Heat kernels on metric spaces with doubling measure

Transcription

1 Heat ernels on metric spaces with doubling measure Alexander Grigor yan, Jiaxin Hu and Ka-Sing Lau Abstract. In this survey we discuss heat ernel estimates of self-similar type on metric spaces with doubling measures. We characterize the tail functions from heat ernel estimates in both non-local and local cases. In the local case we also specify the domain of the energy form as a certain Besov space, and identify the wal dimension in terms of the critical Besov exponent. The techniques used include self-improvement of heat ernel upper bound and the maximum principle for wea solutions. All proofs are completely analytic. athematics Subject Classification Primary: 47D07, Secondary: 28A80, 46E35. Keywords. Doubling measure, heat ernel, maximum principle, heat equation. Contents. Introduction 2 2. What is a heat ernel The notion of a heat ernel Heat semigroup and Dirichlet forms Examples 6 3. Auxiliary material on metric measure spaces Besov spaces Doubling condition and reverse doubling condition 8 4. Consequences of heat ernel estimates Consequences of lower bound Consequences of upper bound Wal dimension Consequence of two-sided estimates non-local case 7 5. A maximum principle and its applications Wea differentiation aximum principle for wea solutions Some applications of the maximum principle Upper bounds in the local case Exponential tail Consequences of two-sided estimates local case 27 References 3 AG was supported by SFB 70 of the German Research Council DFG and the Grants from the Department of athematics and IS in CUHK. JH was supported by NSFC Grant No and the HKRGC Grant in CUHK. KSL was supported by the HKRGC Grant in CUHK.

2 2 Grigor yan, Hu and Lau. Introduction The heat ernel is an important tools in modern analysis, which appears to be useful for applications in mathematical physics, geometry, probability, fractal analysis, graphs, function spaces and in other fields. There has been a vast literature devoted to various aspects of heat ernels see, for example, a collection [29]. It is not feasible to give a full-scale panorama of this subject here. In this article, we consider heat ernels on abstract metric measure spaces and focus on the following questions: Assuming that heat ernel satisfies certain estimates of self-similar type, what are the consequences for the underlying metric measure structure? Developing of the self-improvement techniques for heat ernel upper bounds of subgaussian types. Useful auxiliary tools that we develop here include the family of Besov function spaces and the maximum principle for wea solution for abstract heat equation. Some of these questions have been discussed in various settings, for example, in [, 4, 0, 2, 4, 8, 32, 33, 35, 36, 37, 38, 39] for the Euclidean spaces or Riemannian manifolds, in [5, 7, 25] for torus or infinite graphs, in [9, 27, 4] for metric spaces, in [2, 3, 6, 26] for certain classes of fractals. The contents of this paper are based on the wor [20], [2], [22] and [24]. Similar questions were discussed in the survey [9] when the underlying measure is Ahlfors-regular, while the main emphasis in the present survey is on the case of doubling measures. Notation. The sign below means that the ratio of the two sides is bounded from above and below by positive constants. Besides, c is a positive constant, whose value may vary in the upper and lower bounds. The letters C, C, c, c will always refer to positive constants, whose value is unimportant and may change at each occurrence. 2. What is a heat ernel We give the definition of a heat ernel on a metric measure space, followed by some well-nown examples on Riemannian manifolds and on a certain class of fractals. 2.. The notion of a heat ernel Let, d be a locally compact separable metric space and let μ be a Radon measure on with full support. The triple, d, μ is termed a metric measure space. In the sequel, the norm in the real Banach space L p := L p, μ is defined as usual by /p f p := fx dμx p, p <, and f := esup fx, x where esup is the essential supremum. The inner product of f, g L 2 is denoted by f, g. Definition 2.. A family {p t } t>0 of functions p t x, y on is called a heat ernel if for any t > 0 it satisfies the following five conditions:. easurability: the p t, is μ μ measurable in. 2. arovian property: p t x, y 0 for μ-almost all x, y, and p t x, ydμy, 2. for μ-almost all x. 3. Symmetry: p t x, y = p t y, x for μ-almost all x, y. 4. Semigroup property: for any s > 0 and for μ-almost all x, y, p t+s x, y = p t x, zp s z, ydμz. 2.2

3 Heat ernels on metric measure spaces 3 5. Approximation of identity: for any f L 2, p t x, y f y dμ y L2 f x as t 0 +. We say that a heat ernel p t is stochastically complete if equality taes place in 2., that is, for any t > 0, p t x, ydμy = for μ-almost all x. Typically a heat ernel is associated with a arov process {X t } t 0, {P x } x on, so that p t x, y is the transition density of X t, that is, P x X t A = for any Borel set A see Fig.. A p t x, y dμ y x X t A Here are some examples of heat ernels. Figure. arov process X t hits the set A Example 2.2. The best nown example of a heat ernel is the Gauss-Weierstrass function in R n : x y 2 p t x, y = exp. 2.3 n/2 4πt 4t It satisfies all the conditions of Definition 2. provided μ is the Lebesgue measure. This heat ernel is the transition density of the canonical Brownian motion in R n. Example 2.3. The following function in R n p t x, y = C n x y 2 + t n t 2 n+ 2 where C n = Γ n+ 2 /π n+/2 is nown on the one hand as the Poisson ernel, and on the other hand as the density of the Cauchy distribution. It is not difficult to verify that it also satisfies Definition 2. also with respect to the Lebesgue measure and, hence, is a heat ernel. The associated arov process is the symmetric stable process of index. ore examples will be mentioned in the next section Heat semigroup and Dirichlet forms The heat ernel is an integral ernel of a heat semigroup in L 2. A heat semigroup corresponds uniquely to a Dirichlet form in L 2. A Dirichlet form E, F in L 2 is a bilinear form E : F F R defined on a dense subspace F of L 2, which satisfies in addition the following properties: Positivity: E f := E f, f 0 for any f F. Closedness: the space F is a Hilbert space with respect to the following inner product: Ef, g + f, g. 2.4

