On the length of chains in a metric space

Abstract

We obtain an upper bound on the minimal number of points in an ε-chain joining two points in a metric space. This generalizes a bound due to Hambly and Kumagai (1999) for the case of resistance metric on certain self-similar fractals. As an application, we deduce a condition on ε-chains introduced by Grigor'yan and Telcs (2012). This allows us to obtain sharp bounds on the heat kernel for spaces satisfying the parabolic Harnack inequality without assuming further conditions on the metric. A snowflake transform on the Euclidean space shows that our bound is sharp.

Introduction

The fundamental solution of the heat equation (or heat kernel) on ${\mathbb{R}}^n$ is given by the Gauss Weierstrass kernel

$$p_t(x,y)=\frac{1}{{(4\pi t)}^{n/2}}\exp (-\frac{d{(x,y)}^2}{4t}),\text{for all }x,y\in {\mathbb{R}}^n,t>0\text{.}$$

From a probabilistic viewpoint, the heat kernel can be interpreted as the transition probability density of the diffusion generated by the Laplacian Δ. More generally, for any uniformly elliptic, divergence form operator $\mathcal{L}u={\sum }_{i,j=0}^n\frac{\partial }{\partial x_i}(a_{ij}(x)\frac{\partial u}{\partial x_j})$ on ${\mathbb{R}}^n$, Aronson [1] proved that the heat kernel $p_t(x,y)$ of the corresponding heat equation ${\partial }_{t}u-\mathcal{L}u=0$ satisfies

$$\frac{c_1}{V(x,t^{1/2})}\exp (-\frac{d{(x,y)}^2}{c_{1}t})\le p_t(x,y)\le \frac{C_1}{V(x,t^{1/2})}\exp (-\frac{d{(x,y)}^2}{C_{1}t}),$$

for all $x,y\in {\mathbb{R}}^n,t>0$, where $c_1,C_1\in (0,\infty )$. Here $V(x,r)$ denotes the Lebesgue measure of the Euclidean ball $B(x,r)$ centered at x with radius r.

To prove the above lower bound on $p_t(x,y)$, the first step is to obtain the lower bound under the additional restriction that $d(x,y)\le C{t}^{1/2}$. Such an estimate is called a near diagonal lower bound. In order to obtain the full lower bound from a near diagonal lower bound, one chooses a sequence of points (called a chain) ${\left\{x_i\right\}}_{i=0}^n$ such that $x_0=x,x_n=y$ and $n\in \mathbb{N}$ such that $d(x_i,x_{i+1})\le C{(t/n)}^{1/2}/2$ for all $i=0,1,\dots ,n-1$. Then we use Chapman-Kolmogorov equation to obtain the estimate

$$p_t(x,y)\ge \underset{B(x_{n-1},C{(t/n)}^{1/2}/2)}{\int }\dots \underset{B(x_1,C{(t/n)}^{1/2}/2)}{\int }{\mathrm{\Pi }}_{i=0}^{n-1}{p}_{t/n}(y_i,y_{i+1})d{y}_1\dots d{y}_{n-1},$$

where $y_0=x,y_n=y$. By optimizing over n and the sequence ${\left\{x_i\right\}}_{i=0}^n$ and using the near diagonal lower bound, we obtain the full lower bound on the heat kernel $p_t(x,y)$. This method of obtaining full heat kernel lower bound is called the chaining argument.

The use of chaining argument to obtain heat kernel estimates is classical [2], [1]. Such chaining arguments are also used to obtain heat kernel estimates on fractals; see [3] for a general introduction to diffusions on fractals. In this work, we address the natural converse question: Do heat kernel estimates imply the existence of short chains? Our main result provides an upper bound on the length of chains, which in some sense is the best possible. The goal of this work is to obtain sharp quantitative bounds on the connectivity of a metric space. This will be expressed as bounds on the length of chains.

We recall the definition of a chain in a metric space $(X,d)$.

