Bottom of the L 2 spectrum of the Laplacian on locally symmetric spaces
aa r X i v : . [ m a t h . SP ] J un BOTTOM OF THE L SPECTRUM OF THE LAPLACIANON LOCALLY SYMMETRIC SPACES
JEAN-PHILIPPE ANKER & HONG-WEI ZHANG
Abstract.
We estimate the bottom of the L spectrum of the Laplacianon locally symmetric spaces in terms of the critical exponents of appropriatePoincaré series. Our main result is the higher rank analog of a characterizationdue to Elstrodt, Patterson, Sullivan and Corlette in rank one. It improves uponprevious results obtained by Leuzinger and Weber in higher rank. Introduction
We adopt the standard notation and refer to [12] for more details. Let G be asemi-simple Lie group, connected, noncompact, with finite center, and K be a max-imal compact subgroup of G . The homogenous space X = G/K is a Riemanniansymmetric space of noncompact type. Let g = k ⊕ p be the Cartan decompositionof the Lie algebra of G . Fix a maximal abelian subspace a in p . Let Γ be a dis-crete torsion-free subgroup of G that acts freely and properly discontinuously on X . Then Y = Γ \ X is a locally symmetric space, whose Riemmannian structure isinherited from X . We denote by n the joint dimension of X and Y , and by ℓ thejoint rank of X and Y , which is the dimension of a .Let Σ ⊂ a be the root system of ( g , a ) and let W be the associated Weyl group.Choose a positive Weyl chamber a + ⊂ a and let Σ + ⊂ Σ be the correspondingsubsystem of positive roots. Denote by ρ = P α ∈ Σ + m α α the half sum of positiveroots counted with their multiplicities. Occasionally we shall need the reduced rootsystem Σ red = { α ∈ Σ | α / ∈ Σ } .Consider the classical Poincaré series P s ( xK, yK ) = X γ ∈ Γ e − sd ( xK,γyK ) ∀ s > , ∀ x, y ∈ G (1.1)and denote by δ (Γ) = inf { s > | P s ( xK, yK ) < + ∞} its critical exponent. Recallthat δ (Γ) ∈ [0 , k ρ k ] may be also defined by δ (Γ) = lim sup R → + ∞ log N R ( xK, yK ) R ∀ x, y ∈ G, where N R ( xK, yK ) = |{ γ ∈ Γ | d ( xK, γyK ) ≤ R }| denotes the orbital countingfunction. Finally, let ∆ Y be the Laplace-Beltrami operator on Y and let λ ( Y ) bethe bottom of the L spectrum of − ∆ Y .The following celebrated result, due to Elstrodt [9, 10, 11], Patterson [16], Sul-livan [18] and Corlette [7], expresses λ ( Y ) in terms of ρ and δ (Γ) in rank one. Mathematics Subject Classification.
Key words and phrases.
Locally symmetric space, L spectrum, Poincaré series, critical expo-nent, heat kernel. Theorem 1.1.
In the rank one case, we have λ ( Y ) = ( k ρ k if ≤ δ (Γ) ≤ k ρ k , k ρ k − ( δ (Γ) − k ρ k ) if k ρ k ≤ δ (Γ) ≤ k ρ k . (1.2)This result was extended in higher rank as follows by Leuzinger [15] and Weber[19]. Let ρ min = min H ∈ a + , k H k =1 h ρ, H i ∈ (0 , k ρ k ] . Theorem 1.2.