4 4 Grigor yan, Hu and Lau The arov property: if f F then the function g := min {, max{f, 0}} also belongs to F and E g E f. Here we have used the shorthand E f := E f, f. Any Dirichlet form has the generator L, which is a non-positive definite self-adjoint operator on L 2 with domain D F such that E f, g = Lf, g for all f D and g F. The generator determines the heat semigroup {P t } t 0 defined by P t = e tl. The heat semigroup satisfies the following properties: {P t } t 0 is contractive in L 2, that is P t f 2 f 2 for all f L 2 and t > 0. {P t } t 0 is strongly continuous, that is, for every f L 2, {P t } t 0 is symmetric, that is, P t f L2 f as t 0 +. P t f, g = f, P t g for all f, g L 2. {P t } t 0 is arovian, that is, for any t > 0, if f 0 then P t f 0, and if f then P t f. Here and below the inequalities between L 2 -functions are understood μ-almost everywhere in. The form E, F can be recovered from the heat semigroup as follows. For any t > 0, define a quadratic form E t on L 2 as follows E t f := t f P tf, f. 2.5 It is easy to show that E t f is non-negative and is increasing as t is decreasing. In particular, it has the limit as t 0. It turns out that the limit is finite if and only if f F, and, moreover, cf. [9]. Extend E t to a bilinear form as follows lim E t f = E f t 0+ E t f, g := t f P tf, g. Then, for all f, g F, lim E t f, g = E f, g. t 0+ The arovian property of the heat semigroup implies that the operator P t preserves the inequalities between functions, which allows to use monotone limits to extend P t from L 2 to L and, in fact, to any L q, q. oreover, the extended operator P t is a contraction on any L q cf. [5, p.33]. Recall some more terminology from the theory of the Dirichlet form cf. [5]. The form E, F is called conservative if P t = for every t > 0. The form E, F is called local if Ef, g = 0 for any couple f, g F with disjoint compact supports. The form E, F is called strongly local if Ef, g = 0 for any couple f, g F with compact supports, such that f const in an open neighborhood of supp g. The form E, F is called regular if F C 0 is dense both in F and in C 0, where C 0 is the space of all continuous functions with compact support in, endowed with the sup-norm. For a non-empty open Ω, let FΩ be the closure of F C 0 Ω in the norm of F. It is nown that if E, F is regular, then E, FΩ is also a regular Dirichlet form in L 2 Ω, μ. Assume that the heat semigroup {P t } of a Dirichlet form E, F in L 2 admits an integral ernel p t, that is, for all t > 0 and x, the function p t x, belongs to L 2, and the following identity holds: P t f x = p t x, y f y dμ y, 2.6

5 Heat ernels on metric measure spaces 5 for all f L 2 and μ-a.a. x. Then the function p t is indeed a heat ernel, as we will show below. For this reason, we also call p t the heat ernel of the Dirichlet form E, F or of the heat semigroup {P t }. Observe that if the heat ernel p t of E, F exists, then by 2.5 and 2.6, E t f = fy fx 2 p t x, ydμydμx 2.7 2t + P t x fx 2 dμx 2.8 t for any t > 0 and f L 2. Proposition 2.4. [2] If p t is the integral ernel of the heat semigroup {P t }, then p t is a heat ernel. Proof. We will verify that p t satisfies all the conditions in Definition 2.. Let t > 0 be fixed until stated otherwise. Setting p t,x = p t x,, we see from 2.6 that, for any f L 2, P t fx = p t,x, f for μ-almost all x, whence it follows that the function x p t,x, f is μ-measurable in x. Let {ϕ } = be an orthonormal basis of L 2. Using the identity p t x, y = p t,x y = p t,x, ϕ ϕ y, = we conclude that p t x, y is jointly measurable in x, y, because so are the functions p t,x, ϕ ϕ y. 2 By the arovian property of P t, for any non-negative function f L 2, there is a null set N f such that P t f x 0 for all x \ N f. Let S be a countable family of non-negative functions, which is dense in the cone of all non-negative functions in L 2, and set N = f S so that N is a null set. Then P t f x 0 for all x \ N and for all f S. If f is any other non-negative function in L 2, then f is an L 2 -limit of a sequence {f } S, whence, for any x \ N, p t,x, f = lim p t,x, f = lim P tf x 0. Therefore, for any x \ N, we have that p t,x 0 μ-a.e. in, which proves that p t x, y 0 for μ-a.a. x, y. Let K be compact. Then the indicator function K belongs to L 2 and is bounded by, whence p t x, y dμ y = P t K x K for μ-a.a. x. Choosing an increasing sequence of compact sets {K n } n= that exhausts, we obtain that p t x, y dμ y = lim p t x, y dμ n K n for μ-a.a. x. Consequently, for any compact set K, we obtain by Fubini s theorem p t x, y dμydμx = p t x, y dμ y dμ x K K dμ x = μ K <, which implies that p t x, y L loc. K N f

6 6 Grigor yan, Hu and Lau 3 For all f, g L 2, we have, again by Fubini s theorem, P t f, g = p t x, y f y g x dμydμ x. 2.9 On the other hand, by the symmetry of P t, P t f, g = f, P t g = P t g y f y dμ y = p t y, x f y g x dμydμx. 2.0 Comparing 2.9 and 2.0, we obtain p t x, y = p t y, x for μ-almost all x, y. 4 Using the semigroup identity P t+s = P t P s and Fubini s theorem, we obtain that, for any f L 2 and for μ-a.a. x, whence, for any g L 2, P t+s f, g = Comparing with we obtain 2.2. P t+s f x = P t P s f x = p t x, z p s z, y f y dμ y dμ z = p t x, zp s z, ydμz f y dμ y, P t+s f, g = p t x, zp s z, ydμz f y g x dμydμx. p t+s x, y f y g x dμydμ x, 5 Finally, the approximation of identity property follows immediately from 2.6 and P t f L2 f as t 0. Corollary 2.5. If p t and q t are two integral ernels of a heat semigroup {P t }, then, for any t > 0, Proof. Similarly to 2.9, we have P t f, g = p t x, y = q t x, y for μ-a.a. x, y. 2. Comparing with 2.9, we obtain 2.. q t x, y f y g x dμydμ x. Remar 2.6. Of course, not every heat semigroup possesses a heat ernel. The existence for the heat ernel and results related to these on-diagonal upper bounds can be found in [4, Theorem 2.], [2, Propositions 4.3, 4.4], [8], [9], [], [3], [4, Lemma 2..2], [7], [2], [23], [3], [40], [4], [42], [43] Examples Example 2.7. Let be a connected Riemannian manifold, d be the geodesic distance, and μ be the Riemannian measure. The Laplace-Beltrami operator Δ on can be made into a self-adjoint operator in L 2, μ by appropriately defining its domain. Then Δ generates the heat semigroup P t = e tδ, which is associated with the local Dirichlet form E, F where E f = f 2 dμ, F = W,2 0. The corresponding arov process is a Brownian motion on. It is nown that this {P t } always has a smooth integral ernel p t x, y, which is called the heat ernel of. Although the explicit expression of p t x, y can not be given in general, there are many important classes of manifolds where p t x, y admits certain upper and/or lower bounds.

7 Heat ernels on metric measure spaces 7 For example, as it was proved in [32], if is geodesically complete and its Ricci curvature is non-negative, then p t x, y c V x, t exp dx, y2, x, y, t > 0, 2.2 t where V x, r = μ Bx, r is the volume of the geodesic ball Bx, r = {y : dy, x < r}. Example 2.8. If Δ is a self-adjoint Laplace operator as above then the operator L = Δ β/2 where 0 < β < 2 generates on a arov process with jumps. In particular, if = R n then this is the symmetric stable process of index β, and the corresponding heat ernel admits the following estimate p t x, y + t n/β n+β β x y β. t A particular case β = was already mentioned in Example 2.3. Example 2.9. Let be the Sierpinsi gaset in R n see Fig. 2. Figure 2. Sierpinsi gaset in R 2 It is nown that the Hausdorff dimension of is equal to α := logn + / log 2. Let μ be the α-dimensional Hausdorff measure on, which clearly possesses the same self-similarity properties as the set itself. It is possible to construct also a self-similar local Dirichlet form on which possesses a continuous heat ernel, that is the transition density of a natural Brownian motion on ; moreover, the heat ernel admits the following estimate p t x, y exp tα/β β/β dx, y c, 2.3 where β = logn + 3/ log 2 is the wal dimension see [6], [6], [30]. Similar results hold also for a large family of fractal sets, including p.c.f. fractals and the Sierpinsi carpet in R n see [30] and [3], but with different values of α and β. t /β 3. Auxiliary material on metric measure spaces Fix a metric measure space, d, μ and define its volume function V x, r by V x, r := μ B x, r where x and r > 0. For the bacground of fractal sets including the notion of the Sierpinsi gaset, see [2].

