Intrinsic Ultracontractivity for Domains in Negatively Curved Manifolds

Abstract

Let M be a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. A sufficient condition is obtained for open and connected sets D in M for which the corresponding Dirichlet heat semigroup is intrinsically ultracontractive. That condition is formulated in terms of capacitary width. It is shown that both the reciprocal of the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(D)$, and the supremum of the torsion function for D are comparable with the square of the capacitary width for D if the latter is sufficiently small. The technical key ingredients are the volume doubling property, the Poincaré inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel at finite scale.

1 Main Results

Let M be a complete, non-compact, $n$-dimensional connected Riemannian manifold, without boundary, and with Ricci curvature bounded below by a negative constant, i.e., ${{\,\mathrm{Ric}\,}}\ge -K$ with non-negative constant K. Throughout the paper, K is reserved for this constant. In this article, we investigate domains (open, and connected sets) in M for which the heat semigroup is intrinsically ultracontractive.

For a domain $D\subset M$ we denote by $p_D(t,x,y)$, $t>0$, $x,y\in D$, the Dirichlet heat kernel for in $D$, i.e., the fundamental solution to subject to the Dirichlet boundary condition $u(t,x)=0$ for $x\in {\partial D}$ and $t>0$. Davies and Simon [12] introduced the notion of intrinsic ultracontractivity. There are several equivalent definitions for intrinsic ultracontractivity ( [12, p. 345]). The following is in terms of the heat kernel estimate.

Definition 1.1

Let $D\subset M$. We say that the semigroup associated with $p_D(t,x,y)$ is intrinsically ultracontractive (abbreviated to IU) if the following two conditions are satisfied:

• (i) The Dirichlet Laplacian $-\Delta$ has no essential spectrum and has the first eigenvalue $\lambda _D>0$ with corresponding positive eigenfunction $\varphi _D$ normalized by $\Vert \varphi _D\Vert _2=1$.

• (ii) For every $t>0$, there exist constants $0<c_t<C_t$ depending on t such that

  $$\begin{aligned} c_t\varphi _D(x)\varphi _D(y) \le p_D(t,x,y) \le C_t\varphi _D(x)\varphi _D(y) \quad \text{ for } \text{ all } x,y\in D. \end{aligned}$$

  (1.1)

For simplicity, we say that $D$ itself is IU if the semigroup associated with $p_D(t,x,y)$ is IU.

Both the analytic and probabilistic aspects of IU have been investigated in detail. For example it turns out that IU implies the Cranston-McConnell inequality, while IU is derived from very weak regularity of the domain. Davis [13] showed that a bounded Euclidean domain above the graph of an upper semi-continuous function is IU; no regularity of the boundary function is needed. There are many results on IU for Euclidean domains. Bañuelos and Davis [5, Thm. 1, Thm. 2] gave conditions characterizing IU and the Cranston-McConnell inequality when restricting to a certain class of plane domains, which illustrate subtle differences between IU and the Cranston-McConnell inequality. Méndez-Hernández [16] gave further extensions. See also [1, 4, 7, 8, 13], and references therein.

There are relatively few results for domains in a Riemannian manifold. Lierl and Saloff-Coste [15] studied a general framework including Riemannian manifolds. In that paper, they gave a precise heat kernel estimate for a bounded inner uniform domain, which implies IU ([15, Thm. 7.9]). In view of [13], however, the requirement of inner uniformity for IU to hold can be relaxed. See Sect. 7.

Our main result is a sufficient condition for IU for domains in a manifold, which is a generalization of the Euclidean case [1]. Our condition is given in terms of capacity. It is applicable not only to bounded domains but also to unbounded domains. Let $\Omega \subset M$ be an open set. For $E\subset \Omega$ we define relative capacity by

$$\begin{aligned} {{\,\mathrm{Cap}\,}}_\Omega (E) =\inf \left\{ \int _\Omega |\nabla \varphi |^2d\mu : \varphi \ge 1~\text {on} ~E, \varphi \in C_0^\infty (\Omega )\right\} , \end{aligned}$$

where $\mu$ is the Riemannian measure in M and $C_0^\infty (\Omega )$ is the space of all smooth functions compactly supported in $\Omega$. Let d(x, y) be the distance between x and y in M. The open geodesic ball with center x and radius $r>0$ is denoted by $B(x,r)=\{y\in M: d(x,y)<r\}$. The closure of a set E is denoted by $\overline{E}$, and so $\overline{B}(x,r)$ stands for the closed geodesic ball of center x and radius r.

Definition 1.2

Let $0<\eta <1$. For an open set $D$ we define the capacitary width $w_\eta (D)$ by

$$\begin{aligned} w_\eta (D) =\inf \left\{ r>0: \frac{{{\,\mathrm{Cap}\,}}_{B(x,2r)}(\overline{B}(x,r)\setminus D)}{{{\,\mathrm{Cap}\,}}_{B(x,2r)}(\overline{B}(x,r))} \ge \eta \quad \text{ for } \text{ all } x\in D\right\} . \end{aligned}$$

The next theorem asserts that the parameter $\eta$ has no significance.

Theorem 1.3

Let $0<R_0<\infty$. If $0<\eta '<\eta <1$, then

$$\begin{aligned} w_{\eta '}(D)\le w_\eta (D)\le Cw_{\eta '} (D) \quad \text{ for } \text{ all } \text{ open } \text{ sets } D\text { with}~ w_\eta (D)<R_0 \end{aligned}$$

with $C>1$ depending only on $\eta ,\eta '$, $\sqrt{K}\, R_0$ and $n$.

The first condition for IU has a characterization in terms of capacitary width. This is straightforward from Persson’s argument [17], and Theorem 1.6 below. Hereafter we fix $o\in M$.

Theorem 1.4

Let $D$ be a domain in M. Then $D$ has no essential spectrum if and only if $\lim _{R\rightarrow \infty }w_\eta (D\setminus \overline{B}(o,R))=0$.

We shall prove the following sufficient condition for IU, which looks the same as in the Euclidean case [1]. Nevertheless, the proof is significantly different for negatively curved manifolds. See the remark after Theorem A.

Theorem 1.5

Suppose M has positive injectivity radius. Then a domain $D\subset M$ is IU if the following two conditions are satisfied:

• (i) $\lim _{R\rightarrow \infty }w_\eta (D\setminus \overline{B}(o,R))=0$.

• (ii) For some $\tau >0$

  $$\begin{aligned} \int _0^\tau w_\eta (\{x\in D: G_D(x,o)<t\})^2\frac{dt}{t}<\infty , \end{aligned}$$

  (1.2)

  where $G_D$ is the Green function for $D$.

Our results are based on the relationship between the torsion function

$$\begin{aligned} v_D(x)=\int _DG_D(x,y)d\mu (y) \end{aligned}$$

and the bottom of the spectrum

$$\begin{aligned} \lambda _{\text {min}}(D) =\inf \Biggl \{\frac{\Vert \nabla f\Vert _2^2}{\Vert f\Vert _2^2}:f \in C_0^\infty (D)\text { with } \Vert f\Vert _2\ne 0\Biggr \}. \end{aligned}$$

(1.3)

We note that $\lambda _{\text {min}}(D)$ is the first eigenvalue $\lambda _D$ if $D$ has no essential spectrum. This is always the case for a bounded domain $D$. Theorem 1.4 asserts that the same holds even for an unbounded domain $D$ whenever $\lim _{R\rightarrow \infty }w_\eta (D\setminus \overline{B}(o,R))=0$. We also observe that the torsion function is the solution to the de Saint-Venant problem:

$$\begin{aligned} \begin{aligned} -\Delta v_D=1&\quad \text{ in } D,\\ v_D=0&\quad \text{ on } {\partial D}, \end{aligned} \end{aligned}$$

where the boundary condition is taken in the Sobolev sense. The second named author [19] proved the following theorem.

Theorem A

Let $K=0$. If $D\subset M$ satisfies $\lambda _{\mathrm{min}}(D)>0$, then

$$\begin{aligned} \lambda _{\mathrm{min}}(D)^{-1}\le \Vert v_{D}\Vert _\infty \le C\lambda _{\mathrm{min}}(D)^{-1}, \end{aligned}$$

(1.4)

where $C$ depends only on M.

The second inequality of (1.4) does not necessarily hold for negatively curved manifolds. Let ${\mathbb {H}}^n$ be the $n$-dimensional hyperbolic space of constant curvature $-1$. It is known that

$$\begin{aligned} \lambda _{\text {min}}({\mathbb {H}}^n)=\frac{(n-1)^2}{4}, \end{aligned}$$

whereas $v_{{\mathbb {H}}^n}\equiv \infty$ as ${\mathbb {H}}^n$ is stochastically complete. Hence the second inequality of (1.4) fails to hold if $D$ is the whole space ${\mathbb {H}}^n$.

The point of this paper is that (1.4) still holds if $D$ is limited to a certain class. We make use of (1.4) with this limitation to derive Theorems 1.4 and 1.5 . We have the following theorem, which is a key ingredient in their proofs.

Theorem 1.6

Let $K\ge 0$ and let $0<\eta <1$. Then there exist $R_0>0$ and $C>1$ depending only on K, $\eta$ and $n$ such that if $D\subset M$ satisfies $w_\eta (D)<R_0$, then

$$\begin{aligned} \frac{C^{-1}}{w_\eta (D)^2} \le \frac{1}{\Vert v_D\Vert _\infty }\le \lambda _{\mathrm{min}}(D) \le \frac{C}{\Vert v_D\Vert _\infty } \le \frac{C^2}{w_\eta (D)^2}. \end{aligned}$$

(1.5)

Remark 1.7

We actually find $\Lambda _0>0$ depending only on K and $n$ such that (1.4) holds for $D$ with $\lambda _{\text {min}}(D)> \Lambda _0$ (Lemma 3.2 below). This is a generalization of Theorem A as $\Lambda _0=0$ for $K=0$. In practice, however, the condition $w_\eta (D)<R_0$ in Theorem 1.6 is more convenient since the capacitary width $w_\eta (D)$ can be more easily estimated than the bottom of the spectrum $\lambda _{\text {min}}(D)$.