In the general case, the following estimates hold: • Upper bound: λ ( Y ) ≤ ( k ρ k if ≤ δ (Γ) ≤ k ρ k , k ρ k − ( δ (Γ) − k ρ k ) if k ρ k ≤ δ (Γ) ≤ k ρ k . • Lower bound: λ ( Y ) ≥ ( k ρ k if ≤ δ (Γ) ≤ ρ min , max { k ρ k − ( δ (Γ) − ρ min ) } if ρ min ≤ δ (Γ) ≤ k ρ k . In other terms, λ ( Y ) = k ρ k if δ (Γ) ∈ (cid:2) , ρ min (cid:3) ,λ ( Y ) ∈ (cid:2) k ρ k − ( δ (Γ) − ρ min ) , k ρ k (cid:3) if δ (Γ) ∈ (cid:2) ρ min , k ρ k (cid:3) ,λ ( Y ) ∈ (cid:2) k ρ k − ( δ (Γ) − ρ min ) , k ρ k − ( δ (Γ) −k ρ k ) (cid:3) if δ (Γ) ∈ (cid:2) k ρ k , k ρ k + ρ min (cid:3) ,λ ( Y ) ∈ (cid:2) , k ρ k − ( δ (Γ) −k ρ k ) (cid:3) if δ (Γ) ∈ (cid:2) k ρ k + ρ min , k ρ k (cid:3) . Notice that ρ min = k ρ k in rank one and thus Theorem 1.2 reduces Theorem 1.1.In this paper, we sharpen Theorem 1.2 by considering appropriate Poincaréseries. We first improve in Theorem 2.9 the lower bound of λ ( Y ) by a slightmodification of the classical Poincaré series (1.1). We obtain next in Theorem 3.1the proper analog of Theorem 1.1 by considering a more involved family of Poincaréseries. In the last section, we use these results to restate the existing heat kernelbounds on Y [19]. 2. First improvement
In this section, we replace the Riemannian distance d on X by a polyhedraldistance d ′ , which reflects the volume growth at infinity and which was alreadyused by the first author in [2, 3, 4]. Specifically, let d ′ ( xK, yK ) = h ρ k ρ k , ( y − x ) + i ∀ x, y ∈ G, (2.1)where ( y − x ) + denotes the a + -component of y − x in the Cartan decomposition G = K (exp a + ) K . Proposition 2.1. d ′ is a G -invariant distance on X .Proof. The G -invariance of d ′ is straightforward from the definition (2.1). Thesymmetry d ′ ( xK, yK ) = d ′ ( yK, xK ) follows from ( y − ) + = − w .y + and − w .ρ = ρ, (2.2) SPECTRUM OF THE LAPLACIAN ON LOCALLY SYMMETRIC SPACES 3 where w denotes the longest element in the Weyl group. Let us check the triangularinequality d ′ ( xK, yK ) ≤ d ′ ( xK, zK ) + d ′ ( zK, yK ) . By G -invariance, we may reduce to the case where zK = eK . According to Lemma2.2 below, x + + ( y − ) + − ( y − x ) + belongs to the cone generated by the positive roots. (cid:3) Lemma 2.2.
For every x, y ∈ G , we have the following inclusion co[ W. ( xy ) + ] ⊂ co[ W. ( x + + y + )] (2.3) between convex hulls.Proof. The inclusion (2.3) amounts to the fact that x + + y + − ( xy ) + belongs to the cone generated by the positive roots or, equivalently, to the inequality h λ, ( xy ) + i ≤ h λ, x + i + h λ, y + i ∀ λ ∈ a + . (2.4)It is enough to prove (2.4) for all highest weights λ of irreducible finite-dimensionalcomplex representations π : G −→ GL ( V ) with K -fixed vectors. As usual, consideran inner product on V such that ( π ( k ) is unitary ∀ k ∈ K,π ( a ) is self-adjoint ∀ a ∈ exp a . Then e h λ, ( xy ) + i = k π ( xy ) k ≤ k π ( x ) kk π ( y ) k = e h λ,x + i e h λ,y + i = e h λ,x + + y + i . (cid:3) Remark 2.3.
The distance d ′ is comparable to the Riemannian distance d . Specif-ically, ρ min k ρ k d ( xK, yK ) ≤ d ′ ( xK, yK ) ≤ d ( xK, yK ) ∀ x, y ∈ G. (2.5)This follows indeed from ρ min k ρ k k H k ≤ h ρ k ρ k , H i ≤ k H k ∀ H ∈ a + . The volume of balls B ′ r ( xK ) = { yK ∈ X | d ′ ( yK, xK ) ≤ r } was determined in [2, Lemma 6]. For the reader’s convenience, we recall the state-ment and its proof. Lemma 2.4.
For every x ∈ G and r > , we have ( ) | B ′ r ( xK ) | ≍ ( r n − if r is small, r ℓ − e k ρ k r if r is large. The symbol f ≍ g between two non-negative expressions means that there exist constants < A ≤ B < + ∞ such that Ag ≤ f ≤ Bg . JEAN-PHILIPPE ANKER & HONG-WEI ZHANG
Remark 2.5.
Notice the different large scale behavior, in comparison with theclassical ball volume | B r ( xK ) | ≍ ( r n − if r is small ,r ℓ − e k ρ k r if r is large . (see for instance [17] or [13]). Proof.