Definition 1.1

We say that a sequence ${\left\{x_i\right\}}_{i=0}^N$ of points in X is an ε-chain between points $x,y\in X$ if

$$x_0=x,x_N=y,\text{ and }d(x_i,x_{i+1})<\epsilon \text{ for all }i=0,1,\dots ,N-1\text{.}$$

For any $\epsilon >0$ and $x,y\in X$, define

$$d_{\epsilon }(x,y)=\underset{\left\{x_i\right\}\text{ is}\mathit{\text{ε}}\text{-chain}}{\inf }\sum _{i=0}^{N-1}d(x_i,x_{i+1}),$$

where the infimum is taken over all ε-chains ${\left\{x_i\right\}}_{i=0}^N$ between $x,y$ with arbitrary N.

Note that if $(X,d)$ is a geodesic space, then $d_{\epsilon }(x,y)=d(x,y)$ for all $\epsilon >0,x,y\in X$.

Throughout this paper, we consider a complete, locally compact separable metric space $(X,d)$, equipped with a Radon measure m with full support, i.e., a Borel measure m on X which is finite on any compact set and strictly positive on any non-empty open set. Such a triple $(X,d,m)$ is referred to as a metric measure space. Then we set $\mathrm{diam}(X,d):={\sup }_{x,y\in X}d(x,y)$ and $B(x,r):=\{y\in X|d(x,y)<r\}$ for $x\in X$ and $r>0$.

Let $(\mathcal{E},\mathcal{F})$ be a symmetric Dirichlet form on $L^2(X,m)$. In other words, the domain $\mathcal{F}$ is a dense linear subspace of $L^2(X,m)$, such that $\mathcal{E}:\mathcal{F}\times \mathcal{F}\to \mathbb{R}$ is a non-negative definite symmetric bilinear form which is closed ($\mathcal{F}$ is a Hilbert space under the inner product ${\mathcal{E}}_1(\cdot ,\cdot ):=\mathcal{E}(\cdot ,\cdot )+{〈\cdot ,\cdot 〉}_{L^2(X,m)}$) and Markovian (the unit contraction operates on $\mathcal{F}$; $(u\vee 0)\wedge 1\in \mathcal{F}$ and $\mathcal{E}((u\vee 0)\wedge 1,(u\vee 0)\wedge 1)\le \mathcal{E}(u,u)$ for any $u\in \mathcal{F}$). Recall that $(\mathcal{E},\mathcal{F})$ is called regular if $\mathcal{F}\cap {\mathcal{C}}_{\mathrm{c}}(X)$ is dense both in $(\mathcal{F},{\mathcal{E}}_1)$ and in $({\mathcal{C}}_{\mathrm{c}}(X),{\Vert \cdot \Vert }_{\sup })$. Here ${\mathcal{C}}_{\mathrm{c}}(X)$ is the space of $\mathbb{R}$-valued continuous functions on X with compact support.

For a function $u\in \mathcal{F}$, let ${\mathrm{supp}}_m[u]$ denote the support of the measure $|u|dm$, i.e., the smallest closed subset F of X with ${\int }_{X\setminus F}|u|dm=0$; note that ${\mathrm{supp}}_m[u]$ coincides with the closure of $X\setminus u^{-1}(\{0\})$ in X if u is continuous. Recall that $(\mathcal{E},\mathcal{F})$ is called strongly local if $\mathcal{E}(u,v)=0$ for any $u,v\in \mathcal{F}$ with ${\mathrm{supp}}_m[u]$, ${\mathrm{supp}}_m[v]$ compact and v is constant m-almost everywhere in a neighborhood of ${\mathrm{supp}}_m[u]$. The pair $(X,d,m,\mathcal{E},\mathcal{F})$ of a metric measure space $(X,d,m)$ and a strongly local, regular symmetric Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^2(X,m)$ is termed a metric measure Dirichlet space, or an MMD space. We refer to [8], [7] for a comprehensive account of the theory of symmetric Dirichlet forms.

We recall the definition of energy measures associated to an MMD space. Note that $fg\in \mathcal{F}$ for any $f,g\in \mathcal{F}\cap L^{\infty }(X,m)$ by [8, Theorem 1.4.2-(ii)] and that ${\{(-n)\vee (f\wedge n)\}}_{n=1}^{\infty }\subset \mathcal{F}$ and ${\lim }_{n\to \infty }(-n)\vee (f\wedge n)=f$ in norm in $(\mathcal{F},{\mathcal{E}}_1)$ by [8, Theorem 1.4.2-(iii)].