In Sect. 2 we summarize the key technical ingredients of the proofs: the volume doubling property, the Poincaré inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel at finite scale. Observe that these fundamental tools are available not only for manifolds with Ricci curvature bounded below by a negative constant but also for unimodular Lie groups and homogeneous spaces. See [15, Ex. 2.11] and [18, Sect. 5.6]. This observation suggests that our approach is also extendable to those spaces.

We use the following notation. By the symbol $C$ we denote an absolute positive constant whose value is unimportant and may change from one occurrence to the next. If necessary, we use $C_0, C_1, \dots$, to specify them. We say that f and g are comparable and write $f\approx g$ if two positive quantities f and g satisfy $C^{-1}\le f/g\le C$ with some constant $C\ge 1$. The constant $C$ is referred to as the constant of comparison.

2 Preliminaries

We recall that M is a manifold of dimension $n\ge 2$ with ${{\,\mathrm{Ric}\,}}\ge -K$ with $K\ge 0$. Let us recall the volume doubling property of the Riemannian measure $\mu$, the Poincaré inequality and the Gaussian estimate for the Dirichlet heat kernel $p_M(t,x,y)$ for M. For $B=B(x,r)$ and $\tau >0$ we write $\tau B=B(x,\tau r)$.

Theorem 2.1

(Volume doubling at finite scale. [18, Thm. 5.6.4]) Let $0<R_0<\infty$. Then for all $B=B(x,r)$ with $0<r<R_0$

$$\begin{aligned} \mu (2B)\le 2^n\exp \bigl (\sqrt{(n-1)K}\, R_0\bigr )\mu (B). \end{aligned}$$

Theorem 2.2

(Poincaré inequality [18, Thm. 5.6.6]) For each $1\le p<\infty$ there exist positive constants $C_{n,p}$ and $C_n$ such that

$$\begin{aligned} \int _B |f-f_B|^p d\mu \le C_{n,p}r^p\exp (C_n\sqrt{K}\, r)\int _{2B} |\nabla f|^pd\mu \end{aligned}$$

for all $B=B(x,r)$. Here $f_B$ stands for the average of f on B.

Corollary 2.3

(Poincaré inequality at finite scale) Let $0<R_0<\infty$. Then for all $B=B(x,r)$ with $0<r<R_0$

$$\begin{aligned} \int _B |f-f_B|^2 d\mu \le C_{n,2} r^2\exp (C_n\sqrt{K}\, R_0)\int _{2B} |\nabla f|^2d\mu . \end{aligned}$$

Remark 2.4

If the Ricci curvature of M is non-negative, i.e., $K=0$, then the estimates in Theorems 2.1, 2.2 and Corollary 2.3 hold with constants independent of $0<r<\infty$.

The Poincaré inequality yields the Sobolev inequality. We see that if $B=B(x,r)$ with $0<r< R_0$, then

$$\begin{aligned} \Biggl (\frac{1}{\mu (B)}\int _B|f|^{2}d\mu \Biggr )^{1/2} \le C_{n,2} r\,\Biggl (\frac{1}{\mu (B)}\int _B|\nabla f|^{2}d\mu \Biggr )^{1/2} \quad \text{ for } \text{ all } f\in C_0^\infty (B) \end{aligned}$$

with different $C_{n,2}$. See [18, Thm. 5.3.3] for a more general Sobolev inequality. Hence the characterization of the bottom of the spectrum in terms of Rayleigh quotients (1.3) gives the following:

Corollary 2.5

Let $0<R_0<\infty$. Then there exists a constant $C>0$ depending only on $\sqrt{K}\,R_0$ and $n$ such that

$$\begin{aligned} \lambda _\mathrm{min}(B(x,r))\ge Cr^{-2} \quad \text{ for } 0<r< R_0. \end{aligned}$$

The celebrated theorem by Grigor’yan and Saloff-Coste gives the relationship between the Poincaré inequality, the volume doubling property of the Riemannian measure, the Li-Yau Gaussian estimate for the heat kernel, and the parabolic Harnack inequality. Let $V(x,r)=\mu (B(x,r))$.

Theorem B

([18, Thm. 5.5.1, Thm. 5.5.3]) Let $0<R_0\le \infty$. Consider the following conditions:

• (i) (PI) There exists a constant $P_0>0$ such that for all $B=B(x,r)$ with $0<r<R_0$ and all $f\in C^\infty (B)$,

  $$\begin{aligned} \int _B |f-f_B|^2 d\mu \le P_0r^2\int _{2 B} |\nabla f|^2d\mu . \end{aligned}$$

• (ii) (VD) There exists a constant $D_0>0$ such that for all $B=B(x,r)$ with $0<r<R_0$

  $$\begin{aligned} \mu (2B)\le D_0\mu (B). \end{aligned}$$

• (iii) (PHI) There exists a constant $A>0$ such that for all $B=B(x,r)$ with $0<r<R_0$ and all $u>0$ with $(\partial _t-\Delta )u=0$ in $(s-r^2,s)\times B$

  $$\begin{aligned} \sup _{Q_-} u \le A\inf _{Q_+}u, \end{aligned}$$

  where $Q_-=(s-3r^2/4,s-r^2/2)\times B(x,r/2)$ and $Q_+=(s-r^2/4,s)\times B(x,r/2)$.

• (iv) (GE) There exists a finite constant $C>1$ such that for $0<t<R_0^2$ and $x,y\in M$,

  $$\begin{aligned} \frac{1}{CV(x,\sqrt{t})}\exp \left( -\frac{Cd(x,y)^2}{t}\right) \le p_M(t,x,y) \le \frac{C}{V(x,\sqrt{t})}\exp \left( -\frac{d(x,y)^2}{Ct}\right) . \end{aligned}$$

  (2.1)

Then

$$\begin{aligned} (i)+(ii)\iff (iii) \iff (iv). \end{aligned}$$

Theorem 2.1 and Corollary 2.3 assert that (i) and (ii) of Theorem B hold true for $0<R_0<\infty$ with constants depending only on K, $R_0$ and $n$. Hence, the Li-Yau Gaussian estimate of the heat kernel for the whole manifold M and the parabolic Harnack inequality up to scale $R_0$ are available in our setting. Observe that the volume doubling inequality $\mu (B(x,2r))\le D_0\mu (B(x,r))$ implies

$$\begin{aligned} \mu (B(x,r))\ge C\left( \frac{r}{R}\right) ^\alpha \mu (B(x,R)) \quad \text{ for } 0<r<R<R_0 \end{aligned}$$

(2.2)

with $\alpha =\log D_0/\log 2$. We also have the following elliptic Harnack inequality since positive harmonic functions are time-independent positive solutions to the heat equation.

Corollary 2.6

(Elliptic Harnack inequality) Let $0<r_1<r_2<R_0<\infty$. If h is a positive harmonic function in $B(x,r_2)$, then

$$\begin{aligned} C^{-1}\le \frac{h(y)}{h(x)} \le C\quad \text{ for } y\in B(x,r_1) \end{aligned}$$

where $C>1$ depends only on $\sqrt{K}\,R_0$, $r_1/r_2$ and $n$.

3 Torsion Function and the Bottom of Spectrum

In this section we obtain estimates between the bottom of the spectrum and the torsion function $v_D$. We shall prove the second and the third inequalities of (1.5).

Since the Green function $G_D(x,y)$ is the integral of the heat kernel $p_D(t,x,y)$ with respect to $t\in (0,\infty )$, we have

$$\begin{aligned} v_D(x) =\int _0^\infty P_D(t,x) dt, \end{aligned}$$

where

$$\begin{aligned} P_D(t,x)=\int _Dp_D(t,x,y)d\mu (y). \end{aligned}$$

We note that $P_D(t,x)={\mathbb {P}}_x[\tau _D>t]$, i.e., the survival probability that the Brownian motion $(B_t)_{t\ge 0}$ started at x stays in $D$ up to time t, where $\tau _D$ is the first exit time from $D$. We also observe that $P_D(t,x)$ is considered to be the (weak) solution to

Let $\pi _D(t)=\sup _{x\in D} P_D(t,x)$. Let us begin with the proof of the second inequality of (1.5).

Lemma 3.1

If $\lambda _{\text {min}}(D)>0$, then $\lambda _{\text {min}}(D)\,\Vert v_D\Vert _\infty \ge 1$.

Proof

We follow [1, Lem. 3.2, Lem. 3.3]. Without loss of generality we may assume that $\Vert v_D\Vert _\infty <\infty$. It suffices to show the following two estimates:

$$\begin{aligned}&\exp (-\lambda _{\text {min}}(D)\,t)\le \pi _D(t) \quad \text{ for } \text{ all } ~t>0. \end{aligned}$$

(3.1)

$$\begin{aligned}&\text {If}~ C>1\text {, then } \pi _D(t)\le \frac{C}{C-1}\exp \left( -\frac{t}{C\Vert v_D\Vert _\infty }\right) \quad \text{ for } \text{ all }~ t>0. \end{aligned}$$

(3.2)

In fact, we obtain from (3.1) and (3.2) that

$$\begin{aligned} \exp \left( -\lambda _{\text {min}}(D)\,t+\frac{t}{C\Vert v_D\Vert _\infty }\right) \le \frac{C}{C-1}, \end{aligned}$$

which holds for all $t>0$ only if

$$\begin{aligned} \lambda _{\text {min}}(D)\ge \frac{1}{C\Vert v_D\Vert _\infty }. \end{aligned}$$

Since $C>1$ is arbitrary, we have $\lambda _{\text {min}}(D)\,\Vert v_D\Vert _\infty \ge 1$.