8 8 Grigor yan, Hu and Lau 3.. Besov spaces Here we introduce function spaces W β/2,p on. Choose parameters p <, β > 0 and define the functional E β,p u for all functions u L p as follows: E β,p u = sup r pβ/2 0<r For simplicity, if p = 2, denote it by The Besov space W β/2,p is defined by with the norm [ V x, r Bx,r E β u := E β,2 u. uy ux p dμy ] dμx. 3. W β/2,p := {u L p : E β,p u < } 3.2 u W β/2,p := u p + E β,p u /p. For Ahlfors regular 2 measures μ, the Besov space W β/2,p was introduced in [28, 34, 22] although using different notation. It is not difficult to verify that for any p < and β > 0, the space W β/2,p is a Banach space. Note that the space W β/2,p decreases as β increases; it may happen that this space becomes trivial for large enough β. For example, W β/2,2 R n = {0} for β > 2. Define the critical Besov exponent β by { } β := sup β > 0 : W β/2,2 is dense in L 2, μ. 3.3 Lemma 3.. We have β 2. Proof. It suffices to show that W,2 is dense in L 2 = L 2, μ. Let u be a Lipschitz function with a bounded support A and let A r be the closed r-neighborhood of A. If L is the Lipschitz constant of u, then E 2 u = sup r 2 uy ux 2 dμydμx 0<r A r V x, r Bx,r sup r 2 L 0<r A 2 r 2 dμx r L 2 μ A. It follows that E 2 u < and hence u W,2. We are left to show that the class Lip of all Lipschitz functions with bounded supports is dense in L 2. Indeed, let now A be any bounded closed subset of. For any positive integer n, consider the function on f n x = nd x, A +, which is Lipschitz and is supported in A /n. Clearly, f n A in L 2 as n, whence it follows that A Lip, where the bar means the closure in L 2. Since the linear combinations of the indicator functions of bounded closed sets form a dense subset in L 2, it follows that Lip = L 2, which was to be proved Doubling condition and reverse doubling condition The measure μ on is said to be doubling if there is a constant C D such that for all x and r > 0. V x, 2r C D V x, r 3.4 Proposition 3.2. If 3.4 holds on, then there exists α > 0 depending only on the doubling constant C D such that α V x, R dx, y + R V y, r C D for all x, y and 0 < r R. VD r 2 A measure μ on a metric space, d is said to be Ahlfors-regular if there exist α, c > 0 such that V x, r r α for all balls Bx, r in with r 0,.

9 Heat ernels on metric measure spaces 9 Hence, the inequality of Proposition 3.2 can be used as an alternative definition of the doubling property of μ and will be referred to as V D volume doubling. The advantage of this definition is that it introduces a parameter α that will frequently be used. Proof. If x = y, then R 2 n r where whence, it follows from 3.4 that n = [ ] R R log 2 log r 2 r +, V x, R V x, r V x, 2n log2 C r C D n C D log 2 R r + R D = C D. 3.5 V x, r r If x y, then B x, R B y, R + r 0 where r 0 = d x, y. By 3.5, V x, R V y, r V y, R + r log2 C 0 R + D r0 C D, V y, r r which finishes the proof. The measure μ satisfies a reverse volume doubling condition if there exist positive constants α and c such that α V x, R R V x, r c for all x and 0 < r R. RVD r Proposition 3.3. If, d is connected and μ satisfies 3.4, then there exist positive constants α and c such that RVD holds, provided B x, R c is non-empty. Proof. The condition B x, R c implies that B x, ρ \ B x, ρ 3.6 for all 0 < ρ < R and ρ > ρ. Indeed, otherwise splits into disjoint union of two open sets: B x, ρ and B x, ρ c. Since is connected, the set B x, ρ c must be empty, which contradicts the non-emptiness of B x, R c. If 0 < ρ R/2, then we have by 3.6 B x, 53 ρ \ B x, 43 ρ. Let y be a point in this annulus. It follows from VD that for some constant C > 0, whence V x, ρ CV y, ρ/3 V x, 2ρ V x, ρ + V y, ρ/3 + ε V x, ρ, 3.7 where ε = C. For any 0 < r R, we have that 2 n r R where [ ] R R n := log 2 log r 2 r. For any 0 n, we have 2 r R/2, and whence by 3.7, Iterating this inequality, we obtain V x, 2 + r + ε V x, 2 r. V x, R V x, r V x, 2n r + ε n V x, r + ε log 2 R r = + ε R r log2 +ε, thus proving RVD.

10 0 Grigor yan, Hu and Lau Remar 3.4. As one can see from the argument after 3.7, the measure μ is reverse doubling whenever the following inequality holds for some C >, ε > 0 and all x, r > 0. V x, Cr + ε V x, r 3.8 Corollary 3.5. Assume that, d is connected and μ satisfies VD. Then μ = diam = RVD. Proof. If μ =, then diam = ; indeed, otherwise would be a ball of a finite radius and its measure would be finite by VD. If diam =, then B c x, R for any ball B x, R, and RVD holds by Proposition 3.3. Finally, RVD implies μ = by letting R in RVD. 4. Consequences of heat ernel estimates We give here some consequences of the heat ernel estimates dx, y V x, t /βφ p t /β t x, y V x, t /βφ 2 dx, y for all t > 0 and μ-almost all x, y. Functions Φ s and Φ 2 s are always assumed to be non-negative and monotone decreasing on [0, +, the constant β is positive. We prove that The lower estimate of the heat ernel implies that the measure μ is doubling; the space F is embedded in W β/2,2 ; the lower tail function Φ s is controlled from above by a negative power of s. The upper estimate of the heat ernel implies that the space W β/2,2 is embedded in F; if the Dirichlet form is non-local then the upper tail function Φ 2 s is controlled from below by a negative power of s for large s. 4.. Consequences of lower bound Let p t be a heat ernel on a metric measure space, d, μ. Consider the lower estimate of p t of the form: dx, y p t x, y V x, t /βφ 4. t /β for all t > 0 and μ-almost all x, y. Lemma 4.. Assume that the heat ernel p t satisfies the lower bound 4.. If Φ s 0 > 0 for some s 0 >, then μ is doubling. Proof. Fix r, t > 0 and consider the following integral p t x, ydμy := p t x, y Bx,r y dμy. Bx,r The right-hand side is understood as follows: the function F x, y := p t x, y Bx,r y is measurable jointly in x, y so that, by Fubini s theorem, the integral p t x, y Bx,r y dμy is well-defined for μ-almost all x and is a measurable function of x. Choose any pointwise version of p t x, y as a function of x, y. By Fubini s theorem, there is a subset 0 of full t /β,

