Harmonic Analysis for Resistance
Forms
Jun Kigami
Graduate School of Informatics
Kyoto University
Kyoto 606-8501, Japan
e-mail:[email protected]
Abstract
In this paper, we define the Green functions for a resistance form
by using effective resistance and harmonic functions. Then the Green
functions and harmonic functions are shown to be uniformly Lipschitz
continuous with respect to the resistance metric. Making use of this fact,
we construct the Green operator and the (measure valued) Laplacian. The
domain of the Laplacian is shown to be a subset of uniformly Lipschitz
continuous functions while the domain of the resistance form in general
consists of uniformly 1/2-Hölder continuous functions.
1
Introduction
The theory of resistance forms has been developed as the foundation of analysis
on post critically finite self-similar sets. See [16] for example. It should correspond to a part of potential theory where each point has a positive capacity.
In this paper, for a resistance form, we give a simple definition of the Green
function associated with a boundary consisting of any finite number of points
and show that the Green function is always uniformly Lipschitz continuous with
respect to the distance given by the effective resistance. Then we will follow
ramifications of this fact.
More precisely, let (E, F) be a resistance form on a set X. Then there exists a
natural distance R on X associated with (E, F). R is called the resistance metric.
See Section 2 for details. In Section 4, we will define the Green function gB :
X×X → [0, +∞), where B is a non-empty finite subset of X. In Proposition 4.3,
gB is characterized as a reproducing kernel of (E, FB ), where FB = {u|u ∈
x
x
: X → [0, +∞) by gB
(y) = gB (x, y), then
F, u|B ≡ 0}: define gB
x
E(u, gB
) = u(x)
for any u ∈ FB and any x ∈ X. Next in Theorem 4.5, the Green function gB
is shown to be uniformly Lipschitz continuous with respect to the resistance
metric R. Precisely,
|gB (x, y) − gB (x, z)| ≤ R(y, z)
1
for any x, y and z. This also implies that harmonic functions are uniformly
Lipschitz continuous with respect to the resistance metric. In Section 5, we
will define the Green operator GB from measures on X to FB . (Assuming that
(X, R) is compact for simplicity, we mean the dual space of the continuous functions on (X, R) by measures.) Then we will define the domain of the Laplacian
in the generalized (or universal) sense, DL , by DL = Im(GB ) ⊕ HB , where HB
is the collection of harmonic functions on X with respect to the boundary B.
(We use the word “generalized” (or universal) sense because the image of the
Laplacian is measures in general. In [22], we can find an idea of the measure
valued Laplacian in the case of post critically finite self-similar sets. ) In fact,
DL is shown to be independent of B in Theorem 5.5. Moreover we will see
that every element of DL is uniformly Lipschitz continuous with respect the
resistance metric. Also in Section 6, any u ∈ DL is shown to have the Neumann
derivative (du)x for any x ∈ X. These facts will lead us to the definition of
Laplacians in the generalized sense and we will have the Dirichlet Laplacian
with boundary B, LB and the Neumann Laplacian L. DL is the domain of
both LB and L. Furthermore, in Theorem 6.8, we have the following expression
of the resistance form:
E(u, v) =
u(p)(dv)p − (LB v)(u)
p∈B
for any u ∈ F and any v ∈ DL . (We also obtain the counterpart of this for the
Neumann Laplacian L.) Note that all the notions (i.e. the Green function gB ,
the Green operator GB , the domain of Laplacians DL , the Neumann derivative
(du)p and the Laplacians LB and L) are independent of measures. In this sense,
these are “universal” objects.
In Section 8, we will introduce a measure µ on X and consider measures
which are absolutely continuous with respect to µ. Then the Green operator
GB is realized as GB,µ : L1 (X, µ) → FB . In the course of discussions, we
will finally show that the restriction of the “universal” Laplacians are the selfadjoint operator coming from the Dirichlet form (the closed form) on L2 (X, µ)
associated with the resistance form (E, F).
In Section 9, we will apply the results in the previous sections to a self-similar
resistance form given by a regular harmonic structure on a post critically finite
(p. c. f. for short) self-similar structure. (Such a resistance form is discussed in
[16] in detail.) Indeed, by using probabilistic method, it has already shown
that the Green function, harmonic functions and the elements in the domain of
the Laplacian are uniformly Lipschitz continuous with respect to the resistance
metric. See [6], [18], [8] and [13] for example. They first establish a detailed
short time offdiagonal estimate of the heat kernel with respect to a special selfsimilar measure ν, which is determined by the harmonic structure, and then
show the uniform Lipschitz continuity of the above mentioned functions. Since
this method depends on the special measure ν, one can only know that the
elements in the domain of the ν-Laplacian are uniformly Lipschitz continuous.
