Intrinsic ultracontractivity for domains in negatively curved manifolds

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 Poincar\'e inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel for finite scale.


M
Let be a complete, non-compact, -dimensional connected Riemannian manifold, without boundary, and with Ricci curvature bounded below by a negative constant, i.e., Ric ≥ − with nonnegative constant . Throughout the paper, is reserved for this constant. In this article, we investigate domains (open, and connected sets) in for which the heat semigroup is intrinsically ultracontractive.
For a domain ⊂ we denote by ( , , ), > 0, , ∈ , the Dirichlet heat kernel for / − Δ in , i.e., the fundamental solution to ( / − Δ) = 0 subject to the Dirichlet boundary condition ( , ) = 0 for ∈ and > 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 ⊂ . We say that the semigroup associated with ( , , ) is intrinsically ultracontractive (abbreviated to IU) if the following two conditions are satisfied: (i) The Dirichlet Laplacian −Δ has no essential spectrum and has the first eigenvalue > 0 with corresponding positive eigenfunction normalized by 2 = 1. For simplicity, we say that itself is IU if the semigroup associated with ( , , ) 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, Theorems 1 and 2] gave conditions characterizing IU and the Cranston-McConnell inequality when restricting to a certain class of plane domains, which illustrate subtle difference 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,Theorem 7.9]). In view of [13], however, the requirement of inner uniformity for IU to hold can be relaxed. See Section 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 Ω ⊂ be an open set. For ⊂ Ω we define relative capacity by where is the Riemannian measure in and ∞ 0 (Ω) is the space of all smooth functions compactly supported in . Let ( , ) be the distance between and in . The open geodesic ball with center and radius > 0 is denoted by ( , ) = { ∈ : ( , ) < }. The closure of a set is denoted by , and so ( , ) stands for the closed geodesic ball of center and radius .
The next theorem asserts that the parameter has no significance. 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 ∈ . We shall show 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.
where is the Green function for .
Our results are based on the relationship between the torsion function and the bottom of the spectrum We note that min ( ) is the first eigenvalue if has no essential spectrum. This is always the case for a bounded domain . Theorem 1.4 asserts that the same holds even for an unbounded domain whenever lim →∞ ( \ ( , )) = 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.
The second inequality of (1.4) does not necessarily hold for negatively curved manifolds. Let H be the -dimensional hyperbolic space of constant curvature −1. It is known that whereas H ≡ ∞ as H is stochastically complete. Hence the second inequality of (1.4) fails to hold if is the whole space H . The point of this paper is that (1.4) still holds if 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 ≥ 0 and let 0 < < 1. Then there exist 0 > 0 and > 1 depending only on , and such that if ⊂ satisfies ( ) < 0 , then Remark 1.7. We actually find Λ 0 > 0 depending only on and such that (1.4) holds for with min ( ) > Λ 0 (Lemma 3.2 below). This is a generalization of Theorem A as Λ 0 = 0 for = 0. In practice, however, the condition ( ) < 0 in Theorem 1.6 is more convenient since the capacitary width ( ) can be more easily estimated than the bottom of spectrum min ( ).
In Section 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 for 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,Example 2.11] and [18,Section 5.6]. This observation suggests that our approach is also extendable to those spaces. We use the following notation. By the symbol we denote an absolute positive constant whose value is unimportant and may change from one occurrence to the next. If necessary, we use 0 , 1 , . . . , to specify them. We say that and are comparable and write ≈ if two positive quantities and satisfy −1 ≤ / ≤ with some constant ≥ 1. The constant is referred to as the constant of comparison.
Acknowledgments. The authors would like to thank the referee for his/her careful reading of the manuscript and many useful suggestions.

P
We recall that is a manifold of dimension ≥ 2 with Ric ≥ − with ≥ 0. Let us recall the volume doubling property of the Riemannian measure , the Poincaré inequality and the Gaussian estimate for the Dirichlet heat kernel ( , , ) for . For = ( , ) and > 0 we write = ( , ).
The Poincaré inequality yields the Sobolev inequality. We see that if = ( , ) with 0 < < 0 , then 1 ( ) 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 ( , ) = ( ( , )).
Theorem 2.1 and Corollary 2.3 assert that (i) and (ii) of Theorem B hold true for 0 < 0 < ∞ with constants depending only on , 0 and . Hence, the Li-Yau Gaussian estimate of the heat kernel for the whole manifold and the parabolic Harnack inequality up to scale 0 are available in our setting. Observe that the volume doubling inequality ( ( , 2 )) ≤ 0 ( ( , )) implies We also have the following elliptic Harnack inequality since positive harmonic functions are time-independent positive solutions to the heat equation.

