the supremum given by

sup x {|f (x)g(x)|} ≤ sup x {|f (x)|} sup x {|g(x)|} yield the second inequality in (2.1.7).

Remark 2.1.10. In general, the slope and the asymptotic Lipschitz constant of a function f : X → R do not coincide, not even if we are working in Rnwith a differentiable function. For example: let X = (0, 1) and take a dense subset E ⊂ X withL1(E) < 1. Define the function f (·) as follows: consider the map a(·) given by

a(t) := ( 2 t ∈ E 1 t ∈ Ec and set: f (t) := Z t 0 a(s) ds, ∀t ∈ X.

Then f (·) is everywhere differentiable in X, with

|∇f |(t) = | ˙f (t)| = a(t), for t ∈ X

but we have

Lip_a(f, t) ≡ 2. (2.1.8)

To prove it let’s assume that the function:

x → Lip_a(f, x)

is upper semicontinuous (it will be proved inProposition 2.1.11in the general framework of metric spaces).

Then (2.1.8) comes from the fact that:

2 ≥ Lip_a(f, x) ≥ |∇f |(x), for every x ∈ X. Indeed given z, y ∈ X, x ≤ y, we have:

|f (y) − f (z)| |y − z| =

Ry

z a(s) ds

y − z ≤ ||a||∞= 2.

Hence, it follows that Lip_a(f, x) = 2 = |∇f |(x) for any x ∈ E, which is a dense subset of

X. Hence, upper semincontinuity yields:

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Let’s prove then the upper semicontinuity of the asymptotic Lipschitz constant in general.

Proposition 2.1.11. Given a function f : X → R, we have that the map: x 7→ Lip_a(f, x)

is upper semicontinuous with respect to the metric topology of X, that is:

lim sup

y→x

Lip_a(f, y) ≤ Lip_a(f, x) (2.1.9)

for any x ∈ X.

Proof. We want to prove that if (xn)n is a sequence in X such that d(xn, x) → 0, then:

Lip_a(f, x) ≥ lim sup

n→+∞ Lip_a(f, xn). We have, by definition: Lip_a(f, xn) = inf >0_y,z∈Bsup (xn) , y6=z |f (y) − f (z)| d(y, z) . (2.1.10)

Fix any > 0; since xn is converging to x , from triangular inequality we get: B

3(xn) ⊂ B(x) (2.1.11)

for n big enough.

As a consequence of (2.1.10) and (2.1.11_{), for such n ∈ N it follows:} Lip_a(f, xn) ≤ sup

y,z∈B

3(xn) , y6=z

|f (y) − f (z)|

d(y, z) ≤_y,z∈Bsup_{(x) , y6=z}

|f (y) − f (z)|

d(y, z) .

Eventually taking the limsup in n and then the infimum in , we get (2.1.9).

Even though the slope and the asymptotic Lipschitz constant do not coincide in general, they describe the same concept: their main role is to quantify the incline of a function. The following lemma explains how |∇f | and Lip_a(f, ·) are related to each other.

Firstly, let’s recall the definition of semicontinuous envelope of a function.

Definition 2.1.12. Let f : X → R. We denote by f∗ (f∗ respectively) the upper

semicontinuous (lower semicontinuous) envelope of f , namely:

f∗(x) := inf{h(x) : h ≥ f , h(·) upper semicontinuous},

f∗(x) := sup{h(x) : h ≤ f , h(·) lower semicontinuous}.

Obviously if f (·) is already upper(lower) semicontinuous, its envelope coincides with f itself.

The following result is due to ([1], pag 12).

Lemma 2.1.13. Given a function f : X → R, it holds:

Lip(f ) ≥ Lip_a(f, x) ≥ |∇f |∗(x).

Moreover, if (X, d) is a length space, the second inequality is an equality.

Note that the inequalities in (2.1.13) are both trivial: the first one comes from the very definition of Lip(f ), while the second one is due to the fact that the asymptotic Lipschitz constant of a function is upper semicontinuous (2.1.11) and bigger or equal than the slope.

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2.2. CHEEGER ENERGY OF A METRIC MEASURE SPACE 19

2.2 Cheeger energy of a metric measure space

Slope and asymptotic Lipschitz constant are natural definitions which can be used also in the metric setting, where the classical notion of "gradient" has no meaning.

The next step is to introduce a suitable generalization of the classical energy form we have in the smooth setting, that is:

E(f ) = Z

Rn

|∇f (x)|2dx, f ∈ H1(Rn). (2.2.1) The equivalent of (2.2.1) in a metric measure space (X, d, m) is given by the so-called Cheeger energy (see the works of the namesake author in [11] and [10]).

It is a functional on the space X that can be defined in different (though equivalent) ways; let’s start using the asymptotic Lipschitz constant.