By translation invariance, we may assume that x = e . Recall the integrationformula Z X dx f ( x ) = const . Z K dk Z a + dH ω ( H ) f ( k (exp H ) K ) , (2.6)in the Cartan decomposition, with density ω ( H ) = Y α ∈ Σ + (sinh h α, H i ) m α ≍ Y α ∈ Σ + (cid:18) h α, H i h α, H i (cid:19) m α e h ρ,H i . Thus | B ′ r ( eK ) | ≍ Z { H ∈ a + |k H k≤ r } dH Y α ∈ Σ + h α, H i m α ≍ r n if r is small. Let us turn to r large. On the one hand, we estimate from above | B ′ r ( eK ) | . Z { H ∈ a + |h ρ,H i≤k ρ k r } dH e h ρ,H i ≍ Z k ρ k r ds s ℓ − e s ≍ r ℓ − e k ρ k r . On the other hand, let H ∈ a + . As ω ( H ) ≍ e h ρ,H i ∀ H ∈ H + a + , we estimate from below | B ′ r ( eK ) | & Z { H ∈ H + a + |h ρ,H i≤k ρ k r } dH e h ρ,H i & Z k ρ k rC ds s ℓ − e s ≍ r ℓ − e k ρ k r , where C > is a constant depending on H . (cid:3) Consider now the modified Poincaré series P ′ s ( xK, yK ) = X γ ∈ Γ e − sd ′ ( xK,γyK ) ∀ s > , ∀ x, y ∈ G (2.7)associated with d ′ , its critical exponent δ ′ (Γ) = inf { s > | P ′ s ( xK, yK ) < + ∞} (2.8)and the modified orbital counting function N ′ R ( xK, yK ) = |{ γ ∈ Γ | d ′ ( xK, γyK ) ≤ R }| ∀ R ≥ , ∀ x, y ∈ G. (2.9)The following proposition shows that (2.7), (2.8) and (2.9) share the properties oftheir classical counterparts. Proposition 2.6.
The following assertions hold: (i) δ ′ (Γ) does not depend on the choice of x and y . (ii) ≤ δ ′ (Γ) ≤ k ρ k . (iii) For every x, y ∈ G , δ ′ (Γ) = lim sup R → + ∞ log N ′ R ( xK, yK ) R . (2.10) SPECTRUM OF THE LAPLACIAN ON LOCALLY SYMMETRIC SPACES 5
Remark 2.7.
It follows from (2.5) that P s ( xK, yK ) ≤ P ′ s ( xK, yK ) ≤ P ρ min k ρ k s ( xK, yK ) and N R ( xK, yK ) ≤ N ′ R ( xK, yK ) ≤ N k ρ k ρ min R ( xK, yK ) . Hence ≤ δ (Γ) ≤ δ ′ (Γ) ≤ k ρ k ρ min δ (Γ) . Proof. (i) follows from the triangular inequality. More precisely, let x , y , x , y ∈ G and s > . Then d ′ ( x K, γy K ) ≤ d ′ ( x K, x K ) + d ′ ( x K, γy K ) + d ′ ( ✁ γy K, ✁ γy K ) ∀ γ ∈ Γ , hence X γ ∈ Γ e − s d ′ ( x K,γy K ) | {z } P ′ s ( x K,y K ) ≤ e s { d ′ ( x K,x K )+ d ′ ( y K,y K ) } X γ ∈ Γ e − s d ′ ( x K,γy K ) | {z } P ′ s ( x K,y K ) . (ii) Let us show that P ′ s ( eK, eK ) < + ∞ for every s > k ρ k . According to Lemma2.8 below, there exists r > such that the balls B ′ r ( γK ) , with γ ∈ Γ , are pairwisedisjoint in G/K . Let us apply the integration formula (2.6) to the function f ( xK ) = X γ ∈ Γ e − sd ′ ( xK,eK ) B ′ r ( γK ) ( xK ) . On the one hand, as (cid:12)(cid:12) d ′ ( xK, eK ) − d ′ ( γK, eK ) (cid:12)(cid:12) ≤ r ∀ xK ∈ B ′ r ( γK ) , we have Z X d ( xK ) f ( xK ) ≍ X γ ∈ Γ e − sd ′ ( γK,eK ) | B ′ r ( γK ) | | {z } | B ′ r ( eK ) | ≍ P ′ s ( eK, eK ) . On the other hand, Z X d ( xK ) f ( xK ) ≤ Z X d ( xK ) e − sd ′ ( xK,eK ) ≍ Z a + dH ω ( H ) e − s h ρ k ρ k ,H i . Z a + dH e − ( s k ρ k − h ρ,H i is finite if s > k ρ k . Thus P ′ s ( eK, eK ) < + ∞ and consequently δ ′ (Γ) ≤ k ρ k .