T
In this section we obtain estimates between the bottom of the spectrum and the torsion function . We shall show the second and the third inequalities of (1.5).
Since the Green function ( , ) is the integral of the heat kernel ( , , ) with respect to ∈ (0, ∞), we have We note that , i.e., the survival probability that the Brownian motion ( ) ≥0 started at stays in up to time , where is the first exit time from . We also observe that ( , ) is considered to be the (weak) solution to . Let us begin with the proof of the second inequality of (1.5).
Proof. We follow [1, Lemmas 3.2 and 3.3]. Without loss of generality we may assume that ∞ < ∞. It suffices to show the following two estimates: In fact, we obtain from (3.1) and (3.2) that which holds for all > 0 only if Then Ω has no essential spectrum. Let Ω and Ω be the first eigenvalue and its positive eigenfunction with Ω 2 = 1 for Ω, respectively. By definition Taking the supremum for ∈ Ω, and then dividing by Since > min ( ) is arbitrary, we have (3.1). Let us show (3.2) to complete the proof of the lemma. Let > 1 and = 1/( ∞ ). Put Hence is a super solution to the heat equation. By the comparison principle Dividing the inequality by 0 < ∞ < ∞, and taking the supremum for ∈ , we obtain (3.2).
Next we prove the third inequality of (1.5) under an additional assumption on min ( ).
For simplicity we write for min ( ), albeit min ( ) need not be an eigenvalue. Let 0 < 0 < ∞. By symmetry, the Gaussian estimate (2.1) implies with the same ; and conversely, (3.4) implies (2.1) with different depending only on ( , where takes care of the various volume doubling factors. By the lower estimate of (3.4) with 2 2 in place of and volume doubling, we find 1 ≥ 1 depending only on √ 0 and such that ( , , ) ≤ 1 − /4 (2 2 , , ).
Integrating the inequality with respect to ∈ , we obtain Taking the supremum over ∈ , we obtain Suppose > Λ 0 . Then (3.5) with = yields ( ) ≤ 1/2. Solving the initial value problem from time , we see that Take the supremum for ∈ . We find Repeating the same argument, we obtain by (3.6). Taking the supremum for ∈ , we obtain ∞ ≤ 8 log(2 1 ), as required.

C
By ( , ) we denote the harmonic measure of in evaluated at . 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.
where depends only on √ 0 and . Hence ∞ ≤ 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 ∈ , then for , ∈ and 0 < < 2 valid for some 0 < < 1 and > 0. In fact, this lower estimate is equivalent to the Gaussian estimate (2.1). See e.g. [6, (1.5)]. If ∈ , then For later use we record the above estimate: if 0 < < 0 , then where and 3 depends only on √ 0 and . Since the integrand is less than 1, we have ( , ) ∞ ≤ 1 2 2 for all > 0. Observe that ≤ sinh for > 0 and sinh ≤ cosh 0 for 0 < < 0 . Hence, if 0 < < 0 , then
Next we compare capacity and volume. Observe that Cap ( ) coincides with the Green capacity of with respect to , i.e., for every Borel set ⊂ ( , ).
Proof. Let 0 < < 0 . Lemma 4.2 yields where depends only on Let ( ) = min{2 − ( , )/ , 1}. Observe that ∈ 1 0 ( ( , 2 )), |∇ | ≤ 1/ and = 1 on ( , ). The definition of capacity and the volume doubling property yield This, together with The lower semicontinuous regularization of R is called the regularized reduced function or balayage and is denoted by R . It is known that R is a nonnegative superharmonic function, R ≤ R in with equality outside a polar set. If is a nonnegative superharmonic function in , then R ≤ in . By the maximum principle R is nondecreasing with respect to and . If is the constant function 1, then R 1 ( ) is the probability of Brownian motion hitting before leaving when it starts at . In an almost verbatim way we can extend [1, Lemma F] to the present setting. But, for completeness, we shall provide a proof.
We restate the above lemma in terms of harmonic measure. We recall ( , ) stands for the harmonic measure of in evaluated at . We see that if is a compact subset of ( , ), then Strictly speaking, the harmonic measure is extended by the right-hand side. Lemma 4.5 reads as follows.
(ii) sup Applying Lemma 4.6 repeatedly, we obtain the following estimate of harmonic measure, which is a preliminary version of Theorem 4.1.
It is easy to see that − * is harmonic in ∩ * and = 0 on outside a polar set. Hence the maximum principle yields − * ≤ ∞ ( ∩ * , ∩ * ) in ∩ * .
6. P T 1.5 The crucial step of the proof of Theorem 1.5 is the following parabolic box argument (cf. [ Proof. Without loss of generality we may assume that = 1 in (1.2). For notational convenience we shall prove (6.1) with in place of . For simplicity we write ( ) < 0 < ∞ and choose 6 and 7 as in Theorem 5.2. We find 0 ≥ 0 such that (6.2) and 0 = 0. Then increases and lim →∞ < by (6.2). Observe that We claim that sup ≥ 0 +1 ≤ , which implies (6.1) with in place of , and = max{ , by (6.2). See Figure 1.
By the parabolic comparison principle over 0 +1 we have Divide the both sides by ( , ) and take the supremum over . 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.

R
Once we obtain the theorems in Section 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,Theorem 7.9]. For a curve in we denote the length of and the subarc of between and by ℓ ( ) and ( , ), respectively. For a domain in we define the inner metric in as ( , ) = inf{ℓ( ) : is a curve connecting and in }. (i) We say that is a John domain if there exist ∈ and ≥ 1 such that every ∈ is connected to by a rectifiable curve ⊂ with the property ℓ( ( , )) ≤ ( ) for all ∈ .
If we replace ( , ) by the ordinary metric ( , ) 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: uniform inner uniform John.