Let us prove (3.1). Take $\alpha >\lambda _{\text {min}}(D)$. Then we find $\varphi \in C_0^\infty (D)$ such that $\Vert \nabla \varphi \Vert _2^2\big /\Vert \varphi \Vert _2^2\le \alpha$. Take a bounded domain $\Omega$ such that ${{\,\mathrm{supp}\,}}\varphi \subset \Omega \subset D$. Then $\Omega$ has no essential spectrum. Let $\lambda _\Omega$ and $\varphi _\Omega$ be the first eigenvalue and its positive eigenfunction with $\Vert \varphi _\Omega \Vert _2=1$ for $\Omega$, respectively. By definition

$$\begin{aligned} \lambda _\Omega =\inf \Bigg \{\frac{\Vert \nabla \psi \Vert _2^2}{\Vert \psi \Vert _2^2}:\psi \in C_0^\infty (\Omega )\Bigg \} \le \frac{\Vert \nabla \varphi \Vert _2^2}{\Vert \varphi \Vert _2^2} \le \alpha . \end{aligned}$$

Since $u(t,x)=\exp (-\lambda _\Omega t)\,\varphi _\Omega (x)$ is the solution to the heat equation in $(0,\infty )\times \Omega$ such that $u(0,x)=\varphi _\Omega (x)$ and $u(t,x)=0$ on $(0,\infty )\times \partial \Omega$, it follows from the comparison principle that

$$\begin{aligned} \exp (-\lambda _\Omega t)\,\varphi _\Omega (x) \le \int _{\Omega } p_D(t,x,y)\varphi _\Omega (y)d\mu (y) \le \Vert \varphi _\Omega \Vert _\infty P_D(t,x) \le \Vert \varphi _\Omega \Vert _\infty \pi _D(t) \end{aligned}$$

in $(0,\infty )\times \Omega$. Taking the supremum for $x\in \Omega$, and then dividing by $0<\Vert \varphi _\Omega \Vert _\infty <\infty$, we obtain

$$\begin{aligned} \exp (-\alpha t) \le \exp (-\lambda _\Omega t) \le \pi _D(t). \end{aligned}$$

Since $\alpha >\lambda _{\text {min}}(D)$ is arbitrary, we have (3.1).

Let us prove (3.2) to complete the proof of the lemma. Let $C>1$ and $\beta =1/(C\Vert v_D\Vert _\infty )$. Put

$$\begin{aligned} w(t,x)=e^{-\beta t}(v_D(x)+(C-1)\Vert v_D\Vert _\infty ). \end{aligned}$$

Since $-\Delta v_D=1$ in $D$, it follows that

Hence w is a super solution to the heat equation. By the comparison principle

$$\begin{aligned} (C-1)\Vert v_D\Vert _\infty P_D(t,x)\le & {} w(t,x)\\= & {} e^{-\beta t}(v_D(x)+(C-1)\Vert v_D\Vert _\infty ) \le Ce^{-\beta t}\Vert v_D\Vert _\infty . \end{aligned}$$

Dividing the inequality by $0<\Vert v_D\Vert _\infty <\infty$, and taking the supremum for $x\in D$, we obtain (3.2). $\square$

Next we prove the third inequality of (1.5) under an additional assumption on $\lambda _{\text {min}}(D)$.

Lemma 3.2

There exist $\Lambda _0>0$ and ${}C_{0}>0$ depending only on K and $n$ such that if either $\lambda _{\text {min}}(D)> \Lambda _0$ or $\Vert v_D\Vert _\infty <1/\Lambda _0$, then

$$\begin{aligned} \lambda _{\text {min}}(D)\,\Vert v_D\Vert _\infty \le C_{0}. \end{aligned}$$

(3.3)

Proof

In view of Lemma 3.1, we see that $\Vert v_D\Vert _\infty <1/\Lambda _0$ implies $\lambda _{\text {min}}(D)> \Lambda _0$. So, it suffices to show (3.3) under the assumption $\lambda _{\text {min}}(D)> \Lambda _0$ with $\Lambda _0$ to be determined later.

For simplicity we write $\lambda _D$ for $\lambda _{\text {min}}(D)$, albeit $\lambda _{\text {min}}(D)$ need not be an eigenvalue. Let $0<R_0<\infty$. By symmetry, the Gaussian estimate (2.1) implies

$$\begin{aligned} \begin{aligned}&\frac{1}{CV(x,\sqrt{t})^{1/2}V(y,\sqrt{t})^{1/2}}\exp \left( -\frac{Cd(x,y)^2}{t}\right) \le p_M(t,x,y)\\&\qquad \le \frac{C}{V(x,\sqrt{t})^{1/2}V(y,\sqrt{t})^{1/2}}\exp \left( -\frac{d(x,y)^2}{Ct}\right) \end{aligned} \end{aligned}$$

(3.4)

with the same $C$; and conversely, (3.4) implies (2.1) with different $C$ depending only on $\sqrt{K}\, R_0$ and $n$ by volume doubling. Let $0<t<R_0^2$. By [14, Ex. 10.29] we have

$$\begin{aligned} \begin{aligned} p_D(t,x,y)&\le p_D(t,x,y)^{1/2}p_M(t,x,y)^{1/2}\\&\le \left( e^{-\lambda _D}\sqrt{p_D(t/2,x,x)p_D(t/2,y,y)}\right) ^{1/2} p_M(t,x,y)^{1/2}\\&\le e^{-\lambda _Dt/4}p_M(t/2,x,x)^{1/4}p_M(t/2,y,y)^{1/4}p_M(t,x,y)^{1/2}, \end{aligned} \end{aligned}$$

so that the upper estimates of (2.1) and (3.4), together with volume doubling, show that $p_D(t,x,y)$ is bounded by

$$\begin{aligned} \begin{aligned}&e^{-\lambda _Dt/4}\Big \{\frac{C}{V(x,\sqrt{t/2})}\Big \}^{1/4} \cdot \Big \{\frac{C}{V(y,\sqrt{t/2})}\Big \}^{1/4}\cdot \\&\quad \Bigl \{\frac{C}{V(x,\sqrt{t})^{1/2}V(y,\sqrt{t})^{1/2}} \exp \left( -\frac{d(x,y)^2}{Ct}\right) \Bigr \}^{1/2}\\&\qquad \le e^{-\lambda _Dt/4} \frac{CC'}{V(x,\sqrt{t})^{1/2}V(y,\sqrt{t})^{1/2}} \exp \left( -\frac{d(x,y)^2}{2Ct}\right) , \end{aligned} \end{aligned}$$

where $C'$ takes care of the various volume doubling factors. By the lower estimate of (3.4) with $2C^2t$ in place of t and volume doubling, we find ${}C_{1}\ge 1$ depending only on $\sqrt{K}\,R_0$ and $n$ such that

$$\begin{aligned} p_D(t,x,y)\le C_{1}e^{-\lambda _Dt/4} p_M(2C^2t,x,y). \end{aligned}$$

Integrating the inequality with respect to $y\in D$, we obtain

$$\begin{aligned} P_D(t,x)= & {} \int _Dp_D(t,x,y)d\mu (y)\\\le & {} C_{1} e^{-\lambda _Dt/4} \int _Dp_M(2C^2t,x,y)d\mu (y) \le C_{1} e^{-\lambda _Dt/4}. \end{aligned}$$

Taking the supremum over $x\in D$, we obtain

$$\begin{aligned} \pi _D(t)\le C_{1} \exp \left( -\frac{\lambda _Dt}{4}\right) \quad \text{ for } 0<t<R_0^2. \end{aligned}$$

(3.5)

Let $T=R_0^2/2$. We claim that (3.3) holds with $C_{0}=8\log (2C_{1})$, and with $\Lambda _0=4T^{-1}\log (2C_{1})$ or

$$\begin{aligned} C_{1}\exp \left( -\frac{\Lambda _0 T}{4}\right) =\frac{1}{2}. \end{aligned}$$

(3.6)

Suppose $\lambda _D> \Lambda _0$. Then (3.5) with $t=T$ yields $\pi _D(T)\le 1/2$. Solving the initial value problem from time T, we see that

$$\begin{aligned} P_D(t,x)\le \pi _D(T)\cdot P_D(t-T,x)\le \frac{1}{2} \quad \text{ for } t\ge T. \end{aligned}$$

Take the supremum for $x\in D$. We find

$$\begin{aligned} \pi _D(t)\le \frac{1}{2} \quad \text{ for } t\ge T. \end{aligned}$$

Repeating the same argument, we obtain

$$\begin{aligned} \pi _D(t)\le \frac{1}{2^k} \quad \text{ for }~ kT \le t< (k+1)T~\text { with }~k=0,1,2,\dots . \end{aligned}$$

Hence

$$\begin{aligned} \begin{aligned} v_D(x)&=\int _0^\infty P_D(t,x)\mathrm{d}t =\sum _{k=0}^\infty \int _{kT}^{(k+1)T} P_D(t,x)\mathrm{d}t\\&\le \sum _{k=0}^\infty \int _{kT}^{(k+1)T} \pi _D(t)\mathrm{d}t \le T \sum _{k=0}^\infty \frac{1}{2^k}=2T \le \dfrac{2\Lambda _0 T}{\lambda _D} =\dfrac{8\log (2C_{1})}{\lambda _D} \end{aligned} \end{aligned}$$

by (3.6). Taking the supremum for $x\in D$, we obtain $\lambda _D\Vert v_D\Vert _\infty \le 8\log (2C_{1}),$ as required. $\square$

Remark 3.3

If the Gaussian estimate (2.1) holds uniformly for all $0<t<\infty$, then there exists $C>0$ such that $\lambda _{\text {min}}(D)\,\Vert v_D\Vert _\infty \le C$ for all $D\subset M$. This is the case when $K=0$. See [19].

