HEAT KERNELS AND NON-LOCAL DIRICHLET FORMS ON ULTRAMETRIC SPACES
ALEXANDER BENDIKOV, ALEXANDER GRIGOR’YAN, ERYAN HU, AND JIAXIN HU
Abstract. We consider a class of jump measures on ultrametric spaces and the associated non-local regular Dirichlet forms. We obtain equivalent conditions for certain heat kernel upper and lower estimates in terms of the properties of the jump measure. In particular, heat kernel estimates are given for quite degenerate/singular jump measures as shown in a number of examples.
Contents
1. Introduction and motivation 1
1.1. Jump type Dirichlet forms 2
1.2. Ultrametric spaces 3
1.3. Isotropic Dirichlet forms 3
2. Statement of the main results 4
2.1. Jump kernel and Dirichlet form 5
2.2. Heat kernel estimates 6
2.3. Structure of the paper 9
3. Examples 10
4. Construction of non-local Dirichlet forms 15
5. Nash inequality 17
6. Lemma of growth 19
7. Some auxiliary inequalities 24
8. Weak Harnack inequality 25
9. Oscillation properties for harmonic functions 27
10. Conditions (E) and (S) 31
11. Oscillation inequality for Lu=f 34
12. Heat kernel 35
12.1. Existence and the H¨older continuity of the heat kernel 35
12.2. Near-diagonal lower estimate 35
12.3. Weak upper estimate 36
13. Derivation of (PI) from heat kernel estimates 38
14. Completion of proof of the main results 41
15. Optimality of heat kernel bounds under (T J) and (P I) 41
References 44
1. Introduction and motivation
The purpose of this paper is to obtain upper and lower estimates of heat kernels of certain jump type Dirichlet forms on ultrametric spaces. In particular, our results apply on such
Date: January 2019.
2010Mathematics Subject Classification. Primary: 35K08 Secondary: 28A80, 60J35.
AB was supported by the Polish National Science Center, grant 2015/17/B/ST 1/00062. AG was supported by SFB1283 of the German Research Council and by the Nankai University. EH was supported by SFB1283 of the German Research Council. JH was supported by NSFC No.11371217 and SFB 1283.
1
well-known examples of ultrametric spaces as the fields Qp of p-adic numbers and their self- products Qnp. For general metric spaces most of these results are not known yet.
1.1. Jump type Dirichlet forms. Let (M, d) be a locally compact separable metric space and μ be a Radon measure on M with full support. Let (E,F) be a regular jump type Dirichlet form in L2(M, μ) with the jump kernel J(x, y), that is, for all f, g ∈ F ⊂L2 we have
E(f, g) =ZZ
M×M
(f(x)−f(y)) (g(x)−g(y))J(x, y)dμ(x)dμ(y). (1.1) (see [10] for the theory of Dirichlet forms). The Dirichlet form has a generator Lthat is a non- negative definite self-adjoint operator in L2 and the associated heat semigroup
e−tL t≥0. Theheat kernel pt(x, y) of (E,F) (or ofL) is the integral kernel of the heat operatorPt=e−tL (should the former exist). Equivalently, pt(x, y) is the transition density of the associated jump process.
Given a symmetric, non-negative, measurable function J on M ×M, one may ask if the bilinear form (1.1) becomes a regular Dirichlet form in an appropriate domain F, whether it admits the heat kernel and how to estimate the latter quantitatively.
For example, consider in Rn the jump kernel
J(x, y) =|x−y|−(n+β),
where β is a real parameter. If 0 < β < 2 then E is a regular Dirichlet form with the generator const (−Δ)β/2 (where Δ is the Laplace operator), and the associated jump process is the symmetric stable Levy process of the index β. In the case β = 1 we have
pt(x, y) = cnt
t2+|x−y|2n+12
with some cn >0, which is the Cauchy distribution with the parameter t. For an arbitrary 0 < β < 2, the heat kernel of the symmetric stable process of index β admits the following estimate:
pt(x, y)' t
t1/β +|x−y|n+β = 1 tn/β
1 +|x−y| t1/β
−(n+β)
, (1.2)
where the sign'means that the ratio of two sides is bounded between two positive constants.
The estimate (1.2) is obtained by the subordination techniques from the heat kernel of Δ.
Assume now that
J(x, y)' |x−y|−(n+β).
Then (1.2) is still true, which follows from a result of Chen and Kumagai [6]. One can ask, under what conditions on an arbitrary metric measure space (M, d, μ) and a jump kernel J, the heat kernel of the associated Dirichlet form exists and satisfies the stable-like estimate
pt(x, y)' 1 tα/β
1 +d(x, y) t1/β
−(α+β)
(1.3) with some positive parameters α, β. If the Dirichlet form is conservative then following two conditions are necessary for (1.3) (see [6], [16], and [18]):
• theα-regularity: for any metric ball B(x, r), we have
μ(B(x, r))'rα. (V)
• the jump kernel estimate:
J(x, y)'d(x, y)−(α+β). (J)
It follows from (V) that α is necessarily the Hausdorff dimension of M and μ ' Hα. The value of the parameter β in (1.3) or (J) is called the index of the Dirichlet form or that of the associated jump process.
Chen and Kumagai [5], [6] proved that if 0 < β <2 then (V) and (J) are also sufficient for (1.3), that is,
(V)+(J)⇔(1.3).
There are many examples of fractal spaces where a jump kernel (J) generates a regular Dirichlet form even with β > 2. Indeed, on large families of fractals there are diffusion processes with the walk dimension β∗ > 2. By using the subordination techniques, one obtains a jump process with any index β ∈(0, β∗), in particular, β can be larger than 2 (see [2], [12], [16]).
In the case of β >2, in order to ensure the estimate (1.3), one needs on top of (V) and (J) one more quite complicated condition, which was established independently in [7] (cutoff Sobolev inequality) and [14] (generalized capacity condition). One of the purposes of this paper is to show that in the setting of ultrametric spaces one can manage without the third condition for any β∈(0,+∞).
1.2. Ultrametric spaces. Let (M, d) be a metric space. The metricdis called anultrametric if it satisfies the ultrametric inequality
d(x, y)≤max{d(x, z), d(z, y)}. (1.4) In this case (M, d) is called an ultrametric space.
Consider for anyx∈M and r >0 the metric ball
B(x, r) ={y∈M :d(x, y)≤r}.
It is easy to deduce from the ultrametric inequality (1.4) the following properties of the space in question. These properties will be frequently used in what follows.
• Any two balls of the same radius are either disjoint or coincide. More generally, any two balls are either disjoint or one contains the other. Consequently, the collection of all distinct balls of the same radius r forms a partition of M.