(iii) Denote the right hand side of (2.10) by L ( xK, yK ) and let us first show that L ( xK, yK ) is finite. By applying Lemma 2.8 below to y − Γ y , we deduce that thereexists r > such that the balls B ′ r ( γyK ) , with γ ∈ Γ , are pairwise disjoint. Set Γ ′ R ( xK, yK ) = { γ ∈ Γ | d ′ ( xK, γyK ) ≤ R } ∀ R ≥ , ∀ x, y ∈ G. Then the ball B ′ R + r ( xK ) contains the disjoint balls B ′ r ( γyK ) , with γ ∈ Γ ′ R ( xK, yK ) .By computing volumes, we estimate N ′ R ( xK, yK ) = | Γ ′ R ( xK, yK ) | ≤ | B ′ R + r ( xK ) || B ′ r ( eK ) | ≍ (1 + R ) ℓ − e k ρ k R . Hence L ( xK, yK ) ≤ k ρ k . Let us next show that L ( xK, yK ) is actually indepen-dent of x, y ∈ G . JEAN-PHILIPPE ANKER & HONG-WEI ZHANG
Given x , y , x , y ∈ G and R > , let R = R + d ′ ( x K, x K ) + d ′ ( y K, y K ) . Then the triangular inequality d ′ ( x K, γy K ) ≤ d ′ ( x K, x K ) + d ′ ( x K, γy K ) + d ′ ( ✁ γy K, ✁ γy K ) implies successively Γ ′ R ( x K, y K ) ⊂ Γ ′ R ( x K, y K ) ,N ′ R ( x K, y K ) ≤ N ′ R ( x K, y K ) ,L ( x K, y K ) ≤ L ( x K, y K ) . Let us finally prove the equality between δ ′ (Γ) and L = L ( eK, eK ) . For thispurpose, observe that P ′ s = 1 + X R ∈ N ∗ X γ ∈ Γ ′ R r Γ ′ R − e − s d ′ ( eK,γK ) ≍ X R ∈ N ∗ (cid:0) N ′ R − N ′ R − (cid:1) e − sR ≍ X R ∈ N N ′ R e − sR , (2.11)where we have written for simplicity P ′ s = P ′ s ( eK, eK ) , Γ ′ R = Γ ′ R ( eK, eK ) and N ′ R = N ′ R ( eK, eK ) . One the one hand, let s > L and set ε = s − L . By definition of L , N ′ R . e ( L + ε ) R ∀ R ≥ . Hence P ′ s . X R ∈ N e − εR < + ∞ . One the other hand, let s < L . By definition of L , there exists a sequence of integers < R < R < · · · → + ∞ such that N ′ R j ≥ e sR j ∀ j ∈ N ∗ . Hence the series (2.11) diverges. (cid:3)
Lemma 2.8.
There exists r > such that the balls B ′ r ( γK ) , with γ ∈ Γ , arepairwise disjoint in G/K .Proof.
Let r > . As Γ is discrete in G , its intersection with the compact subsect G ′ r = { y ∈ G | d ′ ( yK, eK ) ≤ r } = K (cid:0) exp { H ∈ a + | h ρ, H i ≤ k ρ k r } (cid:1) K is finite. Moreover, as Γ is torsion-free, γ + = 0 ∀ γ ∈ Γ \{ e } . Hence there exists r > such that Γ ∩ G ′ r = { e } , which implies that the sets γG ′ r are pairwise disjoint in G . In other words, the balls B ′ r ( γK ) are pairwise disjointin G/K . (cid:3) By using δ ′ (Γ) , we now improve the lower bound in Theorem 1.2. SPECTRUM OF THE LAPLACIAN ON LOCALLY SYMMETRIC SPACES 7
Theorem 2.9.
The following lower bound holds for the bottom λ ( Y ) of the L spectrum of − ∆ on Y = Γ \ G/K : λ ( Y ) ≥ ( k ρ k if ≤ δ ′ (Γ) ≤ k ρ k , k ρ k − ( δ ′ (Γ) − k ρ k ) if k ρ k ≤ δ ′ (Γ) ≤ k ρ k . Next statement is obtained by combining this lower bound with the upper boundin Theorem 1.2.