4 Capacitary Width and Harmonic Measure

By we denote the harmonic measure of E in $D$ evaluated at x. In this section we give an estimate for harmonic measure in terms of capacitary width. This will be crucial for the proof of Theorem 1.3.

Theorem 4.1

(cf. [1, Thm. 12.7]) Let $0<R_0<\infty$. Let $D\subset M$ be an open set with $w_\eta (D)<R_0$. If $x\in D$ and $R>0$, then

where ${}C_{2}$ depends only on $\sqrt{K}\,R_0$, $\eta$ and $n$.

Let us begin by estimating the torsion function of a ball.

Lemma 4.2

Let $0<R_0<\infty$. Then there exists a constant $C>1$ depending only on $\sqrt{K}\,R_0$ and $n$ such that

$$\begin{aligned} C^{-1}r^2\le \Vert v_{B(x,r)}\Vert _\infty \le Cr^2 \quad \text{ for } 0<r< R_0. \end{aligned}$$

Proof

Let $0<r< R_0$. Write $B=B(x,r)$ for simplicity. We have $\lambda _{\text {min}}(B)\ge Cr^{-2}$ by Corollary 2.5. Since B is bounded, the bottom of the spectrum is an eigenvalue. So let us write $\lambda _B$ for $\lambda _{\text {min}}(B)$. Let $z\in B$. In view of [14, Ex. 10.29], the Gaussian estimate (2.1) and the volume doubling property, we have

$$\begin{aligned} \begin{aligned} v_B(z)&=\int _B G_B(z,y)d\mu (y)=\int _0^\infty dt\int _B p_B(t,z,y)d\mu (y)\\&=\int _0^{r^2} dt\int _B p_B(t,z,y)d\mu (y)+\int _{r^2}^\infty dt\int _B p_B(t,z,y)d\mu (y)\\&\le r^2+\int _{r^2}^\infty e^{-\lambda _B(t-r^2)}dt \int _B \sqrt{p_B(r^2,z,z)p_B(r^2,y,y)}\,d\mu (y)\\&\le r^2+\frac{1}{\lambda _B} \int _B \frac{Cd\mu (y)}{\sqrt{V(z,r)V(y,r)}} \le r^2+Cr^2, \end{aligned} \end{aligned}$$

where $C$ depends only on $\sqrt{K}\,R_0$ and $n$. Hence $\Vert v_B\Vert _\infty \le Cr^2$.

The opposite inequality is an immediate consequence of the combination of Corollary 2.5 and Lemma 3.1. But for later purpose we give a direct proof based on a lower estimate of the Dirichlet heat kernel of a ball: if $x\in M$, then

$$\begin{aligned} p_B(t,y,z)\ge \frac{C}{V(x,\sqrt{t})} \quad \text{ for }~ y,z\in \varepsilon B \text { and }~0<t<\varepsilon r^2 \end{aligned}$$

valid for some $0<\varepsilon <1$ and $C>0$. In fact, this lower estimate is equivalent to the Gaussian estimate (2.1). See e.g. [6, (1.5)]. If $y\in \varepsilon B$, then

$$\begin{aligned} v_B(y) =\int _B G_B(y,z)d\mu (z) \ge \int _{0}^{\varepsilon r^2} dt\int _{\varepsilon B} p_B(t,y,z)d\mu (z) \ge \frac{\varepsilon r^2C\mu (\varepsilon B)}{V(x,\sqrt{\varepsilon }r)}\ge Cr^2 \end{aligned}$$

by volume doubling. Thus $\Vert v_B\Vert _\infty \ge Cr^2$. $\square$

For later use we record the above estimate: if $0<r<R_0$, then

$$\begin{aligned} v_{B(x,r)}\ge C_{3} r^2 \quad \text{ on } B(x,\varepsilon r), \end{aligned}$$

(4.1)

where $\varepsilon$ and ${}C_{3}$ depends only on $\sqrt{K}\,R_0$ and $n$.

Remark 4.3

In case $K>0$, the inequality (4.1) does not necessarily hold for all $0<r<\infty$ uniformly. Let ${\mathbb {H}}^n$ be the $n$-dimensional hyperbolic space of constant curvature $-1$. Then the torsion function for B(a, r) is a radial function $f(\rho )$ of $\rho =d(x,a)$ satisfying

$$\begin{aligned} -1=\Delta f(\rho ) =\frac{1}{(\sinh \rho )^{n-1}}\frac{d}{d\rho }\Big \{(\sinh \rho )^{n-1}\frac{df}{d\rho }\Big \} \quad \text{ for }~ 0<\rho <r, \end{aligned}$$

$f(r)=0$, $f'(0)=0$ and $f(0)=\Vert v_{B(a,r)}\Vert _\infty$. See [11, pp. 176-177] or [14, (3.85)]. Hence

$$\begin{aligned} \Vert v_{B(a,r)}\Vert _\infty =\int _0^r \int _0^\rho \left( \frac{\sinh t}{\sinh \rho }\right) ^{n-1}dt d\rho . \end{aligned}$$

Since the integrand is less than 1, we have $\Vert v_{B(a,r)}\Vert _\infty \le \frac{1}{2}r^2$ for all $r>0$. Observe that $t\le \sinh t$ for $t>0$ and $\sinh \rho \le \rho \cosh R_0$ for $0<\rho <R_0$. Hence, if $0<r<R_0$, then

$$\begin{aligned} \Vert v_{B(a,r)}\Vert _\infty \ge \int _0^r \int _0^\rho \left( \frac{t}{\rho \cosh R_0}\right) ^{n-1}dt d\rho =\frac{r^2}{2n(\cosh R_0)^{n-1}}, \end{aligned}$$

so that $\Vert v_{B(a,r)}\Vert _\infty \approx r^2$. This gives the estimate in Lemma 4.2 with explicit bounds.

On the other hand, if $r>1$, then $\sinh \rho \ge \frac{1}{2}(1-e^{-2})e^\rho$ for $1<\rho <r$, so that

$$\begin{aligned} \begin{aligned} \Vert v_{B(a,r)}\Vert _\infty&\le \int _0^1 \int _0^\rho dtd\rho +\int _1^r \int _0^\rho \left( \frac{\sinh t}{\sinh \rho }\right) ^{n-1}dt d\rho \\&\le \frac{1}{2}+\int _1^r \int _0^\rho \left( \frac{e^t}{(1-e^{-2})e^\rho }\right) ^{n-1}dt d\rho \\&=\frac{1}{2}+\frac{1}{n-1}\int _1^r\frac{e^{(n-1)\rho }-1}{((1-e^{-2})e^\rho )^{n-1}}d\rho \le \frac{1}{2}+\frac{r-1}{(n-1)(1-e^{-2})^{n-1}}. \end{aligned} \end{aligned}$$

Thus $\Vert v_{B(a,r)}\Vert _\infty =O(r)$ as $r\rightarrow \infty$, so (4.1) fails to hold uniformly for $0<r<\infty$. This example illustrates that the assumption $0<r<R_0$ cannot be dropped in Lemma 4.2.

Next we compare capacity and volume. Observe that ${{\,\mathrm{Cap}\,}}_D(E)$ coincides with the Green capacity of E with respect to $D$, i.e.,

$$\begin{aligned} {{\,\mathrm{Cap}\,}}_D(E) =\sup \Bigl \{\Vert \nu \Vert : {{\,\mathrm{supp}\,}}\nu \subset E \text { and }\int _DG_D(x,y)d\nu (y)\le 1\text { on } D\Bigr \}, \end{aligned}$$

(4.2)

where $\Vert \nu \Vert$ stands for the total mass of the measure $\nu$.

Lemma 4.4

Let $0<R_0<\infty$. There exists a constant ${}C_{4}>0$ depending only on $\sqrt{K}\,R_0$ and $n$ such that if $0<r< R_0$, then

$$\begin{aligned} \frac{\mu (E)}{\mu (\overline{B}(x,r))} \le C_{4}\frac{{{\,\mathrm{Cap}\,}}_{B(x,2 r)}(E)}{{{\,\mathrm{Cap}\,}}_{B(x,2 r)}(\overline{B}(x,r))} \end{aligned}$$

for every Borel set $E\subset \overline{B}(x,r)$.