11 Heat ernels on metric measure spaces measure such that, for any x 0, the function p t x, y is measurable in y and the inequalities 4. and 2. hold for μ-a.a. y. It follows that, for all x 0, p t x, ydμy 4.2 whence Bx,r V x, r einf p tx, y. y Bx,r On the other hand, we have by 4. einf p r tx, y y Bx,r V x, t /β Φ, t /β which together with the previous estimate gives Setting here t = r/s 0 β we obtain V x, r V x, t /β Φ r/t /β. V x, r V x, r/s 0 Φ s Since s 0 > and Φ s 0 > 0, 4.3 implies that measure μ is doubling. Lemma 4.2. Assume that the heat ernel p t satisfies the lower bound 4. with Φ s 0 > 0 for some s 0. Then, there is a constant c > 0 such that for all u L 2, Consequently, the space F embeds into W β/2,2. Eu c E β u. 4.4 Proof. Let t, r > 0. It follows from 2.7 and the lower bound 4. that Eu E t u uy ux 2 p t x, ydμydμx 2t Bx,r r 2t Φ uy ux t 2 dμy dμx, /β V x, t /β Bx,r where we have used the monotonicity of Φ. Choosing t = r/s 0 β and noticing that V x, r/s 0 V x, r by s 0, we obtain Eu sβ 0 2r β Φ s 0 uy ux 2 dμy dμx, V x, r Bx,r whence, by taing supremum in r, Eu 2 sβ 0 Φ s 0 E β u, thus proving 4.4. Finally, we give another consequences of the lower bound 4. of the heat ernel. Lemma 4.3. Assume that the heat ernel p t satisfies the lower bound 4.. If μ satisfies the reverse doubling property RVD, then there is c > 0 such that where α is the same as in RVD. Φ s c + s α +β for all s > 0, 4.5

12 2 Grigor yan, Hu and Lau { } { } Figure 3. Sets A and B Proof. Following [24], let u L 2 be a non-constant function. Choose a ball Bx 0, R where u is non-constant and let a > b be two real values such that the sets A = {x Bx 0, R : ux a} and B = {x Bx 0, R : ux b} both have positive measure see Fig. 3. It follows from 2.7 that Eu uy ux 2 p t x, ydμydμx 2t a b 2 2R 2t A B V x, t /β Φ dμydμx t /β a b2 2R = Φ μb 2t t /β V x, t /β dμx. For x A, we have that Bx, R Bx 0, 3R, and hence, for small enough t > 0, α V x, t /β = V x, R V x, R V x, t /β R V x 0, 3R c, t /β where we have used the reverse doubling property RVD. Therefore, for small t > 0, α +β. Eu c a b 2 2R V x 0, 3RR β μaμb t /β A Φ 2R t /β If 4.5 fails, then there exists a sequence {s } with s as such that Choose t such that s = 2R/t /β. Then α +β 2R Φ t /β s α +β Φ s as. 2R t /β = s α +β Φ s as, and hence Eu =. Hence, we see that F consists only of constants. Since F is dense in L 2, it follows that L 2 also consists of constants only. Hence, there is a point x with a positive mass, that is, μ {x} > 0. Then 2. implies that, for all t > 0, p t x, x μ{x}. 4.6 However, by RVD, we have V x, r 0 as r 0, which together with 4. implies that p t x, x as t 0, thus contradicting 4.6.

13 Heat ernels on metric measure spaces 3 Remar 4.4. The last argument in the above proof can be stated as follows. If RVD is satisfied and 4. holds with a function Φ such that Φ 0 > 0, then μ {x} = 0 for all x. This simple observation will also be used below Consequences of upper bound Consider the upper estimate of p t of the form: p t x, y for all t > 0 and μ-almost all x, y. V x, t /βφ 2 dx, y t /β 4.7 Lemma 4.5. Assume that μ satisfies both VD and RVD, and that the heat ernel p t is stochastically complete and satisfies the upper bound 4.7 with 0 s α+β Φ 2 sds <, 4.8 where α is the same as in VD. Then, there is a constant c > 0 such that for all u L 2, Consequently, the space W β/2,2 embeds into F. Eu CE β u. 4.9 Proof. Fix t 0, and let n be the smallest negative integer such that 2 n+ t /β. Since p t is stochastically complete, we have that for any t > 0, E t u = ux uy 2 p t x, ydμydμx = A 0 t + A t + A 2 t 4.0 2t where A 0 t : = 2t A t : = 2t A 2 t : = 2t Observing that by VD ux uy 2 p t x, ydμydμx, 4. Bx, c ux uy 2 p t x, ydμydμx, 4.2 Bx,\Bx,2 n Bx,2 n V x, 2 + V x, t /β C ux uy 2 p t x, ydμydμx α for all n, 4.4 and using 4.7, we obtain 2 p t x, ydμy Bx, c =0 Bx,2 + \Bx,2 V x, t /β Φ 2 t /β V x, C V x, t /β Φ 2 t /β =0 2 C α 2 Φ 2 =0 C ct t /β t /β t /β dμy 2 t /β s α Φ 2 sds 4.5 s α+β Φ 2 sds t /β

14 4 Grigor yan, Hu and Lau Applying the elementary inequality a b 2 2a 2 + b 2, we obtain from 4. A 0 t t = 2 t ux 2 + uy 2 p t x, ydμydμx Bx, c ux 2 Bx, c p t x, ydμy where we have used 4.8. It follows that By 4.7 and 4.4, we obtain that, for 0 > n, Bx,2 + \Bx,2 dμx 2c u 2 2 s α+β Φ 2 sds 2 t /β = o u 2 2 as t 0, V x, t /β Φ 2 t /β 2 + α 2 c Φ 2 t /β lim t 0+ A 0t = ux uy 2 p t x, ydμy t /β By the definition 3. of E β, for all < 0, Therefore, we obtain ux uy 2 dμy Bx,2 + \Bx,2 V x, 2 + Bx,2 + ux uy 2 dμy. V x, 2 + ux uy 2 dμydμ x 2 + β Eβ u. 4.9 Bx,2 + A t = ux uy 2 p t x, ydμydμx 2t n <0 Bx,2 + \Bx,2 2 + α 2 c Φ 2t t /β 2 t /β n <0 V x, 2 + ux uy 2 dμydμ x 4.20 Bx,2 + c 2 + α+β 2 Φ 2 E β u t /β n <0 ce β u where the latter integral converges due to t /β For < n, we have 2 + < t /β whence by RVD s α+β Φ 2 sds, 4.2 V x, V x, t /β c t /β α. 4.22

15 Heat ernels on metric measure spaces 5 Similarly to the estimate of A, we obtain A 2 t = 2t <n c <n ce β u Bx,2 + \Bx,2 2 + α +β 2 Φ 2 t /β 2 0 t /β ux uy 2 p t x, ydμydμx E β u s α +β Φ 2 sds, 4.23 where the latter integral converges at 0 due to α + β > 0. It follows from 4.0, 4.8, 4.2 and 4.23 that Eu = lim E t u = lim A 0 t + A t + A 2 t CE β u, t 0+ t 0+ which finishes the proof. Lemma 4.6. Assume that μ satisfies VD and that the heat ernel p t satisfies the upper bound 4.7. Then, either E, F is local, or there is c > 0 such that Φ 2 s c + s α+β for all s > Proof. Let u, v F be functions with disjoint compact supports A = supp u and B = supp v see Fig. 4. A R B Figure 4. Functions u and v Noticing that u, v = 0, we obtain, for any t > 0, E t u, v = t u, v P tv = t u, P tv = ux t Setting R = da, B > 0 and using 4.7, we obtain E t u, v R t Φ 2 v t /β A B vyp t x, ydμy dμx. A ux dμx V x, t /β Choose any fixed point x 0 A and let diama = r. Then, using VD, we see that, for all x A and small t > 0, Therefore, by 4.25, V x, t /β = V x 0, r E t u, v t Φ 2 = c V x 0, r V x 0, r V x, t /β α dx0, x + r t /β R v t /β c2r α V x 0, rr α+β u v α c 2r. V x 0, r t /β α c 2r u V x 0, r t /β R t /β α+β Φ 2 R t /β.

