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.
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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
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
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
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
and the bottom of the spectrum
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:
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
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
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
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 \(0<r<R_0\)
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
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 \(0<r<R_0\)
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
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
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
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
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
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
where
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:
In fact, we obtain from (3.1) and (3.2) that
which holds for all \(t>0\) only if
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
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
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
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
Since \(-\Delta v_D=1\) in \(D\), it follows that
Hence w is a super solution to the heat equation. By the comparison principle
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
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
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
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
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
Integrating the inequality with respect to \(y\in D\), we obtain
Taking the supremum over \(x\in D\), we obtain
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
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
Take the supremum for \(x\in D\). We find
Repeating the same argument, we obtain
Hence
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
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
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
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
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
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
\(f(r)=0\), \(f'(0)=0\) and \(f(0)=\Vert v_{B(a,r)}\Vert _\infty \). See [11, pp. 176-177] or [14, (3.85)]. Hence
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
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
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.,
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
for every Borel set \(E\subset \overline{B}(x,r)\).
Proof
Let \(0<r<R_0\). Lemma 4.2 yields
where \(C\) depends only on \(\sqrt{K}\,R_0\) and \(n\). Hence the characterization (4.2) of capacity gives
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
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
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
Thus (i) follows.
Let \(\rho =(r+R)/2\). The elliptic Harnack inequality (Corollary 2.6) implies
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 \(z\in \partial B(x,\rho )\),
so that
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
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
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
\(\square \)
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
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
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
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
with some \(\varepsilon >0\) depending only on a and the doubling constant. By Lemmas 4.4 and 4.6 we have
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
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
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
Let u be the capacitary potential for \(\overline{B}(x,tR)\) in B(x, 2R), i.e.,
Since \(1-u\le s/(CR^\alpha )\) on \(\partial B(x,3R/2)\), it follows from the maximum principle
Hence
with another positive constant \(C\). If \(1-C(1-t)^\alpha >0\), then by definition,
Hence
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
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
If \( w_{\eta '}(D)\ge R_0/(2Nk), \) then \(w_\eta (D)<R_0\le 2Nk w_{\eta '}(D)\), so (5.6) follows. Suppose
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
so that
Multiplying the inequality and (5.5), we obtain
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
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
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
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
as required. Suppose next \(\Vert v_D\Vert _\infty < C_{3}R_0^2/2\). Take R such that
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
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
Dividing by \(CR^2\), we obtain
so that, by Lemma 4.4 and volume doubling
with \(0< \eta '<1\) depending only on \(\sqrt{K}\,R_0\) and \(n\). Thus
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
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
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
Define
and \(t_{j_0}=0\). Then \(t_k\) increases and \(\lim _{k\rightarrow \infty }t_k<T\) by (6.2). Observe that
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
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.
By the parabolic comparison principle over \({{\widetilde{D}}}_{j_0+1}\) we have
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
Let \(k\ge j_0+2\). By the parabolic comparison principle over \({{\widetilde{D}}}_k\) we have
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
Hence we have the claim as
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
Let \(i_0\) be the injectivity radius of M. It is known that
where \(C>0\) depends only on M (Croke [9, Prop. 14]). Hence, the Gaussian estimate (2.1) yields
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
since the heat kernel is symmetric. Moreover,
Hence Lemma 6.1 yields
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
In fact, (2.2) yields
and hence for small \(t>0\),
Replacing (6.5) by this inequality, we obtain
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
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
(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
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:
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
so that (1.2) holds. Therefore Theorem 1.5 asserts that \(D\) is IU. \(\square \)
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The authors would like to thank the referee for his/her careful reading of the manuscript and many useful suggestions.
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Communicated by Tom Carroll.
Dedicated to the memory of Professor Walter K. Hayman.
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This work was supported by JSPS KAKENHI Grant Number 17H01092. MvdB was also supported by The Leverhulme Trust through Emeritus Fellowship EM-2018-011-9.
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Aikawa, H., van den Berg, M. & Masamune, J. Intrinsic Ultracontractivity for Domains in Negatively Curved Manifolds. Comput. Methods Funct. Theory 21, 797–824 (2021). https://doi.org/10.1007/s40315-021-00402-8
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DOI: https://doi.org/10.1007/s40315-021-00402-8
Keywords
- Intrinsic ultracontractivity
- Ricci curvature
- First eigenvalue
- Heat kernel
- Torsion function
- Capacitary width