Definition 2.2.1. Let (X, d, m) be a metric measure space; the Cheeger energy is the

functional defined by:

Ch1_d(f ) := inf lim inf n→+∞ Z X Lip2_a(f_n, x) dm(x) : {fn}n⊂ Lipb(X, d) , fn→ f in L2(X, m) . (2.2.2) The domain of the Cheeger energy is the subset of L2(X, m) defined by:

D(Ch1_d) := {f ∈ L2(X, m) : Ch1_d(f ) < +∞}.

One other possibility could be to use the slope instead of the asymptotic Lipschitz constant, namely: Ch2_d(f ) := inf lim inf n→+∞ Z X |∇f_n|2_{(x) dm(x) : {f} n}n⊂ Lipb(X, d) , fn→ f in L2(X, m) . (2.2.3) Lemma 2.1.13directly implies the easier inequality:

Ch1_d(f ) ≥ Ch2_d(f ), ∀f ∈ D(Ch1)

but it is possible to prove that in fact the two definitions are equivalent. We will return on this point in the next section when we will introduce the theory of weak upper gradients (seeRemark 2.2.25).

From this point onward, we will consider the first one as the main definition, especially when we will prove basic properties of Cheeger energies and we will denote it by Ch_d instead of Ch1

d. We also omit the dependence on the distance d if it is trivial.

The first two definitions of Cheeger energy are built through a relaxation process; indeed Ch1 is the relaxation of the functional

F1(f ) := (R

XLip2a(fn, x) dm(x) if f ∈ Lipb(X, m)

+∞ if f ∈ L2(X, m) \ Lip_b(X, m)

with respect to the L2(X, m) distance, in the sense that it is the biggest lower semicontinuous functional which is less or equal than F1.

Similarly, Ch2 is the relaxation of the functional

F2(f ) := (R

X|∇fn|2(x) dm(x) if f ∈ Lipb(X, m)

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2.2.1 Relaxed slopes

Nevertheless, there are different approaches to define the Cheeger energy; one of those is represented by the theory of relaxed slopes.

Definition 2.2.2. Given f ∈ L2(X, m), a function G ∈ L2(X, m) is said to be a Relaxed

Slope of f if there exists a function H ∈ L2(X, m) such that: 1. H(x) ≤ G(x) for a.e. x ∈ X.

2. There exists a sequence {fn}n∈N⊂ Lipb(X, d) such that:

fn→ f and Lipa(fn, ·) * H in L2(X, m).

The idea is trying to do a "relaxation" directly on the asymptotic Lipschitz constant instead of doing a relaxation on the functional F1.

Definition 2.2.3. Given f ∈ L2(x, m), we denote by R2(f ) the set:

R2(f ) := {G ∈ L2(X, m) : G(·) is a relaxed slope for f }.

The next lemma proves that there is always a minimal element inside the class R2(f ), even in a pointwise sense.

Lemma 2.2.4 (Locality and Minimality). Given a function f ∈ L2(X, m), the following

properties hold:

• R2_{(f ) is a closed and convex subset of L}2_{(X, m).}

• G₁, G2∈ R2(f ) ⇒ min{G1, G2} ∈ R2(f ).

Hence, the L2(X, m)−minimal element in the class R2(f ) is well defined, it is denoted by |Df |∗ and it holds:

|Df |∗ ≤ G m − a.e. in X, ∀G ∈ R2(f ).

Proof. Proposition 2.1.9easily imply that R2(f ) is a closed and convex family of functions. Indeed {G_n}_n ⊂ R2_{(f ) is such that G}

n → G in L2(X, m) then we are able to find two

families {Hn}n⊂ L2(X, m) and {fn,m}n,m ⊂ Lipb(X, d) such that: Hn(x) ≤ Gn(x) for a.e. x ∈ X,

fn,m −→_m f and Lipa(fn,m, ·) −_m*Hn in L2(X, m).

Up to subsequence, using reflexivity of L2(X, m) (Theorem 6.2.3) we can suppose H_n* H ∈ L2(X, m); then using a diagonal argument we find:

fn,m(n)−→_n f and Lipa(fn,m(n), ·) −*_n H in L2(X, m)

with

H(x) ≤ G(x)

that means G ∈ R2_{(f ). In a similar way it is possible to prove the convexity of R}2_{(f ) using}

Proposition 2.1.9.

The second part follows for a more general fact:

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2.2. CHEEGER ENERGY OF A METRIC MEASURE SPACE 21 Recall that given a set X, a Dinkin system F is a family of subsets of X such that X ∈ F and F is stable by complementation and monotone unions from below.

We need to use a classical result in measure theory given by the Dinkin’s lemma: it states that if a Dinkin system cointains the generators of a σ−algebra Σ then it must cointain Σ.

We want to use this result so as to prove (2.2.4); let’s consider the set defined by: F := {B ⊂ X Borel such that (2.2.4) holds for all G₁, G2 ∈ R2(f )}.

We want to prove that F is a Dinkin system which cointains the open sets: therefore by Dinkin’s lemma, it follows that F = B(X).