16 6 Grigor yan, Hu and Lau If 4.24 fails, then there exists a sequence {s } such that s as, and s α+β Φ 2 s 0. Letting t > 0 such that s = R/t /β, we obtain that E t u, v 0 as, showing that Eu, v = 0. Hence, the E, F is local, which was to be proved Wal dimension Here we obtain certain consequence of a two-sided estimate dx, y V x, t /βφ p t /β t x, y V x, t /βφ 2 dx, y t /β The parameter β from 4.26 is called the wal dimension of the associated arov process. Theorem 4.7. Assume that μ satisfies both VD and RVD. Let the heat ernel p t x, y be stochastically complete and satisfy 4.26 where Φ s 0 > 0 for some s 0 and 0 s α+β+ε Φ 2 s ds s < 4.27 for some ε > 0. Then β = β where β is the critical Besov exponent defined in 3.3. Remar 4.8. Assuming in addition that Φ s 0 > 0 for some s 0 > allows to drop VD from the hypothesis, thans to Lemma 4.. If one assumes on top of that, that the metric space, d is connected and has infinite diameter then RVD follows from VD by Corollary 3.5. Hence, in this case RVD can be dropped from the assumptions as well. Proof. By Lemma 4.2, we have the inclusion F W β/2,2. Since F is always dense in L 2, we conclude that W β/2,2 is dense in L 2, whence β β. To prove the opposite inequality, it suffices to verify that, for any β > β, the space W β /2,2 is not dense in L 2. We can assume that β β is sufficiently small so that the condition 4.27 holds with ε = β β. Let us show that u W β /2,2 implies E u = 0. We use again the decomposition E t u = A 0 t + A t + A 2 t where A i t are defined in As in the proof of Lemma 4.5, we have lim A 0 t = 0. t 0 Let us estimate A t similarly to the proof of Lemma 4.5 and using the same notation, but use E β instead of E β. Indeed, using instead of 4.9 the inequality V x, 2 + ux uy 2 dμydμ x 2 + β E β u, 4.28 we obtain from 4.20 that Bx,2 + A t ct β β 2 + α+β 2 Φ 2 t /β n <0 ct β β E β u 0 t /β E β u where the integral converges due to In the same way, one obtains A 2 t ct β β E β u Putting together all the estimates, we obtain s α+β Φ 2 sds s α +β Φ 2 sds. E t u A 0 t + Ct β β E β u 0 as t 0,

17 Heat ernels on metric measure spaces 7 whence E u = lim E t u = 0. t 0 Since E t u E u, this implies bac that E t u 0 for all t > 0. On the other hand, it follows from 2.7 and the lower bound in 4.26 that E t u uy ux 2 p t x, ydμydμx 2t Φ s 0 2t {dx,y s 0 t /β } {dx,y s 0 t /β } ux uy 2 V x, t /β dμydμx, which yields ux = uy for μ-almost all x, y such that dx, y s 0 t /β. Since t is arbitrary, we conclude that u is a constant function. Hence, we have shown that the space W β /2,2 consists of constants. However, it follows from Remar 4.4 that the constant functions are not dense in L 2, which finishes the proof Consequence of two-sided estimates non-local case Lemmas 4.3 and 4.6 of the previous subsections imply immediately the following. Theorem 4.9. Assume that the metric measure space, d, μ satisfies VD and RVD. Let {p t } be a heat ernel on such that, for all t > 0 and almost all x, y, C V d x, y C x, t /βφ 2 C p t /β t x, y V d x, y x, t /βφ C t /β where C, C, C 2, C 2 are positive constants. Then either the associated Dirichlet form E is local or c + s α+β Φ s c 2 + s α +β 4.3 for all s > 0 and some c, c 2 > 0, where α and α are the exponents from VD and RVD, respectively. 5. A maximum principle and its applications 5.. Wea differentiation Let H be a Hilbert space over R and I be an interval in R. We say that a function u : I H is wealy differentiable at t I if for any ϕ H, the function u, ϕ is differentiable at t where the outer bracets stand for the inner product in H, that is, the limit lim ε 0 u t + ε u t, ϕ ε exists. In this case it follows from the principle of uniform boundedness that there is w H such that u t + ε u t lim, ϕ = w, ϕ ε 0 ε for all ϕ H. We refer to the vector w as the wea derivative of the function u at t and write w = u t. Of course, we have the wea convergence u t + ε u t u t as ε 0. ε In the next statement, we collect the necessary elementary properties of wea differentiation. Lemma 5.. i If u is wealy differentiable at t then u is strongly that is, in the norm of H continuous at t. ii The product rule If functions u : I H and v : I H are wealy differentiable at t, then the inner product u, v is also differentiable at t and u, v = u, v + u, v.

18 8 Grigor yan, Hu and Lau iii The chain rule Let, μ be a measure space and set H = L 2, μ. Let u : I L 2, μ be wealy differentiable at t I. Let Φ be a smooth real-valued function on R such that Φ 0 = 0, sup R Φ <, sup Φ <. 5. R Then the function Φ u : I L 2, μ is also wealy differentiable at t and Φ u = Φ u u. Proof. To shorten the notation, we write u t for u t. i It suffices to verify that, for any sequence {ε } 0, we have u t+ε u t 0 as. 5.2 The sequence u t+ε u t ε converges wealy, whence it follows that it is wealy bounded and, hence, also strongly bounded. The latter clearly implies 5.2. ii Let {ε } be as above. We have the identity u t+ε, v t+ε u t, v t ut+ε u t =, v t ε ε + u t, v t+ε v t ε ε ut+ε u t +, v t+ε v t. By the definition of the wea derivative, the first two terms in the right hand side converge to u t, v t and u t, v t respectively. By part i, the sequence ut+ε ut ε is bounded in norm, whereas v t+ε v t 0 as ; hence, the third term goes to 0, and we obtain the desired. iii By 5. the function Φ admits the estimate Φ r C r for all r R, which implies that the function Φ u t belongs to L 2, μ for any t I. By the mean value theorem, for any r, s R, there exists ξ r,s 0, such that Φ r + s Φ r = Φ r + ξ r,s r s s. We have then Φ u t+ε Φ u t = Φ u t + ξ ε u t+ε u t u t+ε u t, ε where we write for simplicity ξ := ξ ut,u t+ε u t. Rewrite this in the form where and Φ u t+ε Φ u t ε = A + B, A = Φ u t + ξ u t+ε u t Φ u t u t+ε u t ε B = Φ u t u t+ε u t. ε Since Φ u t + ξ u t+ε u t Φ u t L 2 sup Φ u t+ε u t L 2 0 and the norm ut+ε ut L ε is uniformly bounded, we obtain that A 2 L 0 as. Since Φ u t + ξ u t+ε u t Φ u t L 2 sup Φ <, we see that the norm A L 2 is uniformly bounded. It follows that A 0 wealy in L 2. We are left to verify that B Φ u t u t wealy in L 2. Indeed, for any ϕ L 2, μ, we have that ut+ε u t B, ϕ =, Φ u t ϕ u t, Φ u t ϕ = Φ u t u t, ϕ, ε where we have used the fact that Φ u t is bounded and, hence, Φ u t ϕ L 2, μ.

