Research and Lecture notes

Contents

• Riemannian manifolds
• Carnot groups
• Sierpinski gasket
• Cheeger space

Riemannian manifolds

Let (M,g) be an n-dimensional Riemannian manifold with Riemannian volume measure μ and Riemannian distance d. We consider the quadratic form ℰ on M, which is obtained as the closure in L²(M,μ) of the quadratic form

∫_M ⟨∇f, ∇g⟩ dμ, f,g ∈ C_c^∞(M).

The domain of the closure ℰ is the Sobolev space W₀^1,2(M) and its generator Δ is a self-adjoint extension of the Laplace-Beltrami operator. If the manifold is complete (which is equivalent to the metric space (M,d) being complete) then ℰ is the unique closed extension and W₀^1,2(M) = W^1,2(M). If we assume further that the Ricci curvature of M is bounded from below then the domain of Δ is the Sobolev space W^2,2(M). If we even assume further that the Ricci curvature of M is non-negative, it is a well-known result by Li and Yau that the heat semigroup P_t admits a smooth heat kernel function p_t(x,y) on [0,∞) × M × M for which there are constants c₁, c₂, C > 0 such that whenever t > 0 and x,y ∈ X,

Carnot groups

A Carnot group of step N is a simply connected Lie group G whose Lie algebra can be stratified as follows:

g = 𝒱₁ ⊕ … ⊕ 𝒱_N,

where

[𝒱_i , 𝒱_j] = 𝒱_i+j

and

𝒱_s = 0, for s > N.

From the above properties, Carnot groups are nilpotent. The number

is called the homogeneous dimension of G.

Let V₁,…,V_d be a basis of the vector space 𝒱₁. The vectors V_i‘s can be seen as left invariant vector fields on G. The left invariant sub-Laplacian on G is the operator:

It is hypoelliptic and essentially self-adjoint on the space of smooth and compactly supported function f : G → ℝ with the respect to the Haar measure μ of G. The heat semigroup (P_t)_t≥0 on G, defined through the spectral theorem, is then seen to be a Markov semigroup. By hypoellipticity of L, this heat semigroup admits a heat kernel denoted by p_t(g,g’). It is then known that p_t satisfies the double-sided Gaussian bounds:

for some constants C,c₁,c₂ > 0. Here d(g,g’) denotes the Carnot-Carathéodory distance from g to g’ on G which is defined by

Sierpiński gasket

A large class of examples for which Dirichlet form theory is useful is the class of p.c.f. fractals. For the sake of presentation we illustrate in detail the case of the Sierpiński gasket, which is one of the most popular examples of a p.c.f. fractal.

One of the classical ways to define the Sierpiński gasket is as follows:
let V₀ = {p₁, p₂, p₃} be a set of vertices of an equilateral triangle of side 1 in ℂ. Define

f_i(z) = (z-p_i)/2 + p_i , for i = 1,2,3.

The Sierpiński gasket K is the unique non-empty compact subset in ℂ such that

Sierpinski gasket

The set V₀ is called the boundary of K, we will also denote it by ∂K. The Hausdorff dimension of K with respect to the Euclidean metric (denoted d(x,y) = |x – y| ) is given by d_h = ln(3)/ln(2). A (normalized) Hausdorff measure on K is given by the Borel measure μ on K such that for every i₁, …, i_n ∈ {1,2,3} ,

μ(f_i1 ∘ … ∘ f_in (K)) = 3^-n.

This measure μ is d_h-Ahlfors regular, i.e., there exist constants c,C > 0 such that for every x ∈ K and r ∈ [0, diam(K)],

cr^d_h ≤ μ(B(x,r)) ≤ C r^d_h.