Proof

Let $0<r<R_0$. Lemma 4.2 yields

$$\begin{aligned} \int _{E}G_{B(x,2 r)}(y,z)d\mu (z)\le & {} \int _{B(x,2 r)}G_{B(x,2 r)}(y,z)d\mu (z) \le \Vert v_{B(x,2r)}\Vert _\infty \\\le & {} Cr^2 \quad \text{ for } \text{ all } y\in M, \end{aligned}$$

where $C$ depends only on $\sqrt{K}\,R_0$ and $n$. Hence the characterization (4.2) of capacity gives

$$\begin{aligned} {{\,\mathrm{Cap}\,}}_{B(x,2 r)}(E)\ge \frac{\mu (E)}{Cr^2}. \end{aligned}$$

(4.3)

Let $\varphi (y)=\min \{2-{d(y,x)}/r,1\}$. Observe that $\varphi \in W_0^1(B(x,2 r))$, $|\nabla \varphi |\le 1/r$ and $\varphi =1$ on $\overline{B}(x,r)$. The definition of capacity and the volume doubling property yield

$$\begin{aligned} {{\,\mathrm{Cap}\,}}_{B(x,2 r)}(\overline{B}(x,r)) \le \int _{B(x,2 r)}|\nabla \varphi |^2d\mu \le \frac{\mu (\overline{B}(x,2r))}{r^2} \le \frac{C\mu (\overline{B}(x,r))}{r^2}. \end{aligned}$$

This, together with (4.3) for $E=\overline{B}(x,r)$, shows that ${{\,\mathrm{Cap}\,}}_{B(x,2 r)}(\overline{B}(x,r))\approx r^{-2}\mu (\overline{B}(x,r))$ with the constant of comparison depending only on $\sqrt{K}\,R_0$ and $n$. Dividing (4.3) by ${{\,\mathrm{Cap}\,}}_{B(x,2 r)}(\overline{B}(x,r))$, we obtain the lemma. $\square$

Let us introduce regularized reduced functions, which are closely related to capacity and harmonic measure. See [3, Sect. 5.3-7] for the Euclidean case. Let $D$ be an open set. For $E\subset D$ and a non-negative function u in E, we define the reduced function ${}^{D}{\mathbf {R}}^{E}_{u}$ by

$$\begin{aligned} {}^{D}{\mathbf {R}}^{E}_{u}(x) =\inf \{v(x): v\ge 0 ~\text { is superharmonic in } D~\text { and }~v\ge u \text { on }~E\} \quad \text{ for } x\in D. \end{aligned}$$

The lower semicontinuous regularization of ${}^{D}{\mathbf {R}}^{E}_{u}$ is called the regularized reduced function or balayage and is denoted by ${}^{D}\widehat{{\mathbf {R}}}^{E}_{u}$. It is known that ${}^{D}\widehat{{\mathbf {R}}}^{E}_{u}$ is a non-negative superharmonic function, ${}^{D}\widehat{{\mathbf {R}}}^{E}_{u}\le {}^{D}{\mathbf {R}}^{E}_{u}$ in $D$ with equality outside a polar set. If u is a non-negative superharmonic function in $D$, then ${}^{D}\widehat{{\mathbf {R}}}^{E}_{u}\le u$ in $D$. By the maximum principle ${}^{D}\widehat{{\mathbf {R}}}^{E}_{u}$ is non-decreasing with respect to $D$ and E. If u is the constant function 1, then ${}^{D}\widehat{{\mathbf {R}}}^{E}_{1}(x)$ is the probability of Brownian motion hitting E before leaving $D$ when it starts at x. In an almost verbatim way we can extend [1, Lem. F] to the present setting. But, for completeness, we shall provide a proof.

Lemma 4.5

Let $0<r<R<R_0<\infty$.

• (i) $\displaystyle \inf _{\overline{B}(x,r)} {}^{B(x,R)}\widehat{{\mathbf {R}}}^{E}_{1} \le \frac{{{\,\mathrm{Cap}\,}}_{B(x,R)}(E)}{{{\,\mathrm{Cap}\,}}_{B(x,R)}(\overline{B}(x,r))}$    for $E\subset B(x,R)$.

• (ii) $\displaystyle \frac{{{\,\mathrm{Cap}\,}}_{B(x,R)}(E)}{{{\,\mathrm{Cap}\,}}_{B(x,R)}(\overline{B}(x,r))} \le C\inf _{\overline{B}(x,r)} {}^{B(x,R)}\widehat{{\mathbf {R}}}^{E}_{1}$    for $E\subset \overline{B}(x,r)$ with $C>1$ depending only on $\sqrt{K}\,R_0$, r/R and $n$.

Proof

Let $\nu _E$ and $\nu _B$ be the capacitary measures of E and $\overline{B}(x,r)$, respectively. Then $\nu _E$ is supported on $\overline{E}$, $G_{B(x,R)}\nu _E={}^{B(x,R)}\widehat{{\mathbf {R}}}^{E}_{1}$ and $\Vert \nu _E\Vert ={{\,\mathrm{Cap}\,}}_{B(x,R)}(E)$; $\nu _B$ is supported on $\overline{B}(x,r)$, $G_{B(x,R)}\nu _B={}^{B(x,R)}\widehat{{\mathbf {R}}}^{\overline{B}(x,r)}_{1}$ and $\Vert \nu _B\Vert ={{\,\mathrm{Cap}\,}}_{B(x,R)}(\overline{B}(x,r))$. In particular, $G_{B(x,R)}\nu _B \le 1$ in B(x, R) and hence

$$\begin{aligned} \begin{aligned} {{\,\mathrm{Cap}\,}}_{B(x,R)}(E)&\ge \int G_{B(x,R)}\nu _B d\nu _E = \int G_{B(x,R)}\nu _Ed\nu _B \\&= \int {}^{B(x,R)}\widehat{{\mathbf {R}}}^{E}_{1} d\nu _B \ge \int \left( \inf _{\overline{B}(x,r)} {}^{B(x,R)}\widehat{{\mathbf {R}}}^{E}_{1}\right) d\nu _B \\&= \left( \inf _{\overline{B}(x,r)} {}^{B(x,R)}\widehat{{\mathbf {R}}}^{E}_{1} \right) {{\,\mathrm{Cap}\,}}_{B(x,R)}(\overline{B}(x,r)). \end{aligned} \end{aligned}$$

Thus (i) follows.

Let $\rho =(r+R)/2$. The elliptic Harnack inequality (Corollary 2.6) implies

$$\begin{aligned}&G_{B(x,R)}(z,y)\approx G_{B(x,R)}(z,x)\quad \text{ for }~ z\in \partial B(x,\rho )~\text {and }~y\in \overline{B}(x,r),\\&{}^{B(x,R)}\widehat{{\mathbf {R}}}^{\overline{B}(x,r)}_{1}\approx 1 \quad \text{ on }~ \partial B(x,\rho ), \end{aligned}$$

where, and hereafter, the constants of comparison depend only on $\sqrt{K}\,R_0$, r/R and $n$. Let $E\subset \overline{B}(x,r)$. Since ${{\,\mathrm{supp}\,}}\nu _E\subset \overline{B}(x,r)$, we have for $z\in \partial B(x,\rho )$,

$$\begin{aligned}&{}^{B(x,R)}\widehat{{\mathbf {R}}}^{E}_{1}(z) =\int G_{B(x,R)}(z,y)d\nu _E(y) \approx G_{B(x,R)}(z,x){{\,\mathrm{Cap}\,}}_{B(x,R)}(E),\\&{}^{B(x,R)}\widehat{{\mathbf {R}}}^{\overline{B}(x,r)}_{1}(z)=\int G_{B(x,R)}(z,y)d\nu _B(y) \approx G_{B(x,R)}(z,x){{\,\mathrm{Cap}\,}}_{B(x,R)}(\overline{B}(x,r)), \end{aligned}$$

so that

$$\begin{aligned} \frac{{{\,\mathrm{Cap}\,}}_{B(x,R)}(E)}{{{\,\mathrm{Cap}\,}}_{B(x,R)}(\overline{B}(x,r))} \approx {}^{B(x,R)}\widehat{{\mathbf {R}}}^{E}_{1}(z). \end{aligned}$$

Since $z\in \partial B(x,\rho )$ is arbitrary, the superharmonicity of ${}^{B(x,R)}\widehat{{\mathbf {R}}}^{E}_{1}$ and the maximum principle yield (ii). $\square$

We restate the above lemma in terms of harmonic measure. We recall stands for the harmonic measure of E in $D$ evaluated at x. We see that if E is a compact subset of B(x, R), then

(4.4)

Strictly speaking, the harmonic measure is extended by the right-hand side. Lemma 4.5 reads as follows.

Lemma 4.6

Let $0<r<R<R_0<\infty$.

• (i)    for $E\subset B(x,R)$.

• (ii)    for $E\subset \overline{B}(x,r)$ with $C>1$ depending only on $\sqrt{K}\,R_0$, r/R and $n$. In particular, if $0<r<R_0/2$, then

  

  where ${}C_{5}>1$ depends only on $\sqrt{K}\,R_0$ and $n$.

Applying Lemma 4.6 repeatedly, we obtain the following estimate of harmonic measure, which is a preliminary version of Theorem 4.1.

Lemma 4.7

Let $0<R_0<\infty$. Let $D\subset M$ be an open set with $w_\eta (D)<R_0$. Suppose $x\in D$ and $R>0$. If k is a non-negative integer such that $R-2kw_\eta (D)>0$, then

Proof

For simplicity let . By definition we find $r> w_\eta (D)$ arbitrarily close to $w_\eta (D)$ such that

$$\begin{aligned} \frac{{{\,\mathrm{Cap}\,}}_{B(y,2r)}(\overline{B}(y,r)\setminus D)}{{{\,\mathrm{Cap}\,}}_{B(y,2r)}(\overline{B}(y,r))}\ge \eta \quad \text{ for } \text{ all } y\in D. \end{aligned}$$

Hence it suffices to show that $\omega _0\le (1-C_{5}^{-1}\eta )^k$ in $D\cap \overline{B}(x,R-2kr)$. Let us prove this inequality by induction on k. The case $k=0$ holds trivially. Let $k\ge 1$ and suppose $\omega _0\le (1-C_{5}^{-1}\eta )^{k-1}$ in $D\cap \overline{B}(x,R-2(k-1)r)$. Take $y\in D\cap \partial B(x,R-2kr)$ and let $E=\overline{B}(y,r)\setminus D$. Since $D\cap B(y,2r)\subset D\cap \overline{B}(x,R-2(k-1)r)$, we have

in $D\cap B(y,2r)$. Since $y\in D\cap \partial B(x,R-2kr)$ is arbitrary, we have $\omega _0\le (1-C_{5}^{-1}\eta )^{k}$ on $D\cap \partial B(x,R-2kr)$, and hence in $D\cap \overline{B}(x,R-2kr)$ by the maximum principle, as required. $\square$

This lemma and the definition of capacitary width yield

Proof of Theorem 4.1

Let k be the integer such that $2kw_\eta (D)<R\le 2(k+1)w_\eta (D).$ Lemma 4.7 gives

which implies the required inequality with

$$\begin{aligned} C_{2}=\frac{1}{2}\log \frac{1}{1-C_{5}^{-1}\eta }. \end{aligned}$$

$\square$

5 Proofs of Theorems 1.3 and 1.6

In this section we prove Theorem 1.3 and complete the proof of Theorem 1.6 by showing

Theorem 5.1

Let $0<R_0<\infty$. If $w_\eta (D)<R_0$, then

$$\begin{aligned} C^{-1}w_\eta (D)^2\le \Vert v_D\Vert _\infty \le Cw_\eta (D)^2 \end{aligned}$$

(5.1)

where $C$ depends only on $\sqrt{K}\, R_0$, $\eta$ and $n$.