19 Heat ernels on metric measure spaces aximum principle for wea solutions As before, let E, F be a regular Dirichlet form in L 2, μ. Consider a path u : I F. We say that u is a wea subsolution of the heat equation in an open set Ω if u is wealy differentiable in the space L 2 Ω at any t I and, for any non-negative ϕ F Ω, u, ϕ + E u, ϕ Similarly one defines the notions of wea supersolution and wea solution. A similar definition was introduced in [20] but with the difference that the time derivative u was understood in the sense of the norm convergence in L 2 Ω. Let us refer to the solutions defined in [20] as semi-wea solutions. Clearly, any semi-wea solution is also a wea solution. It is easy to see that, for any f L 2, the function P t f is a wea solution in 0, Ω for any open Ω cf. [20, Example 4.0]. Proposition 5.2 parabolic maximum principle. Let u be a wea subsolution of the heat equation in 0, T Ω, where T 0, + ] and Ω is an open subset of. Assume in addition that u satisfies the following boundary and initial conditions: u + t, FΩ for any t 0, T ; u + t, L2 Ω 0 as t 0. Then ut, x 0 for any t 0, T and μ-almost all x Ω. Remar 5.3. For semi-wea solutions the maximum principle was proved in [20]. Remar 5.4. It was shown in [20, Lemma 4.4] that the condition u + F Ω is equivalent to the following: u F and u v for some v FΩ. We will use this result to verify the boundary condition of the parabolic maximum principle. Proof. Let Φ be a smooth function on R that satisfies the following conditions for some constant C: i Φ r = 0 for all r 0; ii 0 < Φ r C for all r > 0. iii Φ r C for all r > 0. Then Φ u = Φ u + F Ω so that we can set ϕ = Φ u in 5.3 and obtain u, Φ u + E u, Φ u 0. Since Φ is increasing and Lipschitz, we conclude by [20, Lemma 4.3] that E u, Φ u 0, whence it follows that u, Φ u Since Φ satisfies the conditions 5., we conclude by Lemma 5., that the function t Φ u is wealy differentiable in the space L 2 Ω and and u, Φ u is differentiable in t and Set Ψ r = Φ r r so that Φ u = Φ u u, u, Φ u = u, Φ u + u, Φ u = u, Φ u + u, Φ u u = u, Φ u + u, Φ u u. u, Φ u = u, Φ u + u, Ψ u. Assume for a moment that function Ψ also satisfies the above properties i-iii. Applying 5.4 to Ψ, we obtain from the previous line u, Φ u 0, that is, the function t u, Φ u is decreasing in t. By the properties i-ii, we have Φ r Cr for r > 0, which implies u, Φ u = u +, Φ u + C u as t 0.

20 20 Grigor yan, Hu and Lau Hence, the function t u +, Φ u + is non-negative, decreasing and goes to 0 as t 0, which implies that this function is identical 0. It follows that u + = 0, which was to be proved. We are left to specify the choice of Φ so that the function Ψ r = Φ r r is also in the class i-iii. Let us construct Φ from its derivative Φ that can be chosen to satisfy the following: Φ r = 0 for r 0; Φ r = for r ; Φ r > 0 for r 0,. see Fig. 5. Φr 0 r Figure 5. Function Φ r Clearly, Φ satisfies i-iii. It follows from the identity Ψ r = Φ r r + Φ r that Ψ r is bounded and Ψ r > 0 for r > 0. Finally, we have Ψ r = Φ r r + 2Φ r whence it follows that Ψ r = 0 for large enough r and, hence, Ψ is bounded. We conclude that Ψ satisfies i-iii, which finishes the proof Some applications of the maximum principle Recall that if E, F is a regular Dirichlet form in L 2, μ then, for any open set Ω, E, F Ω is also a regular Dirichlet form in L 2 Ω, μ. Denote by Pt Ω the heat semigroup of E, FΩ. Lemma 5.5. Assume that E, F is a regular and local Dirichlet form. Let u t, x be a wea subsolution of the heat equation in 0, U, where U is an open subset of. Assume further, for any t > 0, u t, is bounded in and is non-negative in U. If u t, L2 U 0 as t then the following inequality holds for all t > 0 and almost all x U: u t, x Pt U U x sup u s, L U <s t Proof. We first assume that U is precompact. Choose an open set W such that W U. Fix a real T > 0 and set m := We show that, for all 0 < t T and μ-almost all x W, sup u s, L U <s T u t, x m P W t W x. 5.8

21 Heat ernels on metric measure spaces 2 Let ζ and η be cut-off functions 3 of the couples W, U and U,, respectively. Consider the function w := ζu m [ η P W t W ]. 5.9 Then 5.8 will follow if we prove that w 0 in 0, T ] W. Claim. The w is a wea subsolution of the heat equation in 0, W. Clearly, Pt W W is a wea solution of the heat equation in 0, W. Let us show that so is ζu. Indeed, the product ζu belongs to F because both ζ and u are in L F. For any test function ψ FW, we have, using ζψ ψ, ζu, ψ = ζ t t u, ψ = t u, ψ = E u, ψ = E ζu, ψ + E ζ u, ψ = E ζu, ψ, where we have used also that ζ u = 0 in W and, hence, E ζ u, ψ = 0, by the locality of E, F. Thus, ζu is a wea solution in 0, W. Finally, the function η x considered as a function of t, x, is a wea supersolution of the heat equation in 0, W, since for any non-negative ψ FW Eη, ψ = lim t 0 t η P t η, ψ = lim t 0 t P t η, ψ 0, whence it follows that w is a wea subsolution. Claim 2. For every t 0, T ], we have w t, + FW. By Remar 5.4, it suffices to prove that in 0, T ] because mp W t w t, mp W t W, 5.0 W F W. In \ U, inequality 5.0 holds trivially because ζ = 0 = P W t W in \ U and, hence, w = mη 0. To prove 5.0 in U, observe that η = in U and 0 u m in 0, T ] U, whence w = ζu m + mp W t W u m + mp W t W mp W t W, which was to be proved. Claim 3. The function w satisfies the initial condition Noticing that η = in W, we see that w t, L2 W 0 as t η P W t W = W P W t W L 2 W 0 as t 0. Combining with 5.5, we obtain 5.. By the parabolic maximum principle cf. Prop. 5.2, we obtain from Claims -3 that w 0 in 0, T ] W, thus proving 5.8. Finally, let U be an arbitrary open subset of. Let {W i } i= and {U i} i= be two increasing sequences of precompact open sets, both of which exhaust U, and such that W i U i for all i. For each i, we have by 5.8 with t = T that in W i [ ] u P W i t Wi sup u s, L U. 5.2 i 0<s t Replacing by the monotonicity in the right hand side U i by U, and noticing that P W i t Wi a.e. P U t U as i, 3 A cut-off function for the couple W, U is a function ζ F C 0 such that 0 ζ in, ζ = on an open neighborhood of W, and supp ζ U. If E, F is a regular Dirichlet form then a cut-off function exists for any couple W, U provided U is open and W is a compact subset of U cf. [5, p.27].