In contrast, our method in this paper do not require any measure and hence
2
the discussions are more direct and simple. Moreover the elements in DL (the
domain of universal Laplacians) are shown to be uniformly Lipschitz continuous.
Also in Section 9, we will obtain relations between F, DL and CL , where
CL is the collection of uniformly Lipschitz continuous functions. In particular,
we show that DL ⊂ F ∩ CL but DL = F ∩ CL .
We can apply the results in this paper to other classes of fractals: the Sierpinski carpets in R2 studied by Barlow-Bass [2, 3, 4, 5] and Kusuoka-Zhou [19],
the randomaized self-similar sets studied by Hambly [10, 11, 12] and the Markov
(graph directed) p. c. f. self-similar sets by Hambly-Nyberg [14] and KigamiStrichartz-Walker [17]. For these three classes of fractals, one can construct
a regular local Dirichlet form with a certain kind of self-similarity and those
forms are known to be resistance forms.
Although we only consider finite sets as boundaries in this paper, it is interesting to study the general case where the boundary can be an infinite set.
For the Sierpinski gasket, such a case has been studied partly in [21]. The first
question should be to determine a proper class of sets which can be thought of
as boundaries. This problem is worth exploring in the future.
2
Resistance form and harmonic functions
In this section, we will briefly review the theory of Dirichlet forms and Laplacians
on finite sets and resistance forms. See [16, Chapter 2] for details and complete
proofs.
Notation. For a set V , we define (V ) = {f |f : V → R}. If V is a finite
set, (V ) is considered
(·, ·)V
to be equipped with the standard inner product
defined by (u, v)V = p∈V u(p)v(p) for any u, v ∈ (V ). Also |u|V = (u, u)V
for any u ∈ (V ).
First we give a definition of Dirichlet forms on a finite set V .
Definition 2.1 (Dirichlet forms). Let V be a finite set. A symmetric bilinear form on (V ), E is called a Dirichlet form on V if it satisfies
(DF1) E(u, u) ≥ 0 for any u ∈ (V ),
(DF2) E(u, u) = 0 if and only if u is constant on V
and
(DF3) For any u ∈ (V ), E(u, u) ≥ E(ū, ū), where ū is defined by


if u(p) ≥ 1,
1
ū(p) = u(p) if 0 < u(p) < 1,


0
if u(p) ≤ 0.
We use DF(V ) to denote the collection of Dirichlet forms on V .
Condition (DF3) is called the Markov property.
3
Notation. Let V be a finite set. The characteristic function χVU of a subset
U ⊆ V is defined by
1 if q ∈ U ,
V
χU (q) =
0 otherwise.
If no confusion can occur, we write χU instead of χVU . If U = {p} for a point
map, then
p ∈ V , we write χp instead of χ{p} . If H : (V ) → (V ) is a linear
we set Hpq = (Hχq )(p) for p, q ∈ V . For f ∈ (V ), (Hf )(p) = q∈V Hpq f (q).
Definition 2.2 (Laplacians). A symmetric linear operator H : (V ) → (V )
is called a Laplacian on V if it satisfies
(L1) H is non-positive definite,
(L2) Hu = 0 if and only if u is a constant on V ,
and
(L3) Hpq ≥ 0 for all p = q ∈ V .
We use LA(V ) to denote the collection of Laplacians on V .
For H ∈ LA(V ), define a bilinear form EH on (V ) by EH (u, v) = −(u, Hv).
Then EH is a Dirichlet form on V . This map from LA(V ) to DF(V ) gives a
natural bijective correspondence between LA(V ) and DF(V ).
We may also associate an electrical network on V consisting of resistances
to a Laplacian H ∈ LA(V ). Let H ∈ LA(V ). For any p, q ∈ V with p = q,
set Rpq = (Hpq )−1 and attach a resistor of resistance Rpq between terminals p
and q. If electrical potentials of p and q are v(p) and v(q) respectively, then the
−1
current
total current at p
from q to p is cpq = (Rpq ) (v(q) − v(p)). Hence the
is q∈V Hpq (v(q) − v(p)) = (Hv)(p). (Note that Hpp = − q∈V \p Hpq for any
Laplacian.)
Definition 2.3 (Effective resistance). Let H ∈ LA(V ). For any p, q ∈ V
with p = q, define
−1
RH (p, q) = min{EH (u, u)|u ∈ (V ), u(p) = 1, u(q) = 0}
.
Also define RH (p, p) = 0 for any p ∈ V . RH (p, q) is called the effective resistance
between p and q with respect to H.
RH (p, q) is the actual resistance between p and q considering all the resistors
associated with a Laplacian H. The remarkable fact is that RH (·, ·) is a distance
on V .