It is obvious by definition of F that X ∈ F and that Bc∈ F for every B ∈ F ; hence we need to prove that F is stable by monotone unions from below, namely:

Bn∈ R2(f ) for n ∈ N, Bn⊂ Bn+1 =⇒ B :=

[

n∈N

Bn∈ R2(f ). (2.2.5)

Let’s take a countable family of monotone sets as in (2.2.5); by definition of F one has: ∀G1, G2∈ R2(f ) →1BnG1+1BcnG2∈ R

2_{(f ).}

Then by construction, it follows1_B_n ↑1_B and1_Bc

n↓1Bc pointwise; therefore by Lebesgue

theorem we obtain:

1BnG1+1BncG2 L2

−→1_BG1+1BcG₂ ∈ R2(f ).

using the fact that R2(f ) is closed.

To conclude is then enough to prove that :

∀B ⊂ X open =⇒ B ∈ R2(f ). (2.2.6)

Take an open set B ⊂ X; fixed a parameter r > 0, consider any function Φr : X → [0, 1]

with the following property:

Φr(x) =

(

1 on B_2r 0 and Bc_r where Br:= {x ∈ X : d(x, Bc) > r}.

Let’s consider G1, G2 ∈ R2(f ); by definition it is possible to find two Lipschitz and

bounded sequence {f_n1}_n, {f_n2}_n which converge to f in L2(x, m) and such that: Lip_a(f_ni, ·) → Gi in L2(X, m), for i = 1, 2.

Now define the sequence:

fn:= Φrfn1+ (1 − Φr)fn2

and note that fn coincides with fn1 on B2r and with fn2 on Brc. This gives us:

Lip_a(fn, ·)|B2r = Lipa(fn1, ·)|B2r , Lipa(fn, ·)|_B¯c

r = Lipa(f

2

n, ·)|B¯c

r. (2.2.7)

Using now the properties inProposition 2.1.9, one gets: Lip_a(fn, ·) ≤ Lipa(fn2, ·) + |fn1− fn2| Lipa(Φr, ·) + Φr

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By construction, f_ni → f in L2_{(X, m) so that (}_2.2.8_{) with (}_2.2.7_{) imply, up to subsequence:}

Lip_a(fn, ·)

L2_(X,m)

−−−−−* ¯G ≤1B2rG1+1B¯c

rG2+1BrrB2r(G1+ 2G2)

which means that by definition of relaxed slope 1B2rG1+1B¯c

rG2+1BrrB2r(G1+ 2G2) ∈ R

2_{(f ).}

Since B is open, we get:

Br → B and ¯Bcr→ Bc.

Passing to the limit as r → 0 and using dominated convergence theorem it follows: 1B2rG1+1B¯c

rG2+1BrrB2r(G1+ 2G2) →1BG1+1B

cG₂, in L2(X, m)

and since R2(f ) is closed, we find that:

1BG1+1BcG₂∈ R2(f ).

We have then proved (2.2.6).

Let’s now prove the second part of the statement: since R2(f ) is a closed and convex subset of L2(X, m), it turns out to be weakly closed.

Therefore weak lower semicontinuity and coercivity of the L2(X, m) norm imply the existence of a minimal element in the weakly closed subset R2(f ) ⊂ L2(X.m).

Note that the pointwise minimality condition follows directly by absurd: suppose it is not true, that is we can find a G ∈ R2(f ) such that:

G < |Df |∗ on a set of m − positive measure.

But then we can define:

H := G ∧ |Df |∗

and by the first part of the statement, it follows H ∈ R2(f ). This gives a contraddiction to the minimality of |Df |∗.

Definition 2.2.5. |Df |∗ is called Minimal Relaxed Slope of f .

An easy consequence of Lemma 2.2.4 is the following relation between the minimal relaxed slope of a Lipschitz function and its asymptotic Lipschitz constant.

Corollary 2.2.6. Let f : X → R be a Lipschitz and bounded function; then we have:

|Df |∗ ≤ Lipa(f, ·) m − a.e. in X. (2.2.9) Proof. It follows immediately from the fact that for a Lipschitz and bounded function f (·), its asymptotic Lipschitz constant is a relaxed slope for f (·). Indeed one can take the

constant approximation

fn:= f, for any n ∈ N

in the definition of relaxed slope.

We have introduced the theory of relaxed slopes because it allows us to characterize the Cheeger energy of a metric measure space in a simpler way.

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Theorem 2.2.7. Let f ∈ L2(X, m) and suppose that R2(f ) 6= ∅, which means that f (·)

has at least one relaxed slope. Then, there exists a sequence {fn}n∈N ⊂ Lipb(X, d) such that:

fn L2

−→ f and Lip_a(f_n, ·)−→ |Df |L2 ∗. (2.2.10)

In particular, we have the following representation formula for the Cheeger energy: Ch(f ) =

Z

X

|Df |2_∗_{dm, ∀f ∈ V.} (2.2.11)

Proof. Note that it is a nontrivial result; we need to prove that is possible to improve the

weak convergence of the asymptotic Lipschitz constant in L2(X, m) inDefinition 2.2.2 so as to obtain a strong L2(X, m) convergence.