• Every point inside a ball is its center. This implies that balls are not only closed sets but also open. Consequently, the topological boundary of any ball is empty. One more consequence is that any ultrametric space is totally disconnected.
• From any covering of a compact set by a family of balls there is a finite subcover that consists of mutually disjoint balls.
A well-known example of an ultrametric space is the field Qp of p-adic numbers, where p is a prime. Recall thatQp is defined as the closure ofQwith respect to thep-adic norm kxkp
that satisfies the ultrametric inequality
kx+ykp ≤maxn
kxkp,kykpo .
Hence, Qp with the metric kx−ykp is an ultrametric space. Analysis on Qp and Qnp was developed, in particular, in [1], [3], [26], [27], [28].
1.3. Isotropic Dirichlet forms. Let (M, d) be an ultrametric space where all balls are compact. Let μbe a Radon measure onM with full support. In [3], the authors introduced an isotropic jump kernel onM given by
J(x, y) =Z ∞
d(x,y)
1 σ(r)
dσ(r)
μ(B(x, r)), (1.5)
whereσis any cumulative probability distribution function on (0,∞) that is strictly monotone increasing and left-continuous. This jump kernel determines a regular Dirichlet form that is referred to as isotropic Dirichlet form, and its heat kernel admits the following explicit formula
pt(x, y) =Z ∞
d(x,y)
tσt−1(r)dσ(r)
μ(B(x, r)) . (1.6)
Assume thatMisα-regular, that is, it satisfies (V). Choosingσto be the Fr´echet distribution:
σ(r) = exp
−c r
β ,
where c, β >0 are arbitrary, we obtain that the jump kernel (1.5) satisfies (J) and the heat kernel (1.7) satisfies the stable-like estimate (1.3).
For example, let M =Qnp wherep is a prime and n∈Z+. This space is n-regular so that the isotropic heat kernel with the above σ admits the estimate
pt(x, y)' 1 tn/β
1 +kx−ykp
t1/β
−(n+β)
. (1.7)
In this case the jump kernel in Qnp can be computed exactly as follows J(x, y) = cn,p,β
kx−ykn+βp . (1.8)
The generator of the Dirichlet form associated with the jump kernel (1.8) coincides with Taibleson operator introduced in [26] (see also [27], [3] and references therein).
Let (M, d) be a general metric space andμbe anα-regular measure onM. For anyβ >0, consider the following quadratic form in L2(M, μ):
Eα,β(f, f) =ZZ
M×M
(f(x)−f(y))2
d(x, y)α+β dμ(x)dμ(y).
Define the the walk dimension β∗ of M by
β∗ = sup{β >0 :Eα,β extends to a regular Dirichlet form}. (1.9) Note that β∗ is an invariant of the metric space (M, d) alone because α = dimHM and μ ' Hα, where dimHM is the Hausdorff dimension of M. It is known that if M carries a diffusion process {Xt} whose transition density satisfies a sub-Gaussian upper and lower estimates, then the walk dimension of {Xt} (a parameter in sub-Gaussian estimates) must be equal to β∗ (see [16]).
It follows from (1.9) that always β∗ ≥2 because for any β < 2, Lipschitz functions with compact supports are in the domain of Eα,β.
If M is a Riemannian manifold then β∗ = 2, while on fractals typically β∗ > 2. On α- regular ultrametric spaces, as it follows from the above construction of isotropic Dirichlet forms, we have always β∗ =∞.Hence, in the family of allα-regular metric spaces, manifolds and ultrametric spaces are extremal opposite cases as far as the walk dimension is concerned.
However, these two extremal classes of metric spaces have something in common: both manifolds and ultrametric spaces possess a priori rich classes of test functions with controlled energy: on manifolds these are usual bump or tent functions, while on ultrametric spaces these are indicators of balls, as we will see below. The existence of such classes of test functions is vital for obtaining upper bounds of the heat kernel.
2. Statement of the main results
In this section we state the main results of this paper, while the proofs will be given in the rest of the paper.
Throughout the paper, (M, d) is a locally compact separable ultrametric space andμ is a Radon measure on M with full support. Denote by B(M) the set of all Borel functions on M. For any open set U ⊂M, denote by C0(U) the space of all continuous functionsf on M with a compact support suppf ⊂U.
2.1. Jump kernel and Dirichlet form. Throughout the paper we fix some parameter R∈(0,diamM] and a kernelJ(x, E) onM×B(M) that satisfies the following two conditions:
for any r ∈ 0, R ,
J(x, B(x, r)c) is a μ-locally integrable function of x∈M, (j.1) and J is symmetric, that is, for all u, v ∈ B+(M),
Z
M
u(x)v(y)J(x, dy)dμ(x) =Z
M
u(y)v(x)J(x, dy)dμ(x). (j.2) For example, the kernel
J(x, E) = Z
E
J(x, y)dμ(y)
satisfies (j.1) and (j.2) providedJ(x, y) is a non-negative symmetric measurable function of (x, y)∈M ×M such that
Z
K
Z
(Kr)c
J(x, y)dμ(x)dμ(y)<∞ (2.1)
for any compact set K ⊂M and any r∈ 0, R
, whereKr is the r-neighborhood of K.
Any kernel J satisfying (j.1) and (j.2) determines a positive symmetric Radon measure j on M×M\diag that is defined by
Z
M×M\diag
f(x, y)dj(x, y) =Z
M
Z
M
f(x, y)J(x, dy)
dμ(x), for any f ∈C0(M×M \diag).
Consider the following bilinear form (E,Fmax) onL2(M, μ):
E(u, v) =R R
M×M\diag(u(x)−u(y)) (v(x)−v(y))dj(x, y) Fmax=
u∈L2(M) :uis Borel measurable and E(u, u)<∞ (2.2) The argument in [10, Example 1.2.4, p. 14] shows that E is well-defined, that is, for any u∈ B(M),
u= 0 μ-a.e.⇒ E(u, u) = 0.
We will prove below that, under conditions (j.1) and (j.2), (E,Fmax) is a Dirichlet form, and construct further a regular Dirichlet form (E,F) with the domainF ⊂ Fmax.
Definition 2.1. A functionf on M is said to be locally constant if, for anyx∈M, there is ε >0 such that f ≡const in B(x, ε).
Denote by D the space of all locally constant functions on M with compact supports.
Clearly, we have D ⊂C0(M).
Since any ball is closed and open, the indicator function 1B of any compact ballB belongs to D. Moreover, using properties of ultrametric balls, it is easy to verify that D consists of finite linear combinations of indicator functions of compact balls:
D= ( n
X
i=0
ci1Bi :n∈N, ci ∈R, Bi is a compact ball )
, (2.3)
where the balls {Bi}ni=0 can be chosen to be mutually disjoint (see the proof of Lemma 4.1).
Theorem 2.2. Assume (j.1) and (j.2) are satisfied.
(I) Then(E,Fmax) is a Dirichlet form onL2(M) and D ⊂ Fmax. (II) Set
F :=DE1, (2.4)
where the closure is taken with respect to the inner product E1 = E + (∙,∙)L2 in F. Then(E,F) is a regular Dirichlet form on L2(M).
In particular, indicator functions 1B of compact balls B belong to F.
The proof of Theorem 2.2is given in Section 4. Unless otherwise stated, in the rest of this paper, (E,F) is always referred to the regular Dirichlet form constructed in Theorem 2.2(II).
Let us emphasize that so far we have not made any additional assumption about μ.
2.2. Heat kernel estimates. Let us now state our main results about the heat kernel estimates. For these results, we always assume the space M is proper, that is, all balls B(x, r) in M are compact. In particular, we have μ(B(x, r))<∞.
Throughout the paper we fix positive reals α, β and R ∈(0,diamM]. Note that R could be ∞if diamM =∞.
Definition 2.3. We say that condition (V≤) is satisfied if, for all x∈M and r∈(0,∞),
μ(B(x, r))≤Crα, (V≤)
for some constant C > 0. We say that condition (V≥) is satisfied if, for all x ∈ M and all r ∈(0, R),
μ(B(x, r))≥C−1rα. (V≥)
We say that (V) is satisfied if both (V≤) and (V≥) are satisfied.
Let us emphasize that (V≤) is assumed to be true for all r > 0 while (V≥) should be satisfied only for r ∈(0, R). This convention allows us to cover compact ultrametric spaces M.
Definition 2.4. We say that the tail condition (T J) is satisfied if there exists C > 0 such that, for any ball B =B(x, r) with x∈M and r∈(0, R),
J(x, Bc)≤Cr−β. (T J)
Clearly, if (T J) is satisfied then J satisfies (j.1) so that Theorem2.2 applies.
For any measurable set A⊂M and any integrable function f on A, set fA :=−
Z
A
f dμ:= 1 μ(A)
Z
A
f dμ.
For any ball B =B(x0, r) and any λ >0, set
λB=B(x0, λr).
Since any point in the ball B can be used as its center, the notation λB is sensitive to the choice of the center of B if λ <1.
Definition 2.5. We say that thePoincar´e inequality (P I) is satisfied if there existκ∈(0,1]
and C > 0 such that, for any ball B := B(x0, r) with x0 ∈ M and r ∈ (0, R), and for any
f ∈ F, Z
κB|f −fκB|2dμ≤Crβ Z
B
Z
B
(f(x)−f(y))2dj(x, y). (P I) Definition 2.6. We say that the weak upper estimate (wU E) is satisfied, if the heat kernel pt(x, y) exists and satisfies the following estimate
pt(x, y)≤ C tα/β
1 +d(x, y)∧R t1/β
−β
, (wU E)
for some C >0, for all t∈(0, Rβ) and forμ-almost all x, y∈M.
In particular, if R≥diamM then (wU E) is equivalent to pt(x, y)≤ C
tα/β
1 +d(x, y) t1/β
−β for all t∈(0, Rβ) and for μ-almost all x, y∈M.
Definition 2.7. We say that near diagonal lower estimate (nLE) is satisfied if the heat kernelpt(x, y) exists and satisfies the following estimate:
pt(x, y)≥ct−α/β, (nLE)
for some c, δ >0, for all t∈(0, Rβ) and μ-almost all x, y∈M such that d(x, y)≤δt1/β.
Our main result is the following theorem.
Theorem 2.8. Let M be a proper ultrametric space. If (V), (T J) and (P I) are satisfied then the heat kernelpt(x, y) of(E,F)exists, is continuous jointly in t, x, y, H¨older continuous jointly in x, y and satisfies (wU E) and (nLE).
Moreover, under the standing assumptions (V) and (T J), the following equivalence takes place:
(P I)⇔(wU E) + (nLE). (2.5)
Note also that the conditions (T J) and (P I) can be satisfied for quite degenerate/singular jump measures as will be shown on examples in Section 15.
Let us emphasize that similar results for general metric spaces are not known and, most probably, they are not true without additional conditions. It would be interesting to obtain a version of Theorem 2.8 for general metric spaces.
Remark 2.9. Let M be a Riemannian manifold with the geodesic distance d and the Rie- mannian measure μ. Let (E,F) be the classical local Dirichlet form
E(u, u) =Z
M|∇u|2dμ.
Then, under the standing assumption (V), the corresponding Poincar´e inequality Z
B|f−fB|2dμ≤Cr2 Z
B|∇f|2dμ,
is equivalent to the following Gaussian estimate of the heat kernel pt(x, y):
pt(x, y) C tα/2 exp
−cd(x, y)2 t
(see [11], [23], [24]). The equivalence (2.5) of Theorem 2.8 can be regarded as a version of this result for ultrametric spaces.
The following stability result is an easy consequence of Theorem 2.8.
Corollary 2.10. Assume that(V)is satisfied. LetJ(1) andJ(2) be two kernels both satisfying (j.1),(j.2). Let(E(1),F(1)),(E(2),F(2)) be two regular Dirichlet forms determined byJ(1)and J(2) respectively (as in Theorem 2.2). Assume that, for some C >0,
C−1J(2) ≤J(1)≤CJ(2).
If (E(1),F(1))satisfies (T J), (wU E) and (nLE), then(E(2),F(2))also satisfies (T J), (wU E) and (nLE).
In Section 15, we give examples with R = ∞ showing that, under the hypotheses (V), (T J) and (P I) of Theorem 2.8, the estimate (wU E) cannot be improved to
pt(x, y)≤ C tα/β
1 +d(x, y) t1/β
−(β+ε)
for any ε >0. Similarly, the lower bound (nLE) cannot be improved to pt(x, y)≥ c
tα/β
1 +d(x, y) t1/β
−N
for any N > 0. Hence, in an ultrametric space, the Poincar´e inequality does not yield matching upper and lower bounds. For the latter one needs stronger assumptions as below.
Definition 2.11. We say that condition (J≤) is satisfied if the jump kernel J has the form:
J(x, dy) =J(x, y)dμ(y), (2.6)
where J(x, y) is a symmetric function of x, y∈M such that, for all distinct x, y∈M,
J(x, y)≤Cd(x, y)−(α+β). (J≤)
Similarly, we say that condition (J≥) is satisfied if, for all distinctx, y∈M,
J(x, y)≥C−1d(x, y)−(α+β). (J≥) We say that condition (J) is satisfied if both (J≤) and (J≥) are satisfied, that is, if, for all x, y∈M,
J(x, y)'d(x, y)−(α+β). (J)
It is easy to see that
(V≤)+(J≤)⇒(T J) (2.7)
(see [14, Prop. 6.4]) and
(V≥)+(J≥)⇒(P I) (2.8)
(see Lemma 3.1and the argument after that).
Hence, (T J) can be regarded as a weak version of the upper bound (J≤), and (P I) can be regarded as a weak version of the lower bound (J≥).
Definition 2.12. We say that the optimalupper estimate (U E) is satisfied if the heat kernel pt(x, y) exists and satisfies the following upper bound:
pt(x, y)≤ C tα/β
1 +d(x, y) t1/β
−(α+β)
, (U E)
for some C >0, for all t∈(0, Rβ) and forμ-almost all x, y∈M.
We say that the optimal lower estimate (LE) is satisfied if the heat kernel pt(x, y) exists and satisfies the following lower bound:
pt(x, y)≥ c tα/β
1 +d(x, y) t1/β
−(α+β)
, (LE)
for some c >0, for all t∈(0, Rβ) and for μ-almost all x, y∈M.
We say that the heat kernel satisfies two-sided stable-like estimate if both (U E) and (LE) are satisfied, that is, if
pt(x, y)' 1 tα/β
1 +d(x, y) t1/β
−(α+β)
, (2.9)
for all t∈(0, Rβ) andμ-a.a. x, y∈M.
Corollary 2.13. Let (V) be satisfied and J have the form (2.6).
(a) We have
(J≤) + (P I)⇔(U E) + (nLE).
(b) If in addition (T J) is satisfied then
(J≥)⇔(wU E) + (LE).
(c)We have
(J)⇔(2.9).
In all the cases (a), (b), (c), the heat kernel pt(x, y) exists, is continuous jointly in t, x, y and H¨older continuous jointly in x, y.
Corollary 2.13(c) recovers the estimate (1.7) of [3] that was proved for the jump kernel (1.8) inQnp. Corollary2.13(c) can be deduced from the previously known results for general metric spaces. Indeed, the case of an arbitrary β > 0 can be reduced to the case β = 1 by a simple change of distance function de(x, y) = d(x, y)β (that is again a metric by the ultrametric property), and then one can apply the results of [6] or [14] to obtain (2.9).
However, Theorem 2.8and parts (a), (b) of Corollary 2.13cannot be obtained in this way.
2.3. Structure of the paper. In Section3we give examples of ultrametric spaces and jump kernels satisfying (T J) and (P I). These examples show, in particular, that our results work for highly anisotropic cases, in particular, the jump measure can vanish on very large area of (M×M)\diag. In Section4we prove Theorem 2.2about construction of a regular Dirichlet form.
The major part of the paper is devoted to the proof of Theorem 2.8. The proof of the key implication
(V) + (T J) + (P I)⇒(wU E) + (nLE), (2.10) is fulfilled in Sections 5–12.
In Section 13 we deduce (P I) from the heat kernel bounds (wU E) and (nLE), and in Section 14 we conclude the proofs of Theorem 2.8 and Corollary 2.13 by combining the results of the previous sections.
In Section 15 we give more examples to show that the heat kernel bounds (wU E) and (nLE) of Theorem2.8 are sharp in certain sense.
Let us now describe the main steps in the proof of the implication (2.10).
Step 1. We show that the Poincar´e inequality (P I) implies the Nash inequality (Lemma 5.2). The latter yields by the well-known argument the existence of the heat kernel and on-diagonal upper bound
pt(x, y)≤Ct−α/β, (2.11)
for all t ∈ (0, Rβ) and μ- a.a. x, y ∈ M (Lemma 5.5). One more consequence of the Nash inequality is the Faber-Krahn inequality (Lemma 5.3).
The on-diagonal upper estimate (2.11) implies the upper bounds of the meat exit time from balls: for any ball B of radius r∈(0, σR)
ess sup
B
GB1≤Crβ, (2.12)
where GB the Green operator inB (Lemma 10.2).
Step 2. This is the largest and most technical part of the proof. We first prove Lemma of growth (Lemma6.4) that is based on the Faber-Krahn inequality, and Lemma 7.2where the Poincar´e inequality is used at full strength. These lemmas imply a weak Harnack inequality for harmonic functions of (E,F) (Lemma 8.1), that in turn yields the oscillation inequalities for harmonic functions (Lemmas 9.1, 9.3, 9.4) and, consequently, the H¨older continuity of harmonic functions.
Step 3. The mean exit time estimate (2.12) implies that GBf
L∞ ≤CrβkfkL∞,
which allows to extends the oscillation inequality to solutions u of Lu=f (Lemma 11.2).
Considering a function u(t,∙) =Ptf as solution toLu=−∂tu and estimatingk∂tukL∞ by means of (2.11), we obtain the oscillation inequality and the H¨older continuity for Ptf and, hence, also for the heat kernel (Lemma 12.1and [14, Lemmas 5.10, 5.11, 5.12, 5.13]).
Step 4. Using one of the consequences of Lemma of growth, we obtain the lower bound for mean exit time in any ball B of radius r∈(0, R):
ess inf
B GB1≥crβ inB, (2.13)
(Lemma 10.4). The estimates (2.12) and (2.13) imply the following survival estimate: for any ball B of radius r∈(0, R),
ess inf
B PtB1≥ε inB, provided t1/β ≤δr (2.14) (Lemma 10.6).
Step 5. The survival estimate (2.14) implies the on-diagonal lower bound pt(x, x)≥ct−α/β,
which together with the oscillation inequality yields the near diagonal lower estimate (nLE) (Lemma 12.2).
Step 6. Here we prove the off-diagonal upper estimate (wU E). The main difficulty is in obtaining the following estimate: for any ball B of radius r < Rand any t >0,
Pt1Bc ≤C t
rβ. (2.15)
It is done by comparingPtto a semigroupQtwith a truncated jump kerneldj(ρ)=1{d(x,y)≤ρ}dj and observing that Qtdoes not propagate from the inside of any ball of radius ρ to the out- side, which follows from the ultrametric property (Lemma 12.3). Combining (2.15) with the on-diagonal upper bound (2.11), we obtain (wU E) (Lemma 12.6).
Notation. The letters C, C0, c, c0,∙ ∙ ∙ denote positive constants whose values are unim- portant and can change at any occurrence. However, the value of all such constants depends only on the parameters in the hypotheses in question. The letters α, β and R denote the global parameters that have the same meaning all over the paper except for Section 15.
The essential supremum and infimum are always taken with respect to measure μ. We use the expression “μ-almost allx, y∈M” as a shorthand for “μ×μ-almost all (x, y)∈M×M”.
We also use Lp as a shorthand for Lp(M, μ).
3. Examples
In this section, we give an example of an ultrametric space (M, d) and a jump kernelJ on M ×M that satisfies (T J) and (P I) but does not satisfy (J≤) or (J≥) (Proposition3.5).
Let (M, d) be so far any metric space with a measure μ that is finite and positive on all balls. LetJ(x, y) be a symmetric non-negative function onM×M, and let Φ be an increasing positive function on (0,+∞).
We say that J satisfies a Φ-Poincar´e inequality if, for any ball B inM of radius r and for any f ∈L2(B),
Z
B×B
(f(x)−f(y))2J(x, y)dμ(x)dμ(y)≥ 1 Φ(r)
Z
B×B
(f(x)−f(y))2dμ(x)dμ(y). (3.1) Lemma 3.1. The inequality (3.1) is equivalent to
Z
B×B
(f(x)−f(y))2J(x, y)dμ(x)dμ(y)≥ 2μ(B) Φ (r)
Z
B
(f−fB)2dμ. (3.2) Clearly, if μ(B) 'rα and Φ (r) 'rα+β then (3.2) coincides with the Poincar´e inequality (P I).
Proof. Let us verify that the right hand sides of (3.1) and (3.2) coincide. We have Z
B
Z
B
(f(x)−f(y))2dμ(x)dμ(y) =Z
B
Z
B
f(x)2−2f(x)f(y) +f(y)2
dμ(x)dμ(y)
= 2μ(B)Z
B
f2dμ−2Z
B
f dμ 2
= 2μ(B)Z
B
f2dμ−fB2μ(B)
and Z
B
(f −fB)2dμ=Z
B
f2dμ−2fBZ
B
f dμ+fB2μ(B) =Z
B
f2dμ−fB2μ(B),
whence Z
B×B
(f(x)−f(y))2dμ(x)dμ(y) = 2μ(B)Z
B
(f−fB)2dμ, (3.3)
which was to be proved.
Remark 3.2. It is clear that if, for all x, y∈M, J(x, y)≥ 1
Φ(d(x, y)),
then (3.1) holds because for all x, y ∈ B we have d(x, y) ≤ r and, hence, J(x, y) ≥ Φ(r)1 . Hence, also (3.2) holds. Consequently, (V≥) and (J≥) imply (P I).
Now let us fix a prime p, consider a finite field Fp :={0,1,2,∙ ∙ ∙ , p−1} and the following set
Mp :={x={xk}k∈Z:xk ∈Fp and xk= 0 for all k <−K for some K ∈Z},
that consists of double sequences of elements of Fp that are vanishing near −∞. Consider Mp as a linear space over Fp with linear operations
x+y={xk+yk}k∈Z and ax={axk}k∈Z, for all x, y∈Mp and a∈Fp. Define in Mp the usual p-adic norm by
kxkp=p−n, where n:= min{k ∈Z:xk 6= 0}.
For all x, y ∈ Mp, set d(x, y) = kx−ykp and observe that (Mp, d) is an ultrametric space.
Furthermore, (Mp, d) is obviously separable and every ball B(x, r) in Mp is compact. As a metric space, Mp coincides withQp, but the operations in Mp are different from those inQp. The Haar measureμon Mp can be constructed as follows. For any n∈Zand for any ball B of radius p−n, set
μ(B) :=p−n. (3.4)
Since each ball of radiusp−nis a disjoint union ofpballs of radiip−(n+1), it is easy to see that μis σ-additive andσ-finite on the semi-ring of all balls in Mp. By Carath´eodory’s extension theorem,μ extends to a Borel measure onMp. It follows easily from (3.4) that measureμis 1-regular, that is,
μ(B(x, r))'r (3.5)
for all x∈Mp and r >0.
For any set A⊂Mp, any u∈Mp and a∈Fp, define the setsA+u={y+u:y ∈A} and aA={ay:y∈A}.
Lemma 3.3. For any a∈Fp\ {0} and u∈Mp, the mapy7→ay+upreserves metric d and measure μ. Consequently, for any nonnegative measurable function f onMp,
Z
Mp
f(y)dμ(y) =Z
Mp
f(ay+u)dμ(y). (3.6)
Proof. For anyy∈Mp we have kykp =kaykp because yk6= 0⇔ayk6= 0. It follows that, for all x, y∈Mp,
d(ax+u, ay+u) =kay−axkp=ky−xkp =d(x, y),
so that the metricdis preserved byy7→ay+u. Let us show that measureμis also preserved, that is, for any Borel set A⊂Mp,
μ(A) =μ(aA+u). (3.7)
It suffices to prove this for A=B(x, r). Since by the first part aA+u is a ball of the same radius r, (3.7) follows from the construction of the Haar measure. The identity (3.6) is a
consequence of (3.7).
Define a function S:Mp 7→Fp as follows: for any x∈Mp with kxkp =p−n, set S(x) =xn.
In other words, S(x) is equal to a non-zero digit xn of x with the smallest n. Define the following subset N of Mp×Mp:
N ={(x, y)∈Mp×Mp:S(x) =S(x+y) = 1 orS(y) =S(x+y) = 1}. (3.8) Fix some n∈ Z and let w∈ Mp be such that wn−1 = 1 and wk = 0 for all k < n−1. We claim that
B(w, p−n)×B(0, p−n)⊂N,
where 0 is the zero element of Mp. Indeed, if x ∈ B(w, p−n) and y ∈ B(0, p−n) then xk−wk = yk = 0 for all k < n, which implies that the first non-zero component of x is xn−1 = 1 and the same is true forx+y, whenceS(x) =S(x+y) = 1 and, hence, (x, y)∈N.
It follows that
(μ×μ)(N)≥μ B(w, p−n)
μ B(0, p−n)
=p−2n. Since n∈Zis arbitrary, we see that (μ×μ)(N) =∞.
Proposition 3.4. Let p >2. Then the jump kernel J(x, y) = 1Nc(x, y)
Φ(d(x, y)) =0, (x, y)∈N,
1
Φ(d(x,y)),(x, y)∈Nc, (3.9) satisfies the 5Φ-Poincar´e inequality.
Proof. Fix a ball B:=B(w, r)⊂Mp. If (x, y)∈(B×B)\N then 1Nc(x, y) = 1 and (f(x)−f(y))2J(x, y) = 1
Φ(d(x, y))(f(x)−f(y))2
≥ 1
Φ(r)(f(x)−f(y))2, which implies
Z
B×B
(f(x)−f(y))2J(x, dy)dμ(x) = Z
(B×B)\N
(f(x)−f(y))2J(x, y)dμ(x)dμ(y)
≥ 1 Φ(r)
Z
(B×B)\N
(f(x)−f(y))2dμ(x)dμ(y).
We will prove that Z
(B×B)∩N
(f(x)−f(y))2dμ(x)dμ(y)≤4 Z
(B×B)\N
(f(x)−f(y))2dμ(x)dμ(y), (3.10) which will then imply that
Z
B×B
(f(x)−f(y))2dμ(x)dμ(y)≤5Z
(B×B)\N
(f(x)−f(y))2dμ(x)dμ(y) and, hence,
Z
B×B
(f(x)−f(y))2J(x, y)dμ(x)dμ(y)≥ 1 5Φ(r)
Z
B×B
(f(x)−f(y))2dμ(x)dμ(y), that is, the 5Φ-Poincar´e inequality.
For any pair x, y∈Mp consider
z=z(x, y) := p+ 1
2 (x+y)∈Mp,
where p+12 ∈Fp sincep > 2. Observe that z−x= p−1
2 x+p+ 1
2 y= p+ 1
2 (y−x) and, hence,
kz−xkp =ky−xkp.
Consequently, if x, y belong to B then also z∈ B because x can be regarded as a center of B.
If (x, y)∈N then we have by (3.8) S(z) =S(p+ 1
2 (x+y)) = p+ 1 2 6= 1.
Therefore, no pair (∙, z) can lie in N, and we conclude that
if (x, y)∈(B×B)∩N then (x, z),(y, z)∈(B×B)\N. (3.11) Next, we have, for all x, y andz= p+12 (x+y),
(f(x)−f(y))2≤2 (f(x)−f(z))2+ 2 (f(y)−f(z))2,
and Z
(B×B)∩N
(f(x)−f(y))2dμ(x)dμ(y)
≤2 Z
Mp×Mp
1{(B×B)∩N}(x, y) (f(x)−f(z))2dμ(x)dμ(y) + 2Z
Mp×Mp
1{(B×B)∩N}(x, y) (f(y)−f(z))2dμ(x)dμ(y). (3.12) Furthermore, by (3.11) and Lemma 3.3with a= p+12 and u=axwe have
Z
Mp×Mp
1{(B×B)∩N}(x, y) (f(x)−f(z))2dμ(x)dμ(y)
≤ Z
Mp
Z
Mp
1(B×B)\N(x, a(x+y)) (f(x)−f(a(x+y)))2dμ(y)
! dμ(x)
= Z
Mp
Z
Mp
1(B×B)\N(x, y) (f(x)−f(y))2dμ(y)
! dμ(x)
= Z
(B×B)\N
(f(x)−f(y))2dμ(x)dμ(y).
Estimating similarly the integral in (3.12), we obtain (3.10), which finishes the proof.
Set now Φ(r) = r1+β. The jump kernel J from (3.9) satisfies (P I) by Lemma 3.1 and Proposition 3.4. By [14, Prop. 6.4], we have, for anyα-regular space,
Z
B(x,r)c
dμ(y)
d(x, y)α+β ≤Cr−β, (3.13)
that is, (J≤) implies (T J). Clearly, J satisfies (J≤) and, hence, (T J) but J obviously does not satisfy (J≥).
Now we construct a new jump kernel Je≥J that satisfies (T J) and (P I) but not (J≤) or (J≥).
For any integer n≥1, define the set En⊂Mp to consist of all sequences of the form x=|{z}...
k≥0
1...1
|{z}
k=−1,...−n
0...0...
| {z }
k<−n
that is, xk = 1 for k = −1, ...,−n, xk = 0 for allk < −n and xk is arbitrary for all k ≥ 0.
Similarly define a set Fn⊂Mp to consist of all the sequences x=|{z}...
k≥0
2...2
|{z}
k=−1,...−n
0...0...
| {z }
k<−n
Clearly, En and Fn are balls of radii 1, so that μ(En) =μ(Fn) = 1,and all the balls En, Fn, n≥1, are pairwise disjoint. It follows that also all the sets
(En×Fn), (Fn×En), n≥1, are pairwise disjoint in Mp×Mp. Define
E= [∞
n=1
(En×Fn)∪(Fn×En), so that the set E is symmetric and (μ×μ) (E) =∞.
It follows from the definition (3.8) of the setNthat, for anyn≥1, the setsEn×FnandFn× En are disjoint with N. Indeed, if x ∈En and y ∈Fn then S(x+y) = 36= 1 in Fp so that (x, y)∈/N. Consequently, the sets N and E are disjoint.
Let J(x, y) be the jump kernel from Proposition 3.4 with Φ(r) =r1+β, that is, J(x, y) = 1
d(x, y)1+β1Nc(x, y), Fix ε >0 and define further the kernels
J0(x, y) := d(x, y)ε
d(x, y)1+β1E(x, y) and Je(x, y) := J(x, y) +J0(x, y).
Proposition 3.5. For anyε∈(0,1), the jump kernel Jesatisfies (T J) and (P I) but neither (J≥) nor (J≤).
Proof. As we have already mentioned, Jesatisfies (P I) since Je≥J and J satisfies (P I).
Since both J and J0 vanish on N, we have Je= 0 on N so thatJedoes not satisfy (J≥).
To disprove (J≤) observe that d(x, y) = pn for all (x, y) ∈ En×Fn, which implies for such pairs (x, y)
J(x, y)de (x, y)1+β ≥J0(x, y)d(x, y)1+β =d(x, y)ε =pεn, that can be arbitrarily large.
It remains to prove that Jesatisfies (T J). Since J satisfies (T J), it suffices to prove that J0 satisfies (T J). By symmetry, it suffices to prove that, for any x∈En and r >0,
Z
B(x,r)c
J0(x, y)dμ(y)≤Cr−β. (3.14)
Consider two cases.
(i) Letr ≥1. By definition ofE, we see that ifx∈Enand (x, y)∈E theny∈Fn.Hence, we have, for x∈En,
Z
B(x,r)c
J0(x, y)dμ(y) =Z
B(x,r)c
1E(x, y) dμ(y) d(x, y)1+β−ε
≤ Z
B(x,r)c∩Fn
dμ(y)
d(x, y)1+β−ε ≤ μ(Fn) r1+β−ε ≤ 1
rβ. (ii) Letr <1. By (i), (3.5) and (3.13), we obtain
Z
B(x,r)c
J0(x, y)dμ(y)≤ Z
B(x,1)c
J0(x, y)dμ(y) +Z
B(x,1)\B(x,r)
dμ(y) d(x, y)1+β
≤1 +Cr−β ≤(1 +C)r−β,
which finishes the proof.
4. Construction of non-local Dirichlet forms The purpose of this section is to prove Theorem 2.2.
For any open set Ω⊂M, we regardL2(Ω) as a subset ofL2(M) by extending any function f ∈L2(Ω) by constant zero outside Ω. Fix a kernel J(x, E) on M × B(M) satisfying (j.1) and (j.2), and consider the bilinear form (E,Fmax) on L2(M, μ) given by (2.2). Let F be defined by (2.4). Recall thatDdenotes the space of all locally constant functions on M with compact supports. Denote byD(Ω) the subspace ofDthat consists of functions with support in Ω.
Lemma 4.1. Under the above hypotheses, the following is true.
(I) For any compact ball B, the indicator function 1B belongs to Fmax. Moreover, D ⊂ Fmax.
(II) For any open set Ω ⊂M, D(Ω) is dense in C0(Ω) with respect to sup-norm and in L2(Ω) with respect to L2-norm. In particular, D is dense inC0(M) and in L2(M).
Proof. (I). Denoteφ=1B for a ball of radius r and prove first that if B is compact and if r < R then E(φ, φ) <∞. Since φ(x)−φ(y) = 0 provided x, y are both in B or in Bc, we obtain by (2.2) and (j.1)
E(φ, φ) = Z
M
Z
M
(φ(x)−φ(y))2dj(x, y)
= 2Z
B
Z
Bc
(φ(x)−φ(y))2dj(x, y)
= 2j(B, Bc) = 2Z
B
J(x, B(x, r)c)dμ(x)<∞, (4.1) where we also use the property that B =B(x, r).
Let f be any function from D(in particular, f can be1B for a compact ball B). Since f is locally constant, for any x ∈M, there exists rx ∈(0, R) such that f = const in B(x, rx).
Since the family {B(x, rx)}x∈B is an open covering of suppf, there exists a finite subcovering {B(xi, rxi)}Ni=1. By the properties of ultrametric balls, we may further assume that all the balls B(xi, rxi) are mutually disjoint. It follows that f is a finite linear combination of functions 1B(xi,rxi) (cf. (2.3)), which implies E(f, f)<∞.
(II) Fix an open set Ω ⊂ M, a function f ∈ C0(Ω) and set K = suppf. Since f is uniformly continuous, for anyε >0 there existsr >0 such that any ballB(x, r) withx∈K lies in V ⊂ Ω, where V is a precompact open set such that K ⊂ V ⊂ V ⊂ Ω, and the oscillation of f in B(x, r) is bounded by ε. Choose a finite covering {B(xi, r)}Ni=1 of K. As above, we can assume that all the balls B(xi, r) are mutually disjoint. Clearly, the function
fε:=
XN i=1
f(xi)1B(xi,r) belongs to D(Ω) and
sup|fε−f|< ε,
which proved that D(Ω) is dense in C0(Ω) in sup-norm. Since also kfε−fk2L2 ≤ε2
XN i=1
μ(B(xi, r))≤ε2μ(V),
D(Ω) is dense in C0(Ω) also in L2-norm, whence all the claims follow.
Proof of Theorem 2.2. (I). Under conditions (j.1) and (j.2), it follows easily from (2.2) that (E,Fmax) is a bilinear, symmetric, non-negative and Markovian form. Moreover, by the arguments in [10, Example 1.2.4, p. 14], (E,Fmax) is also closed. It remains to show that
the domain Fmax is dense in L2(M). Indeed, by Lemma 4.1,Dis a subset of Fmax and Dis dense in L2(M), whence also Fmax is dense in L2(M), and (E,Fmax) is a Dirichlet form.
(II).By Lemma 4.1(II), F is dense inL2(M) so that (E,F) is a Dirichlet form. To prove the regularity of (E,F), we need to verify that F ∩C0(M) is dense in C0(M) in sup-norm and inF inE1-norm. Since F ∩C0(M) contains D, the regularity of (E,F) also follows from
Lemma 4.1(II).
Corollary 4.2. Assume that (T J) is satisfied. Then (E,F) defined in (2.2) and (2.4) is a regular Dirichlet form. Besides, for any compact ball B of radius r ∈ (0, R), the indicator φ:=1B of B belongs to D ⊂ F and satisfies:
E(φ, φ)≤Cμ(B)
rβ . (4.2)
Proof. Clearly, (T J) implies (j.1), and the first claim follows from Theorem 2.2. By (4.1) and (T J), we obtain
E(φ, φ) = 2 Z
B
J(x, B(x, r)c)dμ(x)≤Cμ(B) rβ ,
which proves the second claim.
In the subsequent sections we will need also the following statement.
Proposition 4.3. Under the hypotheses of Theorem2.2, for any open set A ⊂M and for any Borel function v∈ F, that is non-negative on A, we have
Z
A
v(y)J(x, dy)≤ess sup
A
v Z
A
J(x, dy), (4.3)
for μ-a.a. x∈M.
Proof. By [10, Lemma 4.5.4(i), p. 184] and [10, Theorem 4.2.1(ii), p. 161], measure jcharges no part of M×M\diag whose projection on the factor M has capacity 0.
It follows that if
u= 0 q.e.in A
then Z
M×A\diag
u(y)J(x, dy)dμ(x) = 0 and, hence,
Z
A
u(y)J(x, dy) = 0 forμ-a.e. x∈M.
Hence, the function v in (4.3) can be replaced by its quasi-continuous version ev. Set a= ess sup
A
v= ess sup
A ev.
By [10, Lemma 2.1.4, p. 70] we have e
v≤a q.e. in A, which implies
Z
A
v(y)J(x, dy) =Z
Aev(y)J(x, dy)≤a Z
A
J(x, dy).
5. Nash inequality
From this section, we start preparation for the proof of Theorem 2.8. From now on, we always assume (M, d) is an ultrametric space that is proper and separable, and μis a Radon measure on M with full support. Let (E,F) be the Dirichlet form defined by Theorem 2.2.
Other hypotheses will be stated explicitly.
Definition 5.1. We say the Nash inequality (N ash) holds for the Dirichlet form (E,F) if there exist positive constants ν and C such that
kfk2(1+ν)L2 ≤C
E(f, f) +R−βkfk2L2
kfk2νL1 (N ash) for all f ∈ F ∩L1.
The following lemma was proved in [23, Theorem 2.1] for a local Dirichlet form on a Riemannian manifold. We extend this result to non-local Dirichlet forms on ultrametric spaces.
Lemma 5.2. We have (V) + (P I)⇒(N ash) where ν =β/α.
Proof. The proof is divided into three steps. For any f ∈L1(M) ands >0, define a function fs on M by
fs(x) := 1 μ(B(x, s))
Z
B(x,s)
f(z)dμ(z).
Step I. Let us prove that, for any f ∈L1 and for all s∈(0, R),
kfsk2L2 ≤Cs−αkfk2L1, (5.1) where the constant C depends only on the constants in hypotheses. Indeed, for all z ∈M, s >0 andx∈B(z, s), we have B(x, s) =B(z, s) and, hence,
kfskL1 ≤ k|f|skL1 =Z
M
1 μ(B(x, s))
Z
B(x,s)|f(z)|dμ(z)dμ(x)
= Z
M|f(z)|dμ(z)Z
M
1B(z,s)(x) μ(B(z, s))dμ(x)
=kfkL1. On the other hand, we have by (V≥),
kfskL∞ ≤Cs−α sup
x∈M
Z
B(x,s)|f(z)|dμ(z)≤Cs−αkfkL1. It follows that
kfsk2L2 ≤ kfskL∞kfskL1 ≤Cs−αkfk2L1, which proves (5.1).
Step II. Let us prove that, for all f ∈ F ∩L1 and s∈(0, κR),
kf−fsk2L2 ≤CsβE(f, f), (5.2) where κ is the constant from (P I), and C depends only on the constants in hypotheses.
Indeed, since all distinct balls of radius s inM are disjoint and M is separable, there exists a (at most) countable family {Bi:=B(xi, s)}of disjoint balls of radiissuch thatM =tiBi. Note that, for any x∈Bi, we haveB(x, s) =Bi and, hence,
fs(x) =− Z
B(x,s)
f dμ=− Z
Bi
f dμ=fBi. Hence, we obtain by (P I) that
kf −fsk2L2 = X
i
Z
Bi
|f−fs|2dμ= X
i
Z
Bi
|f −fBi|2dμ
≤CsβX
i
Z
(κ−1Bi)×(κ−1Bi)
(f(x)−f(y))2dj(x, y).
Each ball κ−1Bj is a disjoint union of at most N ballsBi whereN depends on the constants in (V). It follows that, for each index i, there is at most N indices j so that Bi ⊂κ−1Bj. Hence, we obtain
kf −fsk2L2 ≤CN sβX
i
Z
Bi×M
(f(x)−f(y))2dj(x, y)
=CN sβ Z
M×M
(f(x)−f(y))2dj(x, y), which proves (5.2).
Step III. Now we can prove (N ash). Indeed, by (5.1) and (5.2), we have, for any f ∈ F ∩L1 and s∈(0, κR),
kfk2L2 ≤2kf−fsk2L2 + 2kfsk2L2 ≤CsβE(f, f) +Cs−αkfk2L1. On the other hand, if s∈[κR,∞) (in the case R <∞) then
kfk2L2 ≤(s/ κR
)βkfk2L2.
Combining the above two inequalities and assuming that C > 1, we obtain that, for any s >0,
kfk2L2 ≤Csβ
E(f, f) +R−βkfk2L2
+Cs−αkfk2L1. Choosing sso that the two terms on the right hand side are equal, that is,
sα+β = kfk2L1 E(f, f) +R−βkfk2L2
, we obtain
kfk2L2 ≤2Cs−αkfk2L1 = 2C
E(f, f) +R−βkfk2L2α+βα kfk2
β α+β
L1 ,
which yields (N ash) with ν =β/α.
Lemma 5.3. Assume that (V≤) and (N ash) hold withν =β/α. Then there existsσ ∈(0,1) such that, for any ball B⊂M of radius R∈(0, σR), for any measurable set E⊂B, and for any function f ∈ F such that f = 0 a.e. in Ec, we have
kfk2L2 ≤Cμ(E)νE(f, f). (5.3) Proof. Indeed, by Cauchy-Schwarz inequality,
kfk2L1 ≤μ(E)kfk2L2 whence by (N ash)
kfk2(1+ν)L2 ≤C
E(f, f) +R−βkfk2L2
kfk2νL2μ(E)ν whence
kfk2L2 ≤CE(f, f)μ(E)ν +CR−βμ(E)νkfk2L2. Choosing σ so small that
CR−βμ(E)ν ≤CR−βμ(B)α/β ≤C0 R
R β
≤C0σβ < 1 2,
we obtain (5.3).
The inequality (5.3) is called theFaber-Krahn inequality. It follows from Lemmas 5.2and 5.3that the hypotheses (V) and (P I) imply the Faber-Krahn inequality (5.3) with ν=β/α and for someσ∈(0,1). This parameterσ will be used in the rest of this paper alongside with α and β. Without loss of generality, we can assume that σ is small enough, in particular, σ ≤κ whereκ is the parameter from (P I).
For a non-empty open set Ω ⊂M, letF(Ω) be the closure of F ∩C0(Ω) inF with respect to the norm E1(u, u) = E(u, u) +kuk2L2. It is well known (see [10]) that if (E,F) is regular, then (E,F(Ω)) is a regular Dirichlet form on L2(Ω). Denote the corresponding generator, heat semigroup and heat kernel (if it exists) respectively by LΩ,{PtΩ}and pΩt(x, y).
Denote by λ1(Ω) the bottom of the spectrum of the operator LΩ in L2(Ω). It is known that
λ1(Ω) = inf
f∈F(Ω)\{0}
E(f, f) kfk2L2
.
It follows from (5.3) that if Ω is contained in a ballB of radius R < σRthen
λ1(Ω)≥cμ(Ω)−ν. (5.4)
Definition 5.4. We say that condition (DU E) is satisfied if the heat kernel pt(x, y) exists and satisfies the following diagonal upper estimate
pt(x, y)≤ C
tα/β, (DU E)
for any t∈(0, Rβ) and for μ-almost all x, y∈M.
A very useful consequence of the Nash inequality is stated in the next lemma.
Lemma 5.5. If (E,F) satisfies (N ash) withν =α/β then, for all t∈(0, Rβ), kPtkL2→L∞ ≤ C
tα/(2β). (5.5)
Consequently, (DU E) is satisfied.
For the proof see [4, Theorem 2.1] and [15, Lemma 3.7].
The converse is also true: (DU E) implies the ultracontractive estimate (5.5), while the latter implies (N ash) (see [8]).
6. Lemma of growth
The main result of this section is Lemma of growth (Lemma 6.4) and its consequences. A similar lemma in general metric spaces was proved [14] but in the present setting we have significant simplifications due to the ultrametric properties. In particular, we do not need to use a generalized capacity condition as in [14].
Consider the space
F0:=F+{const}
and extend E from F toF0 as follows: for all u, v ∈ F and a, b∈R, set E(u+a, v+b) :=E(u, v).
Definition 6.1. Let Ω be an open subset ofM. We say that a functionu∈ F0issubharmonic (resp. superharmonic) in Ω if
E(u, ϕ)≤0 (resp. E(u, ϕ)≥0) (6.1)
for any 0≤ϕ∈ F(Ω). A function u∈ F0 is called harmonic in Ω if it is both subharmonic and superharmonic in Ω.