22 22 Grigor yan, Hu and Lau we obtain 5.6 by letting i in 5.2. Corollary 5.6. Assume that E, F is regular and strongly local. Let U Ω be two open subsets of. Then the following inequality holds for all t > 0 and μ-almost all x U: Pt Ω Ω x Pt U U x sup P Ω s L Ω <s t U Proof. Approximating U by precompact open subsets, it suffices to prove the claim in the case when U Ω. Let ϕ be a cut-off function of the couple U, Ω. Then we can replace the term P Ω t Ω x in the both sides of 5.3 by the function u t, x = ϕ x P Ω t Ω x. Clearly, for any t > 0, the function u t, is bounded in, non-negative in U, and satisfies the initial condition 5.5. Let us verify that u t, x is a wea solution of the heat equation in 0, U. It suffices to show that the function ϕ x as a function of t, x is a wea solution in 0, U. Indeed, since ϕ is constant in a neighborhood of U, the strong locality of E, F yields that Eϕ, ψ = 0 for any ψ FU, which finishes the proof. 6. Upper bounds in the local case 6.. Exponential tail Following [2], we give an analytical approach of how to obtain the exponential tail of the heat ernel upper bound on the doubling space. This is a modification of the argument of [27]. For an alternative approach see [20] and [25] for the case of infinite graphs. The following is a ey technical lemma. Lemma 6.. Let E, F be a regular, strongly local Dirichlet form in L 2, μ. Let ρ : [0, [0, be an increasing function. Assume that there exist ε 0, 2 and δ > 0 such that, for any ball B of radius r and any positive t such that ρ t δr, P t B c ε in B Then, for any t > 0 and any ball B of radius r > 0, cr P t B c C exp c t Ψ t in B, where C, c, c > 0 are constants depending on ε, δ, and function Ψ is defined by { } s Ψs := sup λ>0 ρ/λ λ for all s Remar 6.2. Letting λ 0 in 6.3, one sees that Ψ s 0 for all s 0. It is also obvious from 6.3 that Ψ s is increasing in s. Remar 6.3. If ρt = t /β for β >, then { } Ψs = sup sλ /β λ λ>0 = c β s β/β for all s 0, where c β > 0 depends only on β the supremum is attained for λ = s/β β β. The estimate 6.2 becomes r β β P t B c C exp c in t 4 B.

23 Heat ernels on metric measure spaces 23 Remar 6.4. If the heat semigroup P t possesses the heat ernel p t x, y then the condition 6. can be equivalently stated as follows: If ρ t δr then, for almost all x, Bx,r c p t x, y dμ y ε. 6.4 Indeed, for any ball B x 0, r and for almost all x B x 0, r/4, we have P t Bx0,r Bx c x = p t x, y dμ y p t x, y dμ y, 0,r c Bx,r/2 c so that 6.4 implies 6. although with a different value of δ. Conversely, for almost all x B x 0, r/2, p t x, y dμ y p t x, y dμ y = P t Bx0,r/2 c x, Bx,r c Bx 0,r/2 c so that 6. implies 6.4, for almost all x B x 0, r/8. Covering by a countable family of balls of radius r/8, we obtain that 6.4 holds for almost all x. In the same way, the condition 6.2 is equivalent to the following: For all λ, t, r > 0 and for almost all x, cr p t x, y dμ y C exp c t Ψ. 6.5 Bx,r c t Hence, Lemma 6. in the presence of the heat ernel can be stated as follows: If 6.4 holds for some ε 0, /2, δ > 0 and all r, t > 0 such that ρ t δr then 6.5 holds for all r, t > 0. Proof of Lemma 6.. Let us first show that the hypothesis 6. implies that there exist ε 0, and δ > 0 such that, for any ball B of radius r > 0 and for any positive t such that ρ t δr, Pt B B ε in B Indeed, applying [2, Proposition 4.7] we obtain that, for all t and almost everywhere in, Pt B L 2 B P t 2 B sup P s 2 B 3 4 Bc <s t For any x 4 B, we have that Bx, r/4 2B see Fig. 6. B=Bx 0,r 3 /4 B /2 B Bx, /6 r x /4 B x 0 By, /6 r Bx, /4 r y By, /4 r Figure 6. Illustration to the proof of

24 24 Grigor yan, Hu and Lau Using the identity P t = we obtain, for any x 4 B, Applying 6. for the ball B x, r/4, we obtain provided t satisfies P t 2 B = P t 2 Bc P t Bx,r/4 c. P t Bx,r/4 c ε in B x, r/6, ρ t δ r It follows that, for any x 4 B, P t 2 B ε in B x, r/6, whence P t 2 B ε in B On the other hand, for any y 3 4 B c, we have 2 B B y, r/4c see Fig. 6, whence P s 2 B P s By,r/4 c. Applying 6. for the ball B y, r/4 at time s, we obtain if 6.8 holds then, for all 0 < s t, P s By,r/4 c ε in B y, r/6. It follows that, for any y 3 4 B c, P s 2 B ε in B y, r/6, whence c 3 P s 2 B ε in 4 B. 6.0 Combining 6.7, 6.9 and 6.0, we obtain that, under condition 6.8, Pt B B Pt B 2 B 2ε in B, 6. 4 which is equivalent to 6.6. Now we show that 6.6 implies 6.2. The proof will be split into 5 steps. Step. Assuming that ρ t δr 6.2 and that B is a ball of radius r, rewrite 6.6 in the form For any positive integer, set B = B and we will prove that Pt B B ε in B P B t B ε in B Since is separable, there is a countable dense set of points in B. Let {b j } be a sequence of balls of radii r centered at those points. Clearly, b j B + and the family { 4 b j} covers B see Fig. 7. Due to 6.2, inequality 6.6 is valid for any ball b j, that is, for all 0 < s t, P B + s B+ P bj s bj ε in 4 b j. It follows that P B + s B+ ε in B. Applying the inequality 5.3 of Corollary 5.6 with Ω = B + and U = B, we obtain that the following inequality holds in B : P B + t B+ P B t B sup P B + s L B+ B 0<s t ε P B t B. Iterating in and using 6.3, we obtain 6.4.

