SMALL-TIME ASYMPTOTIC ESTIMATES IN LOCAL DIRICHLET SPACES

. Small-time asymptotic estimates of semigroups on a logarithmic scale are proved for all symmetric local Dirichlet forms on (cid:190) -ﬂnite measure spaces, which is an extension of the work by Hino and Ram¶‡rez [4].


Introduction
Let (X, B, µ) be a σ-finite measure space and (E, D) a symmetric Dirichlet form on the L 2 space of (X, B, µ). Let {T t } denote the semigroup associated with (E, D), and set P t (A, B) = X 1 A · T t 1 B dµ for A, B ∈ B and t > 0. The small-time asymptotic behavior of P t (A, B) on a logarithmic scale is the main interest of this paper. In the paper [4], under assumptions that the total mass of µ is finite and (E, D) is conservative and local, the following small-time asymptotic estimate was proved: where d(A, B) is an intrinsic distance between A and B defined by This result generalizes former works (see [4] and references therein) and can be regarded as an integral version of the small-time asymptotics of the transition density of Varadhan type lim t→0 t log p t (x, y) = − d(x, y) 2 2 , which was proved in [6] for a class of symmetric and non-degenerate diffusion processes on Lipschitz manifolds. In this paper, we further weaken the assumptions in [4] and prove the small-time estimate (1.1) holds for any A, B ∈ B with finite measure, for all local symmetric Dirichlet forms on σ-finite measure spaces. In other words, (1.1) now holds without assuming the finiteness of the total measure nor the conservativeness of (E, D), which may be considered as one of the most general results in this direction. The definition of the intrinsic distance d(A, B) here has to be suitably modified, by introducing the notion of nests. Note that we do not assume any topological structure of the underlying space, as in [4].
The proof is purely analytic and is done by careful modifications of the proof in [4] based on the Ramírez method [7]. In contrast to the simple statement of the result, the proof is rather technical. We will explain an idea of the proof here following the articles [7,4] and how to generalize it. is an easier part and follows from what is called Davies' method. In order to give the outline of the proof of the lower side estimate, let us consider a typical example; suppose that X has a differential structure and a gradient operator ∇ taking values in a Hilbert space with inner product ·, · as in the case of Riemannian manifolds, and E is given by E(f, g) = 1 2 X ∇f, ∇g dµ. Let us further assume that µ(X) is finite. Then, we can deduce that D 0 = {f ∈ D ∩ L ∞ (X) | |∇f | ≤ 1 a.e.}. The function u t = −t log T t 1 A satisfies the equation where L is the generator of {T t }. Letting t → 0, we expect that |∇u 0 | 2 = 2u 0 for a limit u 0 of u t , which implies that |∇ √ 2u 0 | 2 = 1.
(What we can actually expect is |∇ which is close to the lower side estimate. In practice, we cannot prove the convergence of the left-hand side of (1.4) in this form and have to consider the time-averageū t = 1 t t 0 u s ds in place of u t and utilize the Tauberian theorem. Moreover, we have to take the integrability ofū t into consideration. In [7], this was assured by an additional assumption, the spectral gap property. To remove such assumption, a suitable cutoff function φ was introduced in [4] and the proof was done by replacinḡ u t byφ t = 1 t t 0 φ(u s ) ds; bounded functions are always integrable as long as µ is a finite measure. When µ(X) = ∞, this modification is not sufficient. In order to include this case, in this paper, we further introduce a sequence {χ k } of 'cut-off functions in the space-direction' and considerφ t χ k to guarantee the integrability. By such modification, more and more extra terms appear in the argument, which have to be estimated appropriately. This makes the proof rather long.
The organization of this paper is as follows. In Section 2, we state the notion of nests and define the intrinsic distance d, which is naturally consistent with what was given in [4]. Their basic properties are discussed in Section 3. In Section 4, we prove the main theorem. In the last section, we give a few additional claims which have also been discussed in [4].