Definition 1.2 cf. [8, (3.2.14) and (3.2.15)]

Let $(X,d,m,\mathcal{E},\mathcal{F})$ be an MMD space. The energy measure $\mathrm{\Gamma }(f,f)$ of $f\in \mathcal{F}$ is defined, first for $f\in \mathcal{F}\cap L^{\infty }(X,m)$ as the unique ($[0,\infty ]$-valued) Borel measure on X with the property that

$$\underset{X}{\int }gd\mathrm{\Gamma }(f,f)=\mathcal{E}(f,fg)-\frac{1}{2}\mathcal{E}(f^2,g)\text{ for all }g\in \mathcal{F}\cap {\mathcal{C}}_{\mathrm{c}}(X)\text{,}$$

and then by $\mathrm{\Gamma }(f,f)(A):={\lim }_{n\to \infty }\mathrm{\Gamma }((-n)\vee (f\wedge n),(-n)\vee (f\wedge n))(A)$ for each Borel subset A of K for general $f\in \mathcal{F}$.

The notion of energy measure can be extended to the local Dirichlet space ${\mathcal{F}}_{\mathrm{loc}}$, which is defined as

For any $f\in {\mathcal{F}}_{\mathrm{loc}}$ and for any relatively compact open set $V\subset X$, we define

$$\mathrm{\Gamma }(f,f)(V)=\mathrm{\Gamma }(f^{\mathrm{\#}},f^{\mathrm{\#}})(V),$$

where $f^{\mathrm{\#}}$ is as in the definition of ${\mathcal{F}}_{\mathrm{loc}}$. Since $(\mathcal{E},\mathcal{F})$ is strongly local, the value of $\mathrm{\Gamma }(f^{\mathrm{\#}},f^{\mathrm{\#}})(V)$ does not depend on the choice of $f^{\mathrm{\#}}$, and is therefore well defined. Since X is locally compact, this defines a Radon measure $\mathrm{\Gamma }(f,f)$ on X.

Definition 1.3 Capacity between sets

For subsets $A,B\subset X$, we define

$$\mathcal{F}(A,B):=\{f\in \mathcal{F}:f\equiv 1\text{ on a neighborhood of }A\text{ and }f\equiv 0\text{ on a neighborhood of }B\},$$

and the capacity $\mathrm{Cap}(A,B)$ as

$$\mathrm{Cap}(A,B)=\inf \{\mathcal{E}(f,f):f\in \mathcal{F}(A,B)\}.$$

The main result of this work provides upper bounds on $d_{\epsilon }(x,y)$ based on analytic conditions on an MMD space that we introduce now. Henceforth, we fix a continuous increasing bijection $\mathrm{\Psi }:(0,\infty )\to (0,\infty )$ such that for all $0<r\le R$,

$$C^{-1}{\left(\frac{R}{r}\right)}^{{\beta }_1}\le \frac{\mathrm{\Psi }(R)}{\mathrm{\Psi }(r)}\le C{\left(\frac{R}{r}\right)}^{{\beta }_2},$$

for some constants $0<{\beta }_1<{\beta }_2$ and $C>1$. Throughout this work, the function Ψ is meant to denote the space time scaling of the process.

Definition 1.4

We recall the following properties that an MMD space $(X,d,m,\mathcal{E},\mathcal{F})$ may satisfy:

We say that $(X,d,m)$ satisfies the volume doubling Property (VD) if there exists $C_D\ge 1$ such that

$$m(B(x,2r))\le C_{D}m(B(x,r)),\text{ for all }x\in X,r>0\text{.}$$

We say that $(X,d,m,\mathcal{E},\mathcal{F})$ satisfies the Poincaré inequality $\mathrm{PI}(\mathrm{\Psi })$, if there exist constants $C,A\ge 1$ such that for all $x\in X$, $r\in (0,\infty )$ and $f\in \mathcal{F}$

$$\underset{B(x,R)}{\int }{(f-\bar{f})}^{2}dm\le C\mathrm{\Psi }(r)\underset{B(x,Ar)}{\int }d\mathrm{\Gamma }(f,f),$$

where $\bar{f}=m{(B(x,r))}^{-1}{\int }_{B(x,r)}fd\mu$. If all balls are relatively compact, then the above inequality can be extended to all $f\in {\mathcal{F}}_{\mathrm{loc}}$.