This theorem, together with (3.2) in Lemma 3.1, immediately yields the following estimate of the survival probability, which plays a crucial role in the proof of Theorem 1.5.

Theorem 5.2

Let $0<R_0<\infty$. There exist positive constants ${}C_{6}$ and ${}C_{7}$ depending only on $\sqrt{K}\, R_0$, $\eta$ and $n$ such that

$$\begin{aligned} P_D(t,x) \le C_{6} \exp \left( -\frac{C_{7} t}{w_\eta (D)^2}\right) \quad \text{ for } \text{ all }~ t>0 ~\text {and}~ x\in D, \end{aligned}$$

(5.2)

whenever $w_\eta (D)<R_0$.

Let us begin with a uniform estimate of the capacity of balls.

Lemma 5.3

Let $0<R_0<\infty$. For $0<t\le 1$, define

$$\begin{aligned} \kappa (t) =\inf \left\{ \frac{{{\,\mathrm{Cap}\,}}_{B(x,2R)}(\overline{B}(x,tR))}{{{\,\mathrm{Cap}\,}}_{B(x,2R)}(\overline{B}(x,R))}: x\in M,\ 0<R<R_0\right\} . \end{aligned}$$

Then $\lim _{t\rightarrow 1}\kappa (t)=1$.

Proof

Without loss of generality we may assume that $1/2<t\le 1$. Let $\Omega =B(x,2R)\setminus \overline{B}(x,tR)$ and let $E_t=\partial B(x, tR)$. We find $a>0$ such that for each $y\in E_t$ and $0<r<\frac{1}{4}R$ there exists a ball of radius ar lying in $B(y,r)\setminus \Omega$. This means that

$$\begin{aligned} \frac{\mu (B(y,r)\setminus \Omega )}{\mu (B(y,r))}\ge \varepsilon \end{aligned}$$

with some $\varepsilon >0$ depending only on a and the doubling constant. By Lemmas 4.4 and 4.6 we have

(5.3)

with $\varepsilon '>0$ independent of $x,\ R,\ t,\ y$ and r.

The technique in the proof of [2, Thm. 1] yields a positive superharmonic function s in $\Omega$ such that

$$\begin{aligned} s\approx {{\,\mathrm{dist}\,}}(\cdot ,E_t)^\alpha , \end{aligned}$$

(5.4)

where $\alpha >0$ and the constants of comparison are independent of $x,\ R$ and t. In fact, let $r_k=4^k$, $k\in {\mathbb {Z}}$. For each $k\in {\mathbb {Z}}$ choose a locally finite covering of $E_t$ by open balls $B(x_{kj},r_k/4)$, $j\in J_k$; let $B_{kj}=B(x_{kj},r_k)$. By (5.3) we find a positive continuous function $u_{kj}$ in $\Omega \cap \overline{B}_{kj}$, superharmonic in $\Omega \cap B_{kj}$, such that $\varepsilon ''\le u_{kj}\le 2$ in $\Omega \cap B_{kj}$, $u_{kj}\ge 1$ in $\Omega \cap \partial B_{kj}$, $u_{kj}\le 1-\varepsilon ''$ in $\Omega \cap \frac{1}{2}B_{kj}$, where $\varepsilon ''$ is a small positive constant depending only on $\varepsilon '$. Let $A=1-\varepsilon ''/2$ and extend $u_{kj}$ on $\Omega \setminus \overline{B}_{kj}$ by $u_{kj}=\infty$. Then

$$\begin{aligned} s(x)=\inf \{A^{-k}u_{kj}(x): k\in {\mathbb {Z}},\ j\in J_k\},\quad x\in \Omega \end{aligned}$$

is a superharmonic function in $\Omega$ satisfying (5.4) with $\alpha =|\log A|/\log 4$. Actually, we can make s a strong barrier. In the present context, however, superharmonicity is enough.

From (5.4), we find a positive constant $C$ independent of $x,\ R$ and t such that

$$\begin{aligned} \frac{s}{CR^\alpha }\ge 1\quad \text{ on } \partial B(x,3R/2). \end{aligned}$$

Let u be the capacitary potential for $\overline{B}(x,tR)$ in B(x, 2R), i.e.,

$$\begin{aligned} \begin{aligned}&\Delta u=0 \quad \text{ in } B(x,2R)\setminus \overline{B}(x,tR),\\&u=1 \quad \text{ on } \overline{B}(x,tR),\\&u=0 \quad \text{ on } \partial B(x,2R),\\&{{\,\mathrm{Cap}\,}}_{B(x,2R)}(\overline{B}(x,tR))=\int _{B(x,2R)}|\nabla u|^2d\mu . \end{aligned} \end{aligned}$$

Since $1-u\le s/(CR^\alpha )$ on $\partial B(x,3R/2)$, it follows from the maximum principle

$$\begin{aligned} 1-u\le \frac{s}{CR^\alpha } \approx \frac{{{\,\mathrm{dist}\,}}(\cdot ,E_t)^\alpha }{R^\alpha }\quad \text{ in } B(x,3R/2)\setminus \overline{B}(x,tR). \end{aligned}$$

Hence

$$\begin{aligned} u\ge 1-C\frac{((1-t)R)^\alpha }{R^\alpha }=1-C(1-t)^\alpha \quad \text{ in } B(x,R)\setminus \overline{B}(x,tR) \end{aligned}$$

with another positive constant $C$. If $1-C(1-t)^\alpha >0$, then by definition,

$$\begin{aligned} {{\,\mathrm{Cap}\,}}_{B(x,2R)}(\overline{B}(x,R))\le & {} \frac{1}{(1-C(1-t)^\alpha )^2} \int _{B(x,2R)}|\nabla u|^2d\mu \\= & {} \frac{{{\,\mathrm{Cap}\,}}_{B(x,2R)}(\overline{B}(x,tR))}{(1-C(1-t)^\alpha )^2}. \end{aligned}$$

Hence

$$\begin{aligned} \frac{{{\,\mathrm{Cap}\,}}_{B(x,2R)}(\overline{B}(x,tR))}{{{\,\mathrm{Cap}\,}}_{B(x,2R)}(\overline{B}(x,R))} \ge {(1-C(1-t)^\alpha )^2}, \end{aligned}$$

so that the lemma follows as $\lim _{t\rightarrow 1}(1-C(1-t)^\alpha )^2=1$. $\square$

Proof of Theorem 1.3

By definition the first inequality holds for arbitrary open sets $D$. Let us prove the second inequality. In view of Lemma 5.3, we find an integer $N\ge 2$ depending only on $\sqrt{K}\,R_0$ and $n$ such that

$$\begin{aligned} \frac{{{\,\mathrm{Cap}\,}}_{B(x,2R)}(\overline{B}(x,(1-N^{-1})R))}{{{\,\mathrm{Cap}\,}}_{B(x,2R)}(\overline{B}(x,R))} \ge \sqrt{\eta } \end{aligned}$$

(5.5)

uniformly for $x\in M$ and $0<R<R_0$. Let $C_{5}$ be as in Lemma 4.6 and take an integer $k>2$ so large that $(1-C_{5}^{-1}\eta ')^k\le 1-\sqrt{\eta }$.

Let $w_\eta (D)<R_0$. We prove the theorem by showing

$$\begin{aligned} w_\eta (D)\le 2Nk w_{\eta '}(D). \end{aligned}$$

(5.6)

If $w_{\eta '}(D)\ge R_0/(2Nk),$ then $w_\eta (D)<R_0\le 2Nk w_{\eta '}(D)$, so (5.6) follows. Suppose

$$\begin{aligned} w_{\eta '}(D)< \frac{R_0}{2Nk}. \end{aligned}$$

For simplicity we write $\rho =w_{\eta '}(D)$. Apply Lemma 4.7, with $\eta '$ in place of $\eta$, to $x\in D$ and $R=2Nk\rho$. We obtain

Let $E=\overline{B}(x,R)\setminus D$. Then the maximum principle yields

so that

where we use the convention in E. Hence, Lemma 4.6 (i) with $R-2k\rho$ and 2R in place of r and R gives

$$\begin{aligned} 1- \frac{{{\,\mathrm{Cap}\,}}_{B(x,2R)}(E)}{{{\,\mathrm{Cap}\,}}_{B(x,2R)}(\overline{B}(x,R-2k\rho ))} \le 1-\sqrt{\eta }, \end{aligned}$$

so that

$$\begin{aligned} \frac{{{\,\mathrm{Cap}\,}}_{B(x,2R)}(E)}{{{\,\mathrm{Cap}\,}}_{B(x,2R)}(\overline{B}(x,R-2k\rho ))} \ge \sqrt{\eta }. \end{aligned}$$