Let’s suppose to have R2(f ) 6= ∅ that means we are able to find a sequence (fn)n∈N⊂

Lip_b(X, d) such that:

fn→ f & Lipa(fn, ·) * |Df |∗ in L2(X, m).

Applying Mazur’s lemma (Lemma 6.2.1), it is possible to find another sequence (¯gn)n∈N

such that: ¯ gn∈ co +∞ [ i=n {Lip_a(f_n, ·)} ! , ¯gn→ |Df |∗ in L2(X, m). Suppose that ¯ gn(x) = Nn X i=n λn_i Lip_a(fn, x), x ∈ X

for some Nn> n, Nn∈ N and (λni)i,n such that: Nn

X

i=n

λn_i = 1.

Define ¯fn as the same convex linear combination of the sequence fn, namely:

¯ fn(x) := Nn X i=n λn_ifn(x), x ∈ X.

Then it is still true that ¯fn→ f in L2(X, m).

Note that, as a result of properties (2.1.9) we get: lim sup

n→+∞

Z

X

Lip2_a( ¯fn, x) dm(x) ≤ lim sup n→+∞ Z X ¯ g2_ndm(x) = Z X |Df |2 ∗dm. (2.2.12)

Equation (2.2.12) tells us that any limit point in the weak topology of {

Lipa( ¯fn, ·)}n is a minimal relaxed slope of f and then byLemma 2.1.13 coincides almost

everywhere with |Df |∗.

Moreover, if a subsequence of {Lip_a( ¯fn, ·)}n weakly converges in L2(X, m) to |Df |∗,

then by lower semicontinuity of the norm: lim inf n→+∞ Z X Lip2_a( ¯fn, x) dm(x) ≥ Z X |Df |2_∗dm (2.2.13) and then from (2.2.12):

lim n→+∞ Z X Lip2_a( ¯fn, x) dm(x) = Z X |Df |2_∗dm.

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Since L2(X, m) is an Hilbert space, we have in fact that Lip_a( ¯fn, x) → |Df |∗ in L2(X, m)

that is exactly the sought strong convergence.

Now we want to prove that (2.2.11) holds. Without loss of generality, we can suppose

f ∈ D(Ch); as a consequence of what we have just proved, it is possible to find a strong L2(X, m)−approximation {f_n}_n of f such that Lip_a(f_n, ·) → |Df |∗ in L2(X, m). Then by

definition of Cheeger: Ch(f ) ≤ lim inf n→+∞ Z X Lip2_a(fn, x) dm(x) = Z X |Df |2_∗dm (2.2.14) and this provides the upper bound on the Cheeger energy.

On the other hand, take fn→ f in L2(X, m) such that:

lim inf

n→+∞

Z

X

Lip2_a(f_n, x) dm(x) < ∞.

By reflexivity (Theorem 6.2.3), we can find a subsequence of {Lip_a(f_n, ·)}n which converges

weakly in L2(X, m).

Hence, using definition of minimal relaxed slope, we obtain: lim inf n→+∞ Z X Lip2_a(fn, x) dm(x) ≥ Z X |Df |2_∗dm. (2.2.15) Since {fn}n is generic, we end up with:

Ch(f ) ≥

Z

X

|Df |2_∗dm and we are done.

Remark 2.2.8. Note that the strong L2(X, m)− approximation we have found in the first part of the proof (2.2.10) gives us a "recovery sequence" in the relaxation process in the definition of Cheeger energy (that is the a sequence such that (2.2.14) holds). On the other hand, with equation (2.2.15) we proved the "liminf inequality", as a consequence of the minimality of |Df |∗.

Remark 2.2.9. We want to point out that the theory can be done in a very similar way

changing inDefinition 2.2.2 the asymptotic Lipschitz constant with the slope. This means we can speak about relaxed slopes of a function f with respect to |∇f |(·), prove minimality (Lemma 2.2.4) and Cheeger representation (Theorem 2.2.7). When necessary, we will

denote by |Df |∗∗ the minimal relaxed slope of f with respect to |∇f |(·).

2.2.2 Weak upper gradients

Aim of this section is to present the theory of weak upper gradients; we have already seen how to represent the Cheeger energy associated to a metric measure space with the minimal relaxed slope in the previous section. Now we want to present a different notion of "gradient" in this abstract framework, with a more lagrangian approach to the problem.

Let’s start introducing the usual definition of evaluation map, test plans and negligibility in the space of absolutely continuous curves on X. The main references for this part of the work are [1] and [3].

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Definition 2.2.10. Given any t ∈ [0, 1], the evaluation map at time t is the operator defined by:

et: C([0,1], X) → X

γ(·) 7−→ γ(t). (2.2.16)

The previous map turns out to be measurable with respect to the usual borel algebra on the space C([0, 1], X); then it is possible to define the push-forward of a measure on

C([0, 1], X).