It will be useful to approximate the gasket K by a sequence of discrete objects. Namely, starting from the set V₀ = {p₁, p₂, p₃}, we define a sequence of sets {V_m}_m≥0 inductively by

Then we have a natural sequence of Sierpiński gasket graphs (or pre-gaskets) {G_m}_m≥0 whose edges have length 2^-m and whose set of vertices is V_m. Notice that #V_m = 3(3^m + 1)/2. We will use the notations V_* = ⋃_{m ≥ 0} V_m and V_*⁰ = ⋃_m≥0 V_m \ V₀.

The Dirichlet form on the metric space K is defined by approximation. Let m ≥ 1. For any f ∈ ℝ^V_m, we consider the quadratic form

where means that are neighbors in the graph V_m.

We can then define a pre-Dirichlet form (ℰ, ℱ_*) on V_* by setting

ℱ_* = {f ∈ ℝ^V_* , lim_{m → ∞} ℰ_m(f,f) < ∞}

and for f ∈ ℱ_*

ℰ(f,f) = lim_{m → ∞} ℰ_m(f,f).

Each function f ∈ ℱ_* can be uniquely extended into a continuous function defined on K. We denote by ℱ the set of functions with such extensions. (ℰ, ℱ) is a Dirichlet form on L²(K,μ). The generator of the Dirichlet form (ℰ, ℱ), denoted by Δ, corresponds to the Laplacian with Neumann boundary condition.

In this example we have a continuous heat kernel p_t(x,y) satisfying, for some c₁, c₂, c₃, c₄ ∈ (0,∞) and d_H ≥ 1, d_W ∈ [2,+∞),

for μ × μ-a.e. (x,y) ∈ X × X and each t ∈ (0,+∞). Here, d_W = ln(5)/ln(3) is the so-called walk dimension of the Sierpiński gasket. A standard reference are the lecture notes by M. Barlow.

Cheeger metric measure spaces

Consider a locally compact, complete, metric measure space (X,d,μ) where μ is a Radon measure. Any open metric ball centered at x ∈ X with radius r > 0 will be denoted by

B(x,r) = {y ∈ X, d(x,y) < r}.

Definition. The measure μ is said to be doubling (VD) if there exists a constant C > 0 such that for every x ∈ X, r > 0,

0 < μ(B(x,2r)) ≤ C μ(B(x,r)) < +∞.

The Lipschitz constant of a function f ∈ Lip(X) is defined as

Definition. The metric measure space (X,d,μ) is said to satisfy the 2-Poincaré inequality (P) if for any f ∈ Lip(X) and any ball B(x,R) of radius R > 0,

where

The constants C > 0 and λ ≥ 1 are independent from x, R and f.

Definition. A metric measure space satisfying (VD) and (P) is often called a Cheeger space (or PI space).

One can construct a “nice” Dirichlet form and Laplacian on any Cheeger space by the using the technique of Γ-convergence.

Definition. A sequence of forms {ℰ_n}_{n ≥ 1} is said to Mosco-converge to ℰ if

1. For any sequence {f_n}_n≥1 ⊂ L²(X,μ) that converges *weakly* to f ∈ L²(X,μ) in L²(X,μ),

   liminf_n→∞ ℰ_n(f_n, f_n) ≥ ℰ(f,f).

2. For any f ∈ L²(X,μ) there exists a sequence {f_n}_n≥1 ⊂ L²(X,μ) that converges strongly to f in L²(X,μ) and

   limsup_n→∞ ℰ(f_n, f_n) ≤ ℰ(f,f).

The idea is to consider Korevaar-Schoen type energy functionals defined for any f ∈ L²(X,μ) as

and the associated Korevaar-Schoen space

KS^1,2(X) := {f ∈ L²(X,μ), limsup_{r → 0⁺} E(f,r) < +∞}.

One has then the following result:

Theorem. There exists a Dirichlet form (ℰ, ℱ) on L²(X,μ) such that:

1. ℰ has domain ℱ = KS^1,2(X);
2. ℰ is a Γ-limit of E(f,r_n), where r_n is a positive sequence such that r_n → 0;
3. ℰ has a continuous heat kernel p_t(x,y) that satisfies for t > 0 and x,y ∈ X,