Multiplying the inequality and (5.5), we obtain

$$\begin{aligned} \frac{{{\,\mathrm{Cap}\,}}_{B(x,2R)}(E)}{{{\,\mathrm{Cap}\,}}_{B(x,2R)}(\overline{B}(x,R))} \ge \eta , \end{aligned}$$

as $R-2k\rho =(1-N^{-1})R$. Since $x\in D$ is arbitrary, we have $w_\eta (D)<R=2Nk\rho =2Nk w_{\eta '}(D)$. Thus we have (5.6). $\square$

Proof of Theorem 5.1

First, let us prove the second inequality of (5.1), i.e., $\Vert v_D\Vert _\infty \le Cw_\eta (D)^2$. In view of the monotonicity of the torsion function, we may assume that $D$ is bounded and hence $\Vert v_D\Vert _\infty <\infty$. By definition we find r, $w_\eta (D)\le r<2w_\eta (D)<2R_0$, such that

$$\begin{aligned} \frac{{{\,\mathrm{Cap}\,}}_{B(x,2r)}(\overline{B}(x,r)\setminus D)}{{{\,\mathrm{Cap}\,}}_{B(x,2r)}(\overline{B}(x,r))} \ge \eta \quad \text{ for } \text{ every } x\in D. \end{aligned}$$

For a moment we fix $x\in D$ and let $B=B(x,r)$, ${B^*}=B(x,2r)$, and $E=\overline{B}\setminus D$ for simplicity. Then ${{\,\mathrm{Cap}\,}}_{{B^*}}(E)/{{\,\mathrm{Cap}\,}}_{{B^*}}(\overline{B})\ge \eta$. We compare $v_D$ with

$$\begin{aligned} v_{B^*}=\int _{{B^*}}G_{{B^*}}(\cdot ,y)d\mu (y). \end{aligned}$$

It is easy to see that $v_D-v_{B^*}$ is harmonic in $D\cap {B^*}$ and $v_D=0$ on ${\partial D}$ outside a polar set. Hence the maximum principle yields

Since Lemma 4.6 implies that

it follows from Lemma 4.2 that

Taking the supremum with respect to $x\in D$, we obtain

$$\begin{aligned} \Vert v_D\Vert _\infty \le CC_{5}\eta ^{-1}r^2 \le 4CC_{5}\eta ^{-1}w_\eta (D)^2. \end{aligned}$$

Second, let us prove the first inequality of (5.1), i.e. $w_\eta (D)^2\le C\Vert v_D\Vert _\infty$. We distinguish two cases. Suppose first $\Vert v_D\Vert _\infty \ge C_{3} R_0^2/2$ with $C_{3}$ as in (4.1). Then

$$\begin{aligned} \Vert v_D\Vert _\infty \ge \frac{1}{2} C_{3}R_0^2> \frac{1}{2}C_{3} w_\eta (D)^2, \end{aligned}$$

as required. Suppose next $\Vert v_D\Vert _\infty < C_{3}R_0^2/2$. Take R such that

$$\begin{aligned} \Vert v_D\Vert _\infty =\frac{C_{3}R^2}{2}. \end{aligned}$$

(5.7)

Then $0<R<R_0$. Let $x\in D$. This time, we let $B=B(x,R)$, ${B^*}=B(x,2R)$ and $E=\overline{B}\setminus D$ with R as in (5.7). We shall compare $v_D$ with the torsion function

$$\begin{aligned} v_B=\int _{B} G_{B}(\cdot ,y)d\mu (y). \end{aligned}$$

Observe that $v_B-v_D$ is harmonic in $D\cap B$. By the maximum principle and Lemma 4.2

since $\partial (D\cap B) \subset (B\cap {\partial D})\cup (D\cap \partial B) \subset E\cup \partial B,$ and since $v_B=0$ on $\partial B$. Let $0< \varepsilon <1$ be as in (4.1). Taking the infimum over $\overline{B}(x,\varepsilon R)$, we obtain from Lemma 4.6 that

Hence, (4.1) and (5.7) yield

$$\begin{aligned} C_{3} R^2-\frac{C_{3}R^2}{2} \le CR^2\frac{{{\,\mathrm{Cap}\,}}_{{B^*}}(E)}{{{\,\mathrm{Cap}\,}}_{{B^*}}(\overline{B}(x,\varepsilon R))}. \end{aligned}$$

Dividing by $CR^2$, we obtain

$$\begin{aligned} \frac{{{\,\mathrm{Cap}\,}}_{{B^*}}(E)}{{{\,\mathrm{Cap}\,}}_{{B^*}}(\overline{B}(x,\varepsilon R))} \ge \frac{C_{3}}{2C}, \end{aligned}$$

so that, by Lemma 4.4 and volume doubling

$$\begin{aligned} \frac{{{\,\mathrm{Cap}\,}}_{{B^*}}(E)}{{{\,\mathrm{Cap}\,}}_{{B^*}}(\overline{B}(x,R))}= & {} \frac{{{\,\mathrm{Cap}\,}}_{{B^*}}(E)}{{{\,\mathrm{Cap}\,}}_{{B^*}}(\overline{B}(x,\varepsilon R))}\cdot \frac{{{\,\mathrm{Cap}\,}}_{{B^*}}(\overline{B}(x,\varepsilon R))}{{{\,\mathrm{Cap}\,}}_{{B^*}}(\overline{B}(x,R))}\\\ge & {} \frac{C_{3}}{2C}\cdot \frac{C\mu (\overline{B}(x,\varepsilon R))}{\mu (\overline{B}(x,R))} \ge \eta ' \end{aligned}$$

with $0< \eta '<1$ depending only on $\sqrt{K}\,R_0$ and $n$. Thus

$$\begin{aligned} \frac{{{\,\mathrm{Cap}\,}}_{{B^*}}(\overline{B}(x,R)\setminus D)}{{{\,\mathrm{Cap}\,}}_{{B^*}}(\overline{B}(x,R))}\ge \eta '. \end{aligned}$$

Since $x\in D$ is arbitrary, we have $w_{\eta '}(D)<R$ and so $w_\eta (D)\le CR$ by Theorem 1.3. Hence $w_\eta (D)^2\le C\Vert v_D\Vert _\infty$ by (5.7). The proof is complete. $\square$

6 Proof of Theorem 1.5

The crucial step of the proof of Theorem 1.5 is the following parabolic box argument (cf. [1, Lem. 4.1]),

Lemma 6.1

Suppose (1.2) holds. If $t>0$, then

$$\begin{aligned} P_D(t,x)\le C_t G_D(x,o) \quad \text{ for } x\in D\end{aligned}$$

(6.1)

with $C_t$ depending on t.

Proof

Without loss of generality we may assume that $\tau =1$ in (1.2). For notational convenience we shall prove (6.1) with T in place of t. For simplicity we write $w_\eta (G_D^o<s)=w_\eta (\{x\in D: G_D(x,o)<s\}$. Let $\alpha _j=\exp (-2^j)$. Since

$$\begin{aligned} \begin{aligned} \int _{\alpha _j}^{\alpha _{j-1}}w_\eta (G_D^o<s)^2\frac{ds}{s}&\ge w_\eta (G_D^o< \alpha _j)^2 \int _{\alpha _j}^{\alpha _{j-1}}\frac{ds}{s}\\&=w_\eta (G_D^o< \alpha _j)^2(2^j-2^{j-1}) =2^{j-1}w_\eta (G_D^o< \alpha _j)^2, \end{aligned} \end{aligned}$$

it follows from (1.2) that $\sum _{j=0}^\infty 2^{j}w_\eta (G_D^o< \alpha _j)^2< \infty$.

Let $w_\eta (G_D^o<1)<R_0< \infty$ and choose $C_{6}$ and $C_{7}$ as in Theorem 5.2. We find $j_0\ge 0$ such that

$$\begin{aligned} \frac{3}{C_{7}}\sum _{j=j_0+1}^\infty 2^{j}w_\eta (G_D^o< \alpha _j)^2<T. \end{aligned}$$

(6.2)

Define

$$\begin{aligned} t_k =\frac{3}{C_{7}}\sum _{j=j_0+1}^{k} 2^jw_\eta (G_D^o< \alpha _j)^2 \quad \text{ for }~ k\ge j_0+1, \end{aligned}$$

and $t_{j_0}=0$. Then $t_k$ increases and $\lim _{k\rightarrow \infty }t_k<T$ by (6.2). Observe that

$$\begin{aligned} \frac{1}{\alpha _{k+1}}\exp \left( -\frac{C_{7}(t_k-t_{k-1})}{w_\eta (G_D^o< \alpha _k)^2}\right) =\exp (2^{k+1}-3\cdot 2^k) =\exp (-2^k) \end{aligned}$$

(6.3)

for $k\ge j_0+1$.

Let $D_k=\{x\in D: G_D(x,o)< \alpha _k\}$, $E_k=\{x\in D: \alpha _{k+1}\le G_D(x,o)< \alpha _k\}$, ${{\widetilde{D}}}_k=(t_{k-1},\infty )\times D_k$ and ${{\widetilde{E}}}_k=(t_{k},\infty ) \times E_k$. Put

$$\begin{aligned} q_k=\sup _{(t,x)\in {{\widetilde{E}}}_k}\frac{P_D(t,x)}{G_D(x,o)}. \end{aligned}$$

We claim that $\sup _{k\ge j_0+1}q_k\le C$, which implies (6.1) with T in place of t, and $C_T=\max \{C,1/\alpha _{j_0+1}\}$ since by (6.2). See Fig. 1.