Definition 2.2.11. Given a, b ∈ R, a ≤ b, we denote by AC2([a, b], X) the set:

AC2([a, b], X) := ( γ ∈ AC([a, b], X) : Z b a | ˙γ|2(t) dt < +∞ ) .

Definition 2.2.12. µ ∈P(X) is said to be concentrated on a subset C ⊂ X if µ(X\C) = 0. Definition 2.2.13. A probability measure µ ∈PC([0, 1], X) is said to be a (m) test

plan if µ is concentrated on AC2_{([0, 1], X) and there exists a c = c(µ) ∈ R}+ such that: (e_t)_#µ ≤ c(µ)m ∀t ∈ [0, 1]. (2.2.17) Note that by definition of push-forward we have:

(e_t)_#µ(A) = µnγ(·) : γ(t) ∈ Ao. (2.2.18) The idea is that a test plan with respect to a measure m is something that doesn’t see curves passing, for any fixed time t, in a m−negligible set.

Once we have the definition of test plan, it is possible to introduce a notion of negligibility for a subset of AC2([0, 1], X) that is strongly dependent to the reference measure m we have on the space X.

Definition 2.2.14. A set F ⊂ AC2([0, 1], X) is said to be negligible (with respect to the reference measure m on X) if:

µ(F ) = 0 ∀µ (m) test plan . (2.2.19) If a property holds for any γ ∈ AC2([0, 1], X) except for a negligible set F , we will say that it holds for a.e. curve in AC2([0, 1], X) (or simply for a.e. curve)

From this point onward, we will omit the reference measure m speaking about test plans and negligible sets in AC2([0, 1], X).

Remark 2.2.15. Definition 2.2.14is quite natural, under several points of view: for example, given f, g : X → R such that f = g m−a.e. in X, we have:

I ⊂ [0, 1] countable −→ f ◦ γ|I = g ◦ γ|I for a.e. curve γ ∈ AC2([0, 1], X)

as an easy consequence ofDefinition 2.2.14 and subadditivity of the measures µ.

In fact, it is possible to obtain something stronger; for any test plans µ, Fubini’s theorem applied to the product measure µ ⊗L1 on the product space X × [0, 1] yields:

Z C([0,1],X) Z 1 0 1{t : f ◦γ(t)=g◦γ(t)} (t) dt dµ(γ) = Z 1 0 Z C([0,1],X)1{γ : f ◦γ(t)=g◦γ(t)} (γ) dµ(γ) dt = Z 1 0 µnγ(·) : 1{t : f ◦γ(t)=g◦γ(t)} o dt = Z 1 0 (et)#µ({f = g}) dt = 0.

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This gives us a stronger condition, namely:

f ◦ γ = g ◦ γ L1− a.e. in [0, 1], for µ−a.e. curve γ. (2.2.20) We are now ready to introduce the notion of weak upper gradient of a measurable function f : X → R.

Definition 2.2.16. Let f : X → R be a measurable function on X. A function g : X →

[0, +∞) is said to be a weak upper gradient of f if it holds: |f (γ(1)) − f (γ(0))| ≤

Z

γ

g < +∞, for a.e. γ ∈ AC2([0, 1], X) (2.2.21) where the integration along a curve γ(·) of a scalar function g(·) is defined by:

Z γ g := Z 1 0 g(γ(t))| ˙γ|(t) dt.

The notion of test plans brings our attention to the curves instead of working pointwise on the function; then it is natural to use such an approach so as to give a "lagrangian definition" of Sobolev spaces on X.

Definition 2.2.17. A function f : X → R is said to be Sobolev along almost every curve if

for a.e. curve γ we have that f ◦ γ coincides a.e. in [0, 1] and in {0, 1} with an absolutely continuous function f_γ_{: [0, 1] → R.}

Note that by Remark 2.2.15 applied to I = {0, 1}, the previous definition does not depend on the m−representative of the function f .

Let’s recall a basic lemma of functional analysis and Sobolev spaces which will be useful in the following part of this section.

Lemma 2.2.18. Let f : (0, 1) → R a Lebesgue measurable function, q ∈ [1, +∞]; suppose there exists a nonnegative function g ∈ Lq([0, 1]) such that:

|f (s) − f (t)| ≤ Z t

s

g(r) dr, for L2− a.e. (s, t) ∈ (0, 1)2. (2.2.22)

Then f ∈ W1,q_{([0, 1]) and | ˙}_{f | ≤ g a.e. in (0, 1).}

Proof. If q = +∞ then f (·) is Lipschitz and thesis follows directly.