We say that $(X,d,m,\mathcal{E},\mathcal{F})$ satisfies the capacity estimate $\mathrm{cap}{(\mathrm{\Psi })}_{\le }$ if there exist $C_1,A_1,A_2>1$ such that for all $x\in X$, $0<R<\mathrm{diam}(X,d)/A_2$,

$$\mathrm{Cap}(B(x,R),B{(x,A_{1}R)}^c)\le C_1\frac{m(B(x,R))}{\mathrm{\Psi }(R)}.$$

We recall the definition of the heat kernel corresponding to an MMD space.

Definition 1.5

Let $(X,d,m,\mathcal{E},\mathcal{F})$ be an MMD space, and let ${\left\{P_t\right\}}_{t>0}$ denote its associated Markov semigroup. A family ${\left\{p_t\right\}}_{t>0}$ of non-negative Borel measurable functions on $X\times X$ is called the heat kernel of $(X,d,m,\mathcal{E},\mathcal{F})$, if $p_t$ is the integral kernel of the operator $P_t$ for any $t>0$, that is, for any $t>0$ and for any $f\in L^2(X,m)$,

$$P_{t}f(x)=\underset{X}{\int }{p}_t(x,y)f(y)dm(y)\text{for }m\text{-almost every }x\in X\text{.}$$

We remark that not every MMD space $(X,d,m,\mathcal{E},\mathcal{F})$ has a heat kernel. The existence of heat kernel is an issue in general.

Our main result is the following upper bound on $d_{\epsilon }$.

Theorem 1.6

Let $(X,d,m,\mathcal{E},\mathcal{F})$ be an MMD space that satisfies (VD), $\mathrm{PI}(\mathit{\Psi })$, and $\mathrm{cap}{(\mathit{\Psi })}_{\le }$, where Ψ satisfies (1.3). Then there exists $C>1$ such that for all $\epsilon >0$ and for all $x,y\in X$ that satisfy $d(x,y)\ge \epsilon$, we have

$$\frac{d_{\epsilon }{(x,y)}^2}{{\epsilon }^2}\le C\frac{\mathit{\Psi }(d(x,y))}{\mathit{\Psi }(\epsilon )}$$

In particular, for all $x,y\in X$, we have

$$\underset{\epsilon \to 0}{\lim }\mathit{\Psi }(\epsilon )\frac{d_{\epsilon }(x,y)}{\epsilon }=0.$$

Remark 1.7

• If $\mathrm{\Psi }(r)=r^2$, (1.4) implies the chain condition $d_{\epsilon }(x,y)\lesssim d(x,y)$ for all $\epsilon >0,x,y\in X$.

• If $\mathrm{\Psi }(r)=r^{\beta }$, then (1.4) and the triangle inequality $d_{\epsilon }(x,y)\ge d(x,y)$ imply that

  $$\frac{d{(x,y)}^2}{{\epsilon }^2}\le \frac{d_{\epsilon }{(x,y)}^2}{{\epsilon }^2}\le C\frac{d{(x,y)}^{\beta }}{{\epsilon }^{\beta }},$$

  for all $x,y\in X,\epsilon >0$ with $d(x,y)\ge \epsilon$. By letting $\epsilon \to 0$, we give a new proof of the known fact that $\beta \ge 2$ must necessarily hold.

• Let $\mathrm{\Psi }(r)=r^{\beta }$ with $\beta \ge 2$. Consider the Dirichlet form corresponding to the Brownian motion on ${\mathbb{R}}^d$ with Lebesgue measure as the symmetric measure, and the snowflake metric $d(x,y)={\Vert x-y\Vert }_2^{2/\beta }$, where ${\Vert x-y\Vert }_2$ denotes the Euclidean distance (cf. [18, Definition 1.2.8] for the terminology ‘snowflake metric’). In this case, it is easy to obtain

  $$\mathrm{\Psi }(\epsilon )\frac{d_{\epsilon }{(x,y)}^2}{{\epsilon }^2}\asymp \mathrm{\Psi }(d(x,y))\asymp {\Vert x-y\Vert }_2^2$$

  for all $x,y\in {\mathbb{R}}^d,\epsilon >0$ with $\epsilon <d(x,y)$. Hence, the bound (1.4) is sharp for all $\beta \ge 2$.