Fig. 1

Parabolic box argument

By the parabolic comparison principle over ${{\widetilde{D}}}_{j_0+1}$ we have

$$\begin{aligned} P_D(t,x)\le \frac{G_D(x,o)}{\alpha _{j_0+1}}+P_{D_{j_0+1}}(t,x) \quad \text{ for } (t,x)\in {{\widetilde{D}}}_{j_0+1}=(0,\infty )\times D_{j_0+1}. \end{aligned}$$

Divide both sides by $G_D(x,o)$ and take the supremum over ${{\widetilde{E}}}_{j_0+1}$. Then (5.2) and (6.3) yield

$$\begin{aligned} \begin{aligned} q_{j_0+1}&\le \frac{1}{\alpha _{j_0+1}}+\sup _{(t,x)\in {{\widetilde{E}}}_{j_0+1}}\frac{P_{D_{j_0+1}}(t,x)}{G_D(x,o)}\\&\le \frac{1}{\alpha _{j_0+1}}+\frac{C_{6}}{\alpha _{j_0+2}} \sup _{t\ge t_{j_0+1}}\exp \left( -\frac{C_{7}t}{w_\eta (D_{j_0+1})^2}\right) \\&\le \frac{1}{\alpha _{j_0+1}}+\frac{C_{6}}{\alpha _{j_0+2}} \exp \left( -\frac{C_{7}(t_{j_0+1}-t_{j_0+1})}{w_\eta (D_{j_0+1})^2}\right) =\exp (2^{j_0+1})+C_{6}\exp (-2^{j_0+1}). \end{aligned} \end{aligned}$$

Let $k\ge j_0+2$. By the parabolic comparison principle over ${{\widetilde{D}}}_k$ we have

$$\begin{aligned} P_D(t,x)\le q_{k-1} G_D(x,o)+P_{D_k}(t-t_{k-1},x) \quad \text{ for } (t,x)\in {{\widetilde{D}}}_k=(t_{k-1},\infty )\times D_k. \end{aligned}$$

Divide both sides by $G_D(x,o)$ and take the supremum over ${{\widetilde{E}}}_k$. In the same way as above, we obtain from (5.2) and (6.3) that

$$\begin{aligned} q_k \le q_{k-1}+ \frac{C_{6}}{\alpha _{k+1}} \exp \left( -\frac{C_{7}(t_{k}-t_{k-1})}{w_\eta (D_k)^2}\right) \le q_{k-1}+C_{6}\exp (-2^k). \end{aligned}$$

Hence we have the claim as

$$\begin{aligned} \sup _{k\ge j_0+1}q_k\le \exp (2^{j_0+1})+C_{6}\sum _{k=j_0+1}^\infty \exp (-2^k)< \infty . \end{aligned}$$

The lemma is proved. $\square$

Proof of Theorem 1.5

By Theorem 1.4 we have the first condition for IU. Let us show (1.1) for every $t>0$. It is known that the lower estimate of (1.1) follows from the upper estimate. Moreover, if $p_D(t_0,x,y)\le C_{t_0}\varphi _D(x)\varphi _D(y)$ for all $x,y\in D$ with some $t_0>0$, then $p_D(t,x,y)\le C_t\varphi _D(x)\varphi _D(y)$ holds with $C_t\le C_{t_0} e^{-\lambda _D(t-t_0)}$ for $t\ge t_0$ (See e.g. [1, Prop. 2.1]). Hence, it suffices to show the upper estimate of (1.1) for small $t>0$.

Since $\varphi _D$ is superharmonic, and since $G_D(\cdot ,o)$ is harmonic outside $\{o\}$, we have $G_D(\cdot ,o)\le C\varphi _D$ apart from a neighborhood of o. So, it is sufficient to show that if $t>0$ small, then there exists $C_t>0$ such that

$$\begin{aligned} p_D(t,x,y)\le C_t G_D(x,o)G_D(y,o) \quad \text{ for } x,y\in D. \end{aligned}$$

(6.4)

Let $i_0$ be the injectivity radius of M. It is known that

$$\begin{aligned} \mu (B(x,r))\ge Cr^n\quad \text{ for }~ 0<r<i_0/2 \text { and }~x\in M. \end{aligned}$$

where $C>0$ depends only on M (Croke [9, Prop. 14]). Hence, the Gaussian estimate (2.1) yields

$$\begin{aligned} p_M(t,x,y)\le \frac{C}{V(x,\sqrt{t})}\le Ct^{-n/2} \end{aligned}$$

(6.5)

for $0<t< \min \{R_0^2,(i_0/2)^2\}$ and $x,y\in M$. Let $0<t< \min \{R_0^2,(i_0/2)^2\}$ and $x,y,z\in D$. By (6.5) we have

$$\begin{aligned} \begin{aligned} p_D(2t,z,y)&=\int _Dp_D(t,z,w)p_D(t,w,y)d\mu (w)\\&\le \int _Dp_M(t,z,w)p_D(t,w,y)d\mu (w)\\&\le Ct^{-n/2}\int _Dp_D(t,w,y)d\mu (w) =Ct^{-n/2} P_D(t,y), \end{aligned} \end{aligned}$$

since the heat kernel is symmetric. Moreover,

$$\begin{aligned} \begin{aligned} p_D(3t,x,y)&\le \int _Dp_D(t,x,z)p_D(2t,z,y)d\mu (z)\\&\le \int _Dp_D(t,x,z)Ct^{-n/2} P_D(t,y) d\mu (z)\\&=Ct^{-n/2} P_D(t,x)P_D(t,y). \end{aligned} \end{aligned}$$

Hence Lemma 6.1 yields

$$\begin{aligned} p_D(3t,x,y) \le Ct^{-n/2} P_D(t,x)P_D(t,y) \le C_t G_D(x,o)G_D(y,o). \end{aligned}$$

Replacing 3t by t, we obtain (6.4) for small $t>0$. Thus the theorem is proved. $\square$

Remark 6.2

The assumption on the injectivity radius can be replaced by

$$\begin{aligned} \inf _{x\in M} \mu (B(x,R_0))>0. \end{aligned}$$

(6.6)

In fact, (2.2) yields

$$\begin{aligned} \mu (B(x,r))\ge C\left( \frac{r}{R_0}\right) ^\alpha \inf _{x\in M} \mu (B(x,R_0)) \quad \text{ for } \text{ all }~ x\in M ~\text {and}~ 0<r<R_0, \end{aligned}$$

and hence for small $t>0$,

$$\begin{aligned} p_M(t,x,y)\le \frac{C}{V(x,\sqrt{t})}\le Ct^{-\alpha /2}. \end{aligned}$$

Replacing (6.5) by this inequality, we obtain

$$\begin{aligned} p_D(3t,x,y) \le Ct^{-\alpha /2} P_D(t,x)P_D(t,y) \le C_t G_D(x,o)G_D(y,o), \end{aligned}$$

which proves Theorem 1.5. See [10] for further discussion on (6.6).

7 Remarks

Once we obtain the theorems in Sect. 1, we can extend many Euclidean results to the setting of manifolds. Proofs are almost the same as in the Euclidean case. For instance, we relax the requirement of inner uniformity for IU assumed in [15, Thm. 7.9]. For a curve $\gamma$ in M we denote the length of $\gamma$ and the subarc of $\gamma$ between x and y by $\ell (\gamma )$ and $\gamma (x,y)$, respectively. For a domain $D$ in M we define the inner metric in $D$ as

$$\begin{aligned} d_D(x,y)=\inf \{\ell (\gamma ): \gamma \text { is a curve connecting } x~ \text {and }~ y~ \mathrm{in}~ D\}. \end{aligned}$$

Fig. 2

A John domain that is not inner uniform

Definition 7.1

Let $D$ be a domain in M and let $\delta _D(x)={{\,\mathrm{dist}\,}}(x,M\setminus D)$.

(i) We say that $D$ is a John domain if there exist $o\in D$ and $C\ge 1$ such that every $x\in D$ is connected to o by a rectifiable curve $\gamma \subset D$ with the property

$$\begin{aligned} \ell (\gamma (x,z))\le C\delta _D(z)\quad \text{ for } \text{ all } z\in \gamma . \end{aligned}$$

(ii) We say that $D$ is an inner uniform domain if there exists $C\ge 1$ such that every pair of points $x,y\in D$ can be connected by a rectifiable curve $\gamma \subset D$ with the properties $\ell (\gamma )\le Cd_D(x,y)$ and

$$\begin{aligned} \min \{\ell (\gamma (x,z),\ell (\gamma (z,y)\} \le C\delta _D(z)\quad \text{ for } \text{ all } z\in \gamma . \end{aligned}$$

If we replace $d_D(x,y)$ by the ordinary metric d(x, y) in (ii), then we obtain a uniform domain. By definition a John domain is necessarily bounded. We have the following inclusions for these classes of bounded domains:

$$\begin{aligned} \text {uniform}\subsetneqq \text {inner uniform}\subsetneqq \text {John}. \end{aligned}$$

Figure 2 depicts a John domain that is not inner uniform. We find a curve connecting x and o with the property of Definition 7.1 (i); yet there is no curve connecting x and y with the properties of Definition 7.1 (ii) if the gaps on the vertical segment shrink sufficiently fast.

Theorem 7.2

A John domain is IU.

Proof

Let $D$ be a John domain. Observe that $w_\eta (\{x\in D:\delta _D(x)<r\})\le Cr$ for small $r>0$ by definition and $G_D(x,o)\ge C\delta _D(x)^\alpha$ with some $\alpha >0$ by the Harnack inequality. Hence

$$\begin{aligned} w_\eta (\{x\in D: G_D(x,o)<t\}) \le w_\eta (\{x\in D: \delta _D(x)<(t/C)^{1/\alpha }\}) \le Ct^{1/\alpha }, \end{aligned}$$

so that (1.2) holds. Therefore Theorem 1.5 asserts that $D$ is IU. $\square$