Suppose q < +∞ and take a function φ ∈ C_c∞([0, 1]); note that by our hypothesis

f ∈ L∞([0, 1]). Then using dominated convergence theorem, it follows: Z 1 0 f (x) ˙φ(x) dx = Z 1 0 f (x) lim h→0 φ(x + h) − φ(x) h dx = lim h→0 Z 1 0 f (x)φ(x + h) − φ(x) h dx = lim h→0 Z 1 0 f (x − h) − f (x) h φ(x) dx (2.2.23)

where the last equality is obtained using the fact that φ(·) is compactly supported 1.

1

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2.2. CHEEGER ENERGY OF A METRIC MEASURE SPACE 27 Since (2.2.22) holds, Jensen’s inequality yields for h < 1:

Z 1 0 |f (x − h) − f (x)|q |h|q dx ≤ Z 1 0 1 h Z x x−h g(s) dsqdx ≤ Z 1 0 1 h Z x x−h gq(s) ds dx ≤ C||g||q_Lq_(0,1) (2.2.24)

where in the last inequality we have used that L1−almost every x ∈ (0, 1) is a Lebesgue point for the summable function gq (Theorem 6.3.2).

Eventually, using Holder’s inequality we can deduce from (2.2.23) and (2.2.24): Z 1 0 f (x) ˙φ(x) dx ≤ C||g||Lq(0,1)||φ||Lq(0,1).

Riesz representation theorem and usual characterization of the dual of Lp([0, 1]) conclude the proof.

As a consequence of the previous result is then possible to prove a Sobolev regularity for functions which admit a weak upper gradient.

Lemma 2.2.19. Let f : X → R be a measurable function such that there exists a weak upper gradient g : X → R+ for f . Then f is Sobolev along a.e. curve.

Proof. Take f : X → R and suppose that g is a weak upper gradient of f ; then it easy to

see that for t < s in [0, 1]: |f (γ(t) − f (γ(s)| ≤

Z s

t

g(γ(r))| ˙γ|(r) dr, for a.e. γ ∈ AC2([0, 1], X).

Let µ be a test plan; by applying Fubini’s theorem to the product measureL2⊗ µ in the product space (0, 1)2× C([0, 1], X) we obtain that for µ−a.e. γ :

|f (γ(s)) − f (γ(t))| ≤ Z s t g(γ(r))| ˙γ|(r) dr forL2−a.e. (t, s) ∈ (0, 1)2_.

By definition of test plan, we have that:

(g ◦ γ) · | ˙γ| ∈ L1([0, 1]) for µ−a.e. γ;

whereas applying Lemma 2.2.18it follows that f ◦ γ ∈ W1,1([0, 1]) for µ−a.e. γ and d dt(f ◦ γ)(t)

≤ (g ◦ γ) · | ˙γ| a.e. in (0, 1), for µ−a.e. γ. (2.2.26) Since µ is an arbitrary test plan, we have proved that f ◦ γ ∈ W1,1([0, 1]) for a.e. γ.

In conclusion, f ◦ γ admits an absolutely continuous representative fγ which coincides

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In the previous section, we have provedLemma 2.2.4 and then we have used it so as to find an integral representation of the Cheeger energy as inTheorem 2.2.7.

The following lemmas are made in the same spirit in the framework of Sobolev-along-a.e.-curve functions. The first one is a direct consequence of (2.2.26).

Lemma 2.2.20 (Locality). Let f : X → R be Sobolev along a.e. curve, let g, h be weak upper gradients of f . Then min{g, h} is a weak upper gradient of f .

The second one concerns the existence of a minimal element inside the class of weak upper gradients of a function.

Lemma 2.2.21 (Minimality). Let f : X → R be Sobolev along a.e. curve; then there exists a unique (up to m−negligible sets) Borel function |Df |_w characterized by the following

property:

|Df |w≤ g m − a.e. in X (2.2.27)

for every weak upper gradient g of f .

Proof. We make this proof only in the case of a finite measure m.

Uniqueness directly follows from Lemma 2.2.20. In order to prove existence, consider the variational problem:

inf Z

X

arctan(g) dm : g is a weak upper gradient of f

. (2.2.28)

Consider a minimizing sequence {g_n}_n in (2.2.28) and set: |Df |w:= inf

n∈Ngn.

Note that as a result of monotonicity of arctan(·) and Lemma 2.2.20, without loss of generality we can assume that g_n+1≤ g_nm−a.e. in X. Hence, using monotone convergence

we find that |Df |w is a weak upper gradient for f . Again, Lemma 2.2.20 and strict

monotonicity of arctan(·) yield minimality of |Df |_w in (2.2.28) and then (2.2.27).

Definition 2.2.22. |Df |w is called minimal weak upper gradient of f .

One key property of weak upper gradients is given by the following theorem ([3], Theorem 4.19).

Theorem 2.2.23 (m−pointwise stability). Suppose to have a sequence of measurable functions f_n _{: X → R and assume that g}_n _{: X → R}+ are weak upper gradients of f_n. Suppose to have:

fn(x) → f (x) for m−a.e. x ∈ X, gn* g in L2(X, m)

(2.2.29)

for a g : X → R+ in L2_{(X, m) and a measurable function f : X → R; then g is a weak} upper gradient for f .