Corollary 2.10.
The following estimates hold for λ ( Y ) : λ ( Y ) = k ρ k if δ ′ (Γ) ≤ k ρ k , k ρ k − ( δ ′ (Γ) − k ρ k ) ≤ λ ( Y ) ≤ k ρ k if δ (Γ) ≤ k ρ k ≤ δ ′ (Γ) , k ρ k − ( δ ′ (Γ) − k ρ k ) ≤ λ ( Y ) ≤ k ρ k − ( δ (Γ) − k ρ k ) if k ρ k ≤ δ (Γ) . Proof of Theorem 2.9 and Corollary 2.10.
Let us resume the approach in [7, Sec-tion 4] and [15, Section 3]. It consists in studying the convergence of the positiveseries g Γ ζ (Γ xK, Γ yK ) = X γ ∈ Γ g ζ ( Ky − γ − xK ) , (2.12)which expresses the kernel g Γ ζ of ( − ∆ − k ρ k + ζ ) − on the locally symmetric space Y = Γ \ G/K in terms of the corresponding Green function g ζ on the symmetricspace X = G/K . Here ζ > and Γ xK = Γ yK . Recall [5, Theorem 4.2.2] that g ζ (exp H ) ≍ nY α ∈ Σ + red (cid:0) h α, H i (cid:1)o k H k − ℓ − −| Σ + red | e −h ρ,H i− ζ k H k (2.13)for H ∈ a + large, let say k H k ≥ , while g ζ (exp H ) ≍ ( k H k − ( n − if n > k H k if n = 2 for H small, let say < k H k ≤ . Thus (2.12) converges if and only if(2.14) X γ ∈ Γ nY α ∈ Σ + red (cid:0) h α, ( y − γ − x ) + i (cid:1)o ×× d ( xK, γyK ) − ℓ − −| Σ + red | e −k ρ k d ′ ( xK,γyK ) − ζd ( xK,γyK ) converges. Let us compare the series (2.14) with the Poincaré series (1.1) and (2.7).On the one hand, as k ( y − γ − x ) + k = d ( xK, γyK ) , (2.14) is bounded from aboveby P ′k ρ k + ζ ( xK, yK ) . On the other hand, as d ( xK, γyK ) − ℓ − −| Σ + red | & e − εd ( xK,γyK ) for every ε > , (2.14) is bounded from below by P k ρ k + ζ + ε ( xK, yK ) . Hence (2.14)converges if k ρ k + ζ > δ ′ (Γ) , i.e., ζ > δ ′ (Γ) − k ρ k , while (2.14) diverges if ζ <δ (Γ) −k ρ k . We conclude by using the fact [7, Section 4] that λ ( Y ) is the supremumof k ρ k − ζ over all ζ > such that (2.12) converges. (cid:3) JEAN-PHILIPPE ANKER & HONG-WEI ZHANG Second improvement
In this section, we obtain the proper higher rank analog of Theorem 1.1 byconsidering a further family of distances on X , which reflects the large scale behavior(2.13) of the Green function. Specifically, for every s > and x, y ∈ G , let(3.1) d s ( xK, yK ) = min { s, k ρ k} d ′ ( xK, yK ) + max { s − k ρ k , } d ( xK, yK )= ( s d ′ ( xK, yK ) if < s ≤ k ρ k , k ρ k d ′ ( xK, yK ) + ( s − k ρ k ) d ( xK, yK ) if s ≥ k ρ k . Then (3.1) defines a G -invariant distance on X such that s d ′ ( xK, yK ) ≤ d s ( xK, yK ) ≤ s d ( xK, yK ) ∀ s > , ∀ x, y ∈ G. (3.2)Consider the associated Poincaré series P ′′ s ( xK, yK ) = X γ ∈ Γ e − d s ( xK,γyK ) ∀ s > , ∀ x, y ∈ G (3.3)and its critical exponent δ ′′ (Γ) = inf { s > | P ′′ s ( xK, yK ) < + ∞} . It follows from (3.2) that ≤ δ (Γ) ≤ δ ′′ (Γ) ≤ δ ′ (Γ) ≤ k ρ k . (3.4) Theorem 3.1.
The following characterization holds for the bottom λ ( Y ) of the L spectrum of − ∆ on Y = Γ \ G/K : λ ( Y ) = ( k ρ k if ≤ δ ′′ (Γ) ≤ k ρ k , k ρ k − ( δ ′′ (Γ) − k ρ k ) if k ρ k ≤ δ ′′ (Γ) ≤ k ρ k . (3.5) Proof.