25 Heat ernels on metric measure spaces 25 r b j / 4b j B B + Figure 7. Balls {B } and {b j } It follows from 6.4 that P t B c P t B P B t B ε in B Although 6.5 has been proved for any integer, it is trivially true also for = 0, if we define B 0 :=. Step 2. Fix t > 0, x and consider the function E t,x = exp c dx, ρt where the constant c > 0 is to be determined later on. Set and we will prove that r = δ ρ t,, 6.6 P t E t,x C in B x, r/4, 6.7 where C is a constant depending on ε, δ. Set as before B = B x, r,, and B 0 =. Using 6.6 and 6.5, we obtain that in B x, r/4, P t E t,x = P t B+ \B E t,x =0 E t,x L B + P t B+ \B =0 + r exp c P t B c ρt exp c + δ ε. =0 =0 Choosing c < δ log ε we obtain that this series converges, which proves 6.7. Step 3. Let us prove that, for all t > 0 and x, P t E t,x C E t,x, 6.8

26 26 Grigor yan, Hu and Lau for some constant C = C ε, δ. Observe first that, for all y, z, we have by the triangle inequality dx, y E t,x y = exp c ρt dx, z dz, y exp c exp c = E t,x ze t,z y, ρt ρt which can also be written in the form of a function inequality: It follows that By the previous step, we have E t,x E t,x z E t,z. P t E t,x E t,x z P t E t,z. 6.9 P t E t,z C in B z, r, 6.20 where r = 4 δ ρ t. For all y B z, r, we have cr E t,z y exp = exp cδ /4 =: C, ρ t whence Letting y vary, we can write E t,x z E t,x y E t,z y C E t,x y. E t,x z C E t,x in B z, r. Combining this with 6.9 and 6.20, we obtain P t E t,x CC E t,x in B z, r. Since z is arbitrary, covering by a countable sequence of balls lie B z, r, we obtain that 6.8 holds on with C = CC. Step 4. Let us prove that, for all t > 0, x, and for any positive integer, where B = x, δ ρ t. Indeed, by 6.8 P t E t,x C in B, P t E t,x = P t P t E t,x C P t E t,x which implies by iteration that P t E t,x C P t E t,x. Combining with 6.7 and noticing that C C, we obtain 6.2. Step 5. Fix a ball B = B x 0, r and observe that 6.2 is equivalent to the following: for all t, λ > 0, P t B c C exp c λt cr in B, 6.22 ρ/λ 2 where C, c, c > 0 are constants depending on ε, δ. In what follows, we fix also t and λ. Observe first that, for any x 2 B, Hence, it suffices to prove that, for any x 2 B, P t B c P t Bx,r/2 c. P t Bx,r/2 c C exp c λt cr ρ/λ 6.23 in a small ball around x. Covering then 2B by a countable family of such balls, we then obtain Changing t to t/ in 6.2, we obtain that P t Et/,x C in B x, σ

27 Heat ernels on metric measure spaces 27 where σ = 4 δ ρ t/. Since r E t/,x exp c in B x, r c ρ t/ and, hence, Bx,r c exp cr ρt/ we obtain that the following inequality holds in B x, σ P t Bx,r c exp cr ρt/ E t/,x, P t Et/,x exp c where c = log C. Given λ > 0, choose an integer such that cr ρt/ < λ t t. Then we obtain the following inequality in B x, σ P t Bx,r c exp c λt + cr, 6.24 ρ /λ which finishes the proof Consequences of two-sided estimates local case Now we are able to specify the local case in the statement of Theorem 4.9. Given two points x, y, a chain connecting x and y is any finite sequence {x } n =0 of points in such that x 0 = x, x n = y. We say that a metric space satisfies the chain condition if there is a constant C > 0 such that for any positive integer n and for all x, y there is a chain {x } n =0 connecting x and y, such that d x, y d x, x + C for all = 0,,..., n n For example, the geodesic distance on any length space satisfies the chain condition. On the other hand, the combinatorial distance on a graph does not satisfy it. Theorem 6.5. Assume that the metric measure space, d, μ satisfies the chain condition and that μ satisfies VD and RVD. Let E, F be a regular, local and conservative Dirichlet form, and let {p t } be the associated heat ernel such that, for all t > 0 and almost all x, y, C V x, t /βφ C d x, y t /β p t x, y C 2 V x, t /βφ C 2 d x, y t /β 6.26 where C, C, C 2, C 2 are positive constants, α, α are the exponents from VD and RVD, respectively, and β > α α. Then β 2 and the following inequality holds: c exp c s β/β Φs c 2 exp c 2 s β/β 6.27 for some positive constants c, c, c 2, c 2 and all s > 0. Proof. Let us first observe that the locality and the conservativeness imply the strong locality. Indeed, by [5, Lemma , p. 59 and Lemma, p. 6], we have the following identity lim P t u 2 dμ = ũ 2 d t 0 t for any u F where is the illing measure of E, F and ũ is a quasi-continuous version of u. Since P t =, it follows that = 0. Therefore, by the Beurling-Deny formula [5, Theorem 3.2., p.08], E, F is strongly local. This will allow us to apply later Lemma 6.. We split the further proof into five steps. Step. By Lemma 4.3 and the lower bound of p t, we obtain Φs c + s α +β for all s > 0.

28 28 Grigor yan, Hu and Lau Therefore, using the upper bound of p t, we obtain similar to 4.5 that p t x, ydμy c s α Φsds Bx,r c 2 r/t/β c s α α β ds. 2 r/t/β Due to the condition β > α α, the integral in the right hand side converges and, hence, the right hand side can be made arbitrarily small provided rt /β is large enough. We conclude by Lemma 6. cf. Remar 6.4 with ρt = t /β that, for all r, t > 0 and for almost all x, Bx,r c p t x, y dμ y C exp c t Ψ cr t, 6.28 where { } Ψs := sup λ>0 sλ /β λ Step 2. Let us prove that β >. If β < then it follows from 6.29 that Ψ. Substituting into 6.28 and letting r 0, we obtain that, for almost all x, p t x, y dμ y = 0. \{x} It follows from the stochastic completeness that there is a point x of a positive measure, which contradicts Remar 4.4. Assume now that β =. Then by 6.29 { 0, 0 s Φ s =, s >, which implies that, for all t < cr and for almost all x, Bx,r c p t x, y dμ y = 0, that is, p t x, y = 0 for all t < cd x, y and almost all x, y. Together with 6.26, this yields the following bounds of the heat ernel for all t > 0 and almost all x, y : C d x, y V C d x, y x, t /βφ p t /β t x, y V x, t /β Φ 6.30 t /β where { Φ s, s c Φ s = 0, s > c. Clearly, the functions Φ and Φ satisfy the hypotheses of Theorem 4.7. We conclude by Theorem 4.7 that β = β whereas by Lemma 3. β 2, which contradicts to β =. Step 3. Using that β >, let us show that the heat ernel satisfies the following upper bound β/β C d x, y p t x, y V x, t /β exp c. 6.3 t /β Setting in 6.29 λ = s/β β β we obtain as in Remar 6.3 so that 6.28 becomes Ψ s = c β s β/β Bx,r c p t x, y dμ y C exp r β/β c t /β On the other hand, by the upper bound in 6.26, we have, for all t > 0 and almost all x, y, p t x, y C V x, t/β. 6.33