Corollary 2.2.24. Let (X, d, m) be a metric measure space and f ∈ D(Ch1); then f is

Sobolev along a.e. curve and it holds:

|Df |_w≤ |Df |∗∗≤ |Df |∗ m − a.e. in X (2.2.30)

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2.3. HOPF-LAX SEMIGROUP 29

Proof. The proof is direct consequence ofTheorem 2.2.23; indeed for f ∈ D(Ch1) just take a sequence {fn}n⊂ Lipb(X, d) converging to f in L2(X, m) such that |Dfn| → |Df |∗∗ in

L2(X, m) (as in Theorem 2.2.7but using |Df |∗∗ instead of |Df |∗, seeRemark 2.2.9) and

recall that the slope is always a weak upper gradient for Lipschitz functions (it is even a strong one, in the sense that (2.2.21) in Definition 2.2.16holds for any curve).

Remark 2.2.25. In fact, it is possible to prove that:

|Df |w= |Df |∗∗= |Df |∗ m − a.e. in X;

in conclusion, we have that D(Ch1) = D(Ch2) and

Ch1(f ) = Ch2(f ) = Z X |Df |2 wdm = Z X |Df |2 ∗dm, ∀f ∈ D(Ch1).

2.3 Hopf-Lax semigroup

In this section, we will introduce and study the Hopf-Lax semigroup: it is widely used in different situations, from metric measure spaces theory to optimal transportation.

For what concerns this work, it’s going to be really useful in the proof of equivalence theorem inchapter 4 (Theorem 4.0.3). For a reference see, for example [5] or [16].

Let (X, d) be a metric space and t ∈ R+.

Definition 2.3.1. The Hopf-Lax semigroup is the family of operators (Q_t(·))_t>0 given by:

Q_t: C_b(X) → C_b(X) f 7→ Q_tf (x) := inf y∈X f (y) + 1 2td 2_{(x, y)}_. (2.3.1)

Given a function f : X → R, Qtf (x) is then defined by the minimum problem (2.3.1).

We also define: D+f (x, t) := sup{lim sup n→+∞ d(x, yn) : (yn)n is a minimizing sequence in (2.3.1) }, D−f (x, t) := inf{lim inf n→+∞d(x, yn) : (yn)n is a minimizing sequence in (2.3.1) }. (2.3.2)

Remark 2.3.2. D+f and D−f are respectively upper and lower semicontinuous by

construc-tion; moreover they are both non decreasing functions such that: D+f (x, t) ≥ D−f (x, t), ∀x ∈ X, t ∈ R+.

The next theorem treats the main properties of the Hopf-Lax semigroup. Note that no completeness is needed and there is no reference measure.

Theorem 2.3.3. Let (X, d) a metric space and f : X → R. Then we have the following basic properties for Q_tf (x):

1. inf_y∈Xf ≤ Q_tf ≤ f ≤ sup_y∈Xf < +∞. 2. Q_tf ↑ f pointwise as t ↓ 0.

3. (t, x) 7→ Q_tf (x) is Lipschitz in (δ, +∞) × X, for all δ > 0. 4. 2t Lip(f ) ≥ D+f (x, t) ≥ D−f (x, t), for all x ∈ X, t ∈ R+.

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5. ∀x ∈ X and 0 < t < s, it holds: D+f (x, t) ≤ D−f (x, s).

Proof. (1) It is a trivial consequence of the fact that _2t1d2_{(x, y) ≥ 0 and that we can use}

y = x as a competitor in the infimum of (2.3.1).

(2) Fix x ∈ X and take a sequence t_n↓ 0; consider a quasi-minimizing sequence (y_n)_n for Q_t_nf (x), in the sense that:

Q_t nf (x) + 1 n ≥ f (yn) + 1 2t_nd 2_{(x, y} n), for any n ∈ N. (2.3.3)

Firstly, note that the uniform bound given by (1) yields

yn→ x (2.3.4)

in the metric topology, which means d(x, y_n) → 0. Indeed, from (2.3.3) and (1) it follows:

d2(x, xn) ≤ 2tn Q_t_nf (x) + 1 n− f (yn) ≤ 2tn 2||f ||∞+ 1 n → 0 as n → +∞.

Therefore, since f ∈ C_b(X), it follows that f (y_n) → f (x); this and (1) imply:

f (x) ≥ lim

n→+∞Qtnf (x) ≥n→+∞lim f (yn) = f (x).

Thus we get (2).

(3) We are going to prove that Q_tf (x) is uniformly Lipschitz in (δ, ∞) × X with respect to

a variable (in the sense that the Lipschitz constant can be chosen indipendent from the other variable); this easily implies the lipschitzianity in two variables.