Preliminaries
For p ∈ [1, ∞], we denote by L p (µ) the L p -space on the σ-finite measure space (X, B, µ) and its norm by · L p (µ) . The totality of all measurable functions f on X will be denoted by L 0 (µ). Here, as usual, two functions which are equal µ-a.e. are identified. Let L p + (µ) denote the set of all functions f ∈ L p (µ) such that f ≥ 0 µ-a.e. We set f is a C 1 -function on R d and f and ∂f /∂x i (i = 1, 2, . . . , d) are all bounded , This is equivalent to the condition that E(f, g) = 0 if f, g ∈ D and (f + a)g = 0 µ-a.e. for some a ∈ R. (For the proof, see [2, Proposition I.5. 1.3].) The semigroup, the resolvent, and the nonpositive selfadjoint operator on L 2 (µ) associated with (E, D) will be denoted by {T t } t>0 , {G β } β>0 , and L, respectively. {T t } t>0 uniquely extends to a strongly continuous and contraction semigroup on L p (µ) for p ∈ [1, ∞).
Remark 2.3. For every k ∈ N, µ(E k ) < ∞ because of condition (i). By condition (ii), we can prove µ(X \ ∞ k=1 E k ) = 0. We will see in Section 3 that there do exist many nests.
and write I f (h) for I f,f (h). The following are basic properties of I.
Proof. For f, g, h ∈ L 2 (µ) ∩ L ∞ (µ) and t > 0, define  [2] and the limiting argument, the claims follow for I (t) in place of I. Letting t → 0 reaches the conclusion.
By the properties (i) and (iii) above, we can define I f (h) for f ∈ D and h ∈ D b by continuity. Due to the locality of (E, F), I f (h) = 0 if (f + a)h = 0 µ-a.e. for some a ∈ R. This allows us to define Clearly, we can replace D E k ,b by D E k ,b,+ in the definition above. We will show in Proposition 3.9 that the set The following is our main theorem.
Remark 2.8. To make the meaning of D 0 ({E k }) clearer, let us suppose that X is a locally compact separable metric space, µ is a positive Radon measure with supp µ = X, (E, D) is a regular Dirichlet form on L 2 (µ), and there exists a sequence of relatively compact open sets can be described as . Therefore, d is a natural generalization of the usual notion of intrinsic metric.
The corresponding diffusion process is the Brownian motion on X killed at 0. (ii) Let X = R and B = the Borel σ-field on X. Let m be the Lebesgue measure on X and µ a positive Radon measure on X such that supp µ is X, µ and m are mutually singular, and µ ((0, Then, (E, D) is a regular Dirichlet form on L 2 (µ). The corresponding diffusion process is a time changed Brownian motion. The energy measure k=1 . This suggests that considering D loc,b is more natural than D loc in the definition of D 0 .
since ξ is 1-excessive. Therefore, g k := (g − c/k) ∨ 0 belongs to D E k and g k converges to g in D as k → ∞. This means that any element in D − D can be approximated by functions in ∞ k=1 D E k , which proves the claim.
Note that for any function f ∈ L 2 (µ) with f > 0 µ-a.e., ξ = G 1 f satisfies the condition of the lemma above. Therefore, there exist indeed many nests.
The following claim is what is naturally expected. We give a proof for it though it is not needed in the sequel.
It also holds that f k D ≤ g k D + f D and f k converges to f µ-a.e. Therefore, {f k } ∞ k=l converges weakly to f in D. Taking the Cesàro mean of an appropriate subsequence, we obtain a desired approximating sequence.
In order to investigate the space D 0 ({E k }), we will prove some auxiliary properties. (i) If f k converges weakly to f in D and h ≥ 0 µ-a.e., then Proof. The first and the second claims follow from the fact that I ·,· (h) is a nonnegative definite continuous bilinear form on D when h ≥ 0 µ-a.e. For the third one, it is enough to notice that f h k converges weakly to f h in D as k → ∞.
One of the important consequences of the locality of (E, D) is the following.
holds for any F, G ∈Ĉ 1 b (R n ). Moreover, these measures satisfy the following properties.
Proof. By [2, Theorem I.5.2.1], uniquely determined are the family (σ f i,j ) 1≤i, j≤n of signed Radon measures on R n such that (i) and (ii) hold, and (3.1) is true for any F and G ∈ C 1 c (R n ). By the way of construction of σ f i,j (see also [8]), for any F ∈ C 1 c (R n ), Therefore, σ f i,j is of finite variation and |σ f i,j |(R n ) ≤ E(f i ) 1/2 E(f j ) 1/2 . Equation (3.1) now follows for F and G inĈ 1 b (R n ), by taking an approximate sequence from C 1 c (R n ) and using the dominated convergence theorem. Letting This implies the assertion. The following theorem is also necessary for subsequent arguments.
When k = 1 and l = 1, we will write λ f,g,h for λ f,g,h 1,1 . By Proposition 3.6, we have the integral expression for f, g, h ∈ D and F ∈ C 1 b (R) with F (0) = 0. We can define I f,g (F (h)) for F ∈ C 1 b (R) (possibly with F (0) = 0) by the right-hand side of (3.3).