In the proof of Theorem 2.9 and Corollary 2.10, we compared the series(2.12), or equivalently (2.14), with the Poincaré series (1.1) and (2.7). If we considerinstead the Poincaré series (3.3), we obtain in the same way that (2.14) is boundedfrom above by P ′′k ρ k + ζ ( xK, yK ) and from below by P ′′k ρ k + ζ + ε ( xK, yK ) , for every ε > . Hence (2.14) converges if ζ > δ ′′ (Γ) − k ρ k , while (2.14) diverges if ζ <δ ′′ (Γ) − k ρ k . We conclude as in the above-mentioned proof. (cid:3) Remark 3.2. If Γ is a lattice, i.e., Y = Γ \ G/K has finite volume, then λ ( Y ) = 0 and δ ′′ (Γ) = 2 k ρ k , hence δ ′ (Γ) = 2 k ρ k . Furthermore δ (Γ) = 2 k ρ k [1, Theorem 7.4].As pointed out by Corlette [7] in rank one and by Leuzinger [14] in higher rank, if G has Kazhdan’s property (T), then the following conditions are actually equivalent:(a) Γ is a lattice, (b) λ ( Y ) = 0 , (c) δ (Γ) = 2 k ρ k , (d) δ ′′ (Γ) = 2 k ρ k .4. Further results about heat kernel bounds
As for the Green function, the heat kernel h Yt (Γ xK, Γ yK ) = X γ ∈ Γ h t ( Ky − γ − xK ) (4.1)on a locally symmetric space Y = Γ \ G/K can be expressed and estimated byusing the heat kernel h t on the symmetric space X = G/K , whose behavior is wellunderstood [6]. The following Gaussian bounds were obtained this way by Davies & Mandouvalos [8] in rank one and by Weber in higher rank [19]. SPECTRUM OF THE LAPLACIAN ON LOCALLY SYMMETRIC SPACES 9
Theorem 4.1.
The following estimates hold for all t > and all x, y ∈ G : (i) Assume that δ (Γ) < ρ min and let δ (Γ) < s < ρ min . Then h Yt (Γ xK, Γ yK ) . t − n (1 + t ) n − D e −k ρ k t e − d (Γ xK, Γ yK )24 t P s ( xK, yK ) , where D = ℓ + 2 | Σ + red | is the so-called dimension at infinity of X . (ii) Assume that ρ min ≤ δ (Γ) < ρ min + k ρ k and let δ (Γ) − ρ min < s < s < k ρ k .Then h Yt (Γ xK, Γ yK ) . t − n e − ( k ρ k − s ) t P ρ min + s ( xK, yK ) . (iii) Assume that δ (Γ) < ρ min + k ρ k . Let s > δ (Γ) and ε > . Then h Yt (Γ xK, Γ yK ) . t − n e − ( λ ( Y ) − ε ) t e − d (Γ xK, Γ yK )24(1+ ε ) t ×× P s ( xK, xK ) P s ( yK, yK ) . By using Theorem 3.1 instead of Theorem 1.1 and Theorem 1.2, we obtain thefollowing alternative statement in term of δ ′′ (Γ) . Theorem 4.2.
The following estimates hold for all t > and all x, y ∈ G : (i) Assume that δ ′′ (Γ) < k ρ k and let δ ′′ (Γ) < s < k ρ k . Then h Yt (Γ xK, Γ yK ) . t − n (1 + t ) n − D e −k ρ k t e − d (Γ xK, Γ yK )24 t P ′′ s ( xK, yK ) . (ii) Assume that k ρ k ≤ δ ′′ (Γ) < k ρ k and let δ ′′ (Γ) − k ρ k < s < s < k ρ k . Then h Yt (Γ xK, Γ yK ) . t − n e − ( k ρ k − s ) t P ′′k ρ k + s ( xK, yK ) . (iii) Assume that δ ′′ (Γ) < k ρ k . Let s > δ ′′ (Γ) and ε > . Then h Yt (Γ xK, Γ yK ) . t − n e − ( k ρ k − ( δ ′′ (Γ) −k ρ k ) − ε ) t e − d (Γ xK, Γ yK )24(1+ ε ) t ×× P ′′ s ( xK, xK ) P ′′ s ( yK, yK ) . We omit the proof , which is straightforwardly adapted from [8] and [19].
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