Fix x ∈ X, > 0 and consider s, t ∈ R, with s > t. Let’s take an -quasi minimum ys

for Q_sf (x), in the sense that:

Q_sf (x) + ≥ f (ys) +

1 2sd

2_{(x, y}

s). (2.3.5)

Using ys as a competitor for the inf-problem of Qtf (x), we get:

|Q_tf (x) − Q_sf (x)| − = Q_tf (x) − Q_sf (x) − ≤ ≤ f (y_s) + 1 2td 2_{(x, y} s) − f (ys) − 1 2sd 2_{(x, y} s) = = s − t 2st d 2_{(x, y} s) = |s − t| 2st d 2_{(x, y} s). (2.3.6)

We need to have some control on the distance along the quasi minimizing sequence {y_s}_s; to get it, note that we can confine our "attention" in the inf-problem of Q_sf (x) to a subset

of X (dependent on s, x); precisely, we can suppose to work with y inside the set: n

y ∈ X : 1

2sd

2_{(x, y) ≤ sup f − inf f =: Osc(f )}o

. (2.3.7)

Indeed, if we take y such that does not satisfy (2.3.7), we would have:

f (y) + 1

2sd

2_{(x, y) > inf f + sup f − inf f = sup f ≥ f (y) ≥ Q}

sf (x)

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2.3. HOPF-LAX SEMIGROUP 31 Hence, without loss of generality, we can suppose:

ys ∈ B√_{2sOsc(f )}(x). (2.3.8)

From this condition and (2.3.6), it follows: |Q_tf (x) − Q_sf (x)| ≤ 1

2tOsc(f )|s − t| + ≤ 1

2δOsc(f )|s − t| + , ∀ > 0 which gives us the sought uniformly lipschitzianity with respect to t.

On the other hand, fix t ∈ R+, > 0 and take x, z ∈ X. As before, if ys is a

quasi-minimum for Q_tf (z) in the sense of (2.3.5), we obtain: Q_tf (x) − Q_tf (z) − ≤ f (ys) + 1 2td 2_{(x, y} s) − f (ys) − 1 2td 2_{(z, y} s) = 1 2t(d(x, ys) + d(z, ys)) (d(x, ys) − d(z, ys)) ≤ 1 2t(d(x, ys) + d(z, ys)) d(x, z) ≤ 1 2t(d(x, z) + 2d(z, ys)) d(x, z) ≤ 1 t q 2tOsc(f )d(x, z) + 1 2td 2_{(x, z)} ≤ √1 δ q 2Osc(f )d(x, z) + 1 2δd 2_{(x, z)}

where in the last inequalities we have used twice the triangular inequality and the fact that we can suppose to work under the additional assumption (2.3.7).

Since is arbitrary, it follows: |Q_tf (x) − Q_tf (z)| ≤ √1 δ q 2Osc(f )d(x, z) + 1 2δd 2_{(x, z),} ∀x, z ∈ X, t ∈ R+ and then |Q_tf (x) − Q_tf (z)| d(x, z) ≤ 1 √ δ q 2Osc(f ) + 1 2δd(x, z), ∀x, z ∈ X, t ∈ R +_.

To conclude it is sufficient to note that this estimate and the fact that Q_tf (x) is uniformly

bounded from (1) imply a uniformly lipschitzianity with respect to x.

(4) Without loss of generality, we can suppose that Q_tf (x) < f (x); if not, it must be

Q_tf (x) = f (x) and then D+f (x, t) = 0.

Take a minimizing sequence (yn)n for Qtf (x) , so that definitively: f (yn) + 1 2td 2_{(x, y} n) ≤ f (x). (2.3.9) It follows that: d2(x, yn) ≤ 2t · (f (x) − f (yn)) ≤ 2t Lip(f )d(x, yn).

Finally, dividing by d(x, y_n) and taking the limsup in n, we find the thesis.

(5) Fix 0 < t < s and x ∈ X ; let’s make this proof under the additional condition that the infimum in Q_tf (x) and in Q_sf (x) are both attained and so they are minima (if not one

should arrange a bit the proof but it is mainly the same idea).

Hence take yt, ys minima related respectively to Qtf (x) and Qsf (x). By definition of

the Hopf Lax semigroup, we get:

f (yt) + 1 2td 2_(y t, x) ≤ f (ys) + 1 2td 2_(y s, x)

Metric Measure Spaces and Upper Regularity of Dirichlet Forms.

Università degli Studi di Pisa

Corso di Laurea Magistrale in Matematica

Tesi di Laurea Magistrale

Metric Measure Spaces and Upper

Regularity of Dirichlet Forms

22 September 2017

CANDIDATO

Lorenzo Portinale

RELATORE

Prof. Luigi Ambrosio

CONTRORELATORE

Prof. Dario Trevisan

Contents

Notations

Chapter 1

Introduction

Chapter 2

Metric Measure Spaces

2.1

"Derivatives" in metric setting

2.2

Cheeger energy of a metric measure space

2.3

Hopf-Lax semigroup