In other words, when we setD
for f, g ∈ D and h = h 0 + α ∈D b is well-defined. Then, Lemma 2.5 and Lemma 3.3 (i) (ii) are true for h, h 1 , h 2 ∈D b .
Proof. We can take sequences g L ∞ (µ) for every k, and g k converges to g in D and µ-a.e. as k → ∞. By Lemma 3.7 (ii) and Lemma 2.5 (i), we have Since f k g k converges to f g µ-a.e., we obtain that f g ∈ D and As in the proof of Lemma 3.2, we can take {h l } ∞ l=1 such that h l ∈ D E k ∩E l , 0 ≤ h l ≤ h µ-a.e. for all l and h l converges to h in D as l → ∞. Then, for all l, Letting l → ∞, we obtain from Lemma 3.3 (iii) that Hence, we conclude f ∈ D 0 ({E k }).
By this proposition, we will use the notation D 0 as well as D 0 ({E k }) from now on. Note that 1 ∈ D 0 . When µ(X) < ∞ and 1 ∈ D, the space D 0 is the same as given in [4], namely (1.3), because we can take E k = X (k ∈ N) as a nest.
Proof. It is trivial that f ∈ D 0 implies −f ∈ D 0 . Let f and g be in D 0 . Take an arbitrary nest Then, h 1,l , h 2,l ∈ D E k ,b,+ , h = h 1,l +h 2,l , h 1,l = 0 µ-a.e. on {f k ∨g k = f k }, and h 2,l ≤ F l (f k ∨ g k − g k ) h L ∞ (µ) for every l. Here, F l is an arbitrary C 1 -function on R such that 0 ≤ F l ≤ 1, F l (x) = 0 on (−∞, −1/l] ∪ [2/l, ∞), and F l (x) = 1 on [0, 1/l]. Then, we have for any ε > 0, where σ f k ∨g k −g k 1,1 is a measure on R given in Theorem 3.4 with n = 1. Since Theorem 3.5 implies σ f k ∨g k −g k 1,1 Proof. Take a finite measure ν on X such that ν and µ are mutually absolutely continuous. Fix M > 0 and let a = sup{ It also holds that f k converges to some f µ-a.e. and X f dν = a. We will prove f ∈ D A,M . Take a nest {E k } ∞ k=1 and functions {χ k } ∞ k=1 as in Definition 2.1 and Remark 2.2. By Lemma 3.8, f k χ l ∈ D for every k and l, and {f k χ l } ∞ k=1 is bounded in D for each l. Therefore, f χ l ∈ D and f k χ l converges to f χ l weakly in D as k → ∞. For any h ∈ D E l ,b,+ , we have Taking a supremum over f , we obtain d(A, B) ≤ essinf x∈B d A (x).

Proof of Theorem 2.7
We first prove the upper side estimate.
Proof. Let w ∈ D A,M and let {E k } ∞ k=1 be an arbitrary nest. There exists {w k } ∞ k=1 ⊂ D such that w k L ∞ (µ) ≤ M and w k = w µ-a.e. on E k for all k. Note that w k converges to w µ-a.e.
e. for each k, and u k converges to v t in D and µ-a.e. as k → ∞. Then, In the inequality the right-hand side converges to 0 as k → ∞, since Lemma 2.5 (i) and Lemma 3.7 (i)) which is bounded in k. Thus we have . Solving this differential inequality, we have Therefore, The conclusion follows by optimizing the right-hand side in α and letting M → ∞.
We turn to the lower side estimate. Fix a nest {E k } ∞ k=1 and associated functions {χ k } ∞ k=1 as in Definition 2.1 and Remark 2.2. Take functions φ K , Φ K , Ψ K for K > 0 as in Section 2.1 of [4]. That is, using an arbitrary concave function g : R + → R + such that • g is bounded and three times continuously differentiable; • g(x) = x for x ≤ 1 and 0 < g (x) ≤ 1 for any In what follows, we suppress the symbol K from the notation since K is fixed in most of the argument. The following are some basic properties for these functions, proved in [4].
Letting ε → 0 and dividing by t, we obtain lim k →∞ A φ s k dµ = 0. This means that lim t→0 A φ t dµ = 0. Then, by letting t → 0 along the sequence {t n } in the identity we obtain Aφ 0 dµ = 0, which implies the claim.
From these arguments, we conclude that φ 0 ∈ D A, √ KL and therefore,φ 0 ≤ d 2 A µ-a.e. But this inequality is not optimal; a sharper estimate is obtained by the following lemma.  2 2 holds true µ-a.e. for some c > 1 for every K and every limitφ K 0 , then Proof. The proof is a modification of Lemma 2.12 in [4]. Given K, we Let D be a measurable set with 0 < µ(D) < ∞. Using the convexity of Φ(−t log(·)) for small t (see Lemma 2.1 in [4]), we have Also, by Theorem 4.1, Therefore, in the limit, Since Φ K is concave, Lemma 2.2 in [4] is applied to obtain that Φ K (φ M 0 ) ≥Ψ K 0 µ-a.e. Therefore, We will prove from the estimate (4.6). If (4.7) is false, there exists some D ∈ B with 0 < µ(D ) < ∞ and ε > 0 such that Then µ(D) > 0 and Φ K 0 ≤ c −1 essinf x∈DΨ K 0 − ε/2 µ-a.e. on D. This is contradictory to (4.6).
Proof. This is almost the same as Lemma 2.16 in [4]. By Theorem 4.1, it holds that {T t 1 A = 0} ⊃ {d A = ∞} µ-a.e. Let 0 < s < t and suppose P t (A, B) = 0. Then we have Therefore, 1 A · T s 1 B = 0, in particular, P s (A, B) = 0. By Theorem 2.7, we obtain {T t 1 A = 0} ⊂ {d A = ∞} µ-a.e. The second assertion follows from the first one.
The proof of Theorem 1.3 of [4] is also valid in our setting here with slight modification, and we have the following counterpart.
Theorem 5.2. Let A ∈ B with 0 < µ(A) < ∞ take any probability measure ν which is mutually absolutely continuous with respect to µ. Then, the functions u t = −t log T t 1 A converges to d 2 A /2 as t → 0 in the following senses.