• Theorem 1.6 provides a new proof to an estimate due to Hambly and Kumagai [11, Lemma 3.3]. Based on the results in [11], the estimate (1.5) was introduced by Grigor'yan and Telcs in [10, (1.8)] to obtain sharp estimates of the heat kernel (cf. Corollary 2.9).

By [10, Theorem 6.5] along with (1.5), we have the following corollary (see Theorem 2.11 for generalization to arbitrary scale functions Ψ).

Corollary 1.8

Let $(X,d,m,\mathcal{E},\mathcal{F})$ be an MMD space that satisfies the following sub-Gaussian estimate on its heat kernel $p_t(\cdot ,\cdot )$: there exists $\beta \ge 2,C,c>0$ such that

$$\frac{c}{V(x,t^{1/\beta })}\exp (-{\left(\frac{d{(x,y)}^{\beta }}{ct}\right)}^{1/(\beta -1)})\le p_t(x,y)\le \frac{C}{V(x,t^{1/\beta })}\exp (-{\left(\frac{d{(x,y)}^{\beta }}{Ct}\right)}^{1/(\beta -1)})$$

for all $x,y\in X,t>0$, where $V(x,r)=m(B(x,r))$. Then the metric d satisfies the chain condition: there exists $K>1$ such that

$$d(x,y)\le d_{\epsilon }(x,y)\le Kd(x,y)\mathit{\text{for all }}\epsilon >0\mathit{\text{ and for all }}x,y\in X.$$

Remark 1.9

• The chain condition (1.7) admits the following characterization. Let $(X,d)$ be metric space such that the open balls $B(x,r)$ are relatively compact for all $x\in X,r>0$. Then $(X,d)$ satisfies the chain condition if and only if there exists a geodesic metric ρ such that d is bi-Lipschitz equivalent to ρ [15, Proposition A.1]. Recall that $(X,\rho )$ is geodesic, if for any two points $x,y\in X$, there exists a function $\gamma :[0,\rho (x,y)]\to X$ such that $\rho (\gamma (s),\gamma (t))=|s-t|$ for all $s,t\in [0,\rho (x,y)]$.

• Corollary 1.8 was previously known only for the case $\beta =2$. By a version of Varadhan's asymptotic formula in [14, Theorem 1.1], we obtain that d is bi-Lipschitz equivalent to the intrinsic metric. Hence by the remark above, d satisfies the chain condition. However, the intrinsic metric vanishes identically for the case $\beta >2$. This suggests the need for a completely different approach when $\beta >2$.

• The chain condition plays an essential role in the proof of singularity of energy measures in [15] for spaces satisfying the sub-Gaussian heat kernel estimate.

Recall that the parameter β in Corollary 2.9 is called the walk dimension. The following result can be viewed as a generalization of the result that the walk dimension is at least two.

Corollary 1.10

Let $(X,d,m,\mathcal{E},\mathcal{F})$ be an MMD space that satisfies (VD), $\mathrm{PI}(\mathit{\Psi })$, and $\mathrm{cap}{(\mathit{\Psi })}_{\le }$, where Ψ satisfies (1.3). Then there exists $C_1\ge 1$ such that

$$\frac{\mathit{\Psi }(r)}{\mathit{\Psi }(s)}\ge C_1^{-1}{\left(\frac{r}{s}\right)}^2,\mathit{\text{for all }}0<s\le r<\mathrm{diam}(X,d)\mathit{\text{.}}$$

Notation. In the following, we will use the notation $A\lesssim B$ for quantities A and B to indicate the existence of an implicit constant $C\ge 1$ depending on some inessential parameters such that $A\le CB$. We write $A\asymp B$, if $A\lesssim B$ and $B\lesssim A$.