Proposition 2.4. Let H ∈ LA(V ), then RH (·, ·) is a distance on V .
Definition 2.5. (1) Let V1 and V2 be finite sets and let Hi ∈ LA(Vi ) for i =
1, 2. We write (V1 , H1 ) ≤ (V2 , H2 ) if and only if V1 ⊆ V2 and, for any u ∈ (V1 ),
EH1 (u, u) = min{EH2 (v, v)|v ∈ (V2 ), v|V1 = u}.
(2) Let Vi be a finite set for i = 0, 1, 2, · · · and let Hi ∈ LA(Vi ) for any i ≥ 0.
S = {(Vm , Hm )}m≥0 is called a compatible sequence if and only if (Vm , Hm ) ≤
(Vm+1 , Hm+1 ) for any m ≥ 0.
4
If (V1 , H1 ) ≤ (V2 , H2 ), then it is easy to see that RH1 (p, q) = RH2 (p, q) for
any p, q ∈ V1 . In fact, the converse is also true.
Proposition 2.6. Let V1 and V2 be finite sets and let Hi ∈ LA(Vi ) for i = 1, 2.
Assume that V1 ⊆ V2 . Divide H2 into four parts:
T tJ
H2 =
,
J X
where T : (V1 ) → (V1 ), J : (V1 ) → (V2 \V1 ) and X : (V2 \V1 ) → (V2 \V1 ).
Then the following three conditions are equivalent.
(1) (V1 , H1 ) ≤ (V2 , H2 ).
(2) H1 = T − tJX −1 J.
(3) RH2 |V1 ×V1 = RH1 .
Remark. X in the above proposition is known to be negative definite. See [16,
Lemma 2.1.5] for details.
Definition 2.7. Let S = {(Vm , Hm )}m≥0 be a compatible sequence. Then,
define V∗ = ∪m≥0 Vm ,
F(S) = {u|u : V∗ → R, lim EHm (u|Vm , u|Vm ) < ∞}.
m→∞
For any u, v ∈ F(S), define
ES (u, v) = lim EHm (u|Vm , u|Vm ).
m→∞
Also for any p, q ∈ V∗ , define
RS (p, q) = RHm (p, q),
where m is chosen so that p, q ∈ Vm .
Remark. Since S is a compatible sequence, EHm (u|Vm , u|Vm ) is monotonically
increasing. So the limit appearing in the definition of F(S) does exists if we
allow ∞ as the value of the limit.
By Proposition 2.6, the definition of RS is well-defined. Also Proposition 2.4,
implies that RS (·, ·) is a distance on V∗ . Note that V∗ is merely a countable set.
Considering the completion of a metric space (V∗ , RS ), however, we may get an
uncountable set. In fact, (ES , F(S)) is a resistance form on V∗ defined below.
Hence Theorem 2.12 justifies the completion of (V∗ , RS ).
Definition 2.8 (Resistance form). Let X be a set. A pair (E, F) is called
a resistance form on X if it satisfies the following conditions (RF1) through
(RF5).
(RF1) F is a linear subspace of (X) containing constants and E is a nonnegative symmetric quadratic form on F. E(u, u) = 0 if and only if u is constant
on X.
(RF2) Let ∼ be an equivalent relation on F defined by u ∼ v if and only if
5
u − v is constant on X. Then (F/∼, E) is a Hilbert space.
(RF3) For any finite subset V ⊂ X and for any v ∈ (V ), there exists u ∈ F
such that u|V = v.
(RF4) For any p, q ∈ X,
sup{
|u(p) − u(q)|2
: u ∈ F, E(u, u) > 0}
E(u, u)
is finite. The above supremum is denoted by R(E,F ) (p, q).
(RF5) If u ∈ F, then ū ∈ F and E(ū, ū) ≤ E(u, u), where ū is defined in the
same manner as (DF3) in Definition 2.1.
We use RF(X) to denote the collection of resistance forms on X.
Condition (RF5) is called the Markov property. By (RF5), we obtain the
following lemma.
Lemma 2.9. For real valued functions u and v on X, define u ∨ v and u ∧ v by
(u ∨ v)(x) = max{u(x), v(x)}
and
(u ∧ v)(x) = min{u(x), v(x)}
for any x ∈ X. Let (E, F) be a resistance form on X. Then u ∨ v and u ∧ v
belong to F for any u, v ∈ F.
Proposition 2.10. Let (E, F) be a resistance form on a set X. Then, for any
p, q ∈ X, the supremum in (RF4) is the maximum. Moreover, for any finite set
V ⊆ X, there exists a unique HV ∈ LA(V ) such that RHV = R(E,F ) |V ×V . In
particular, R(E,F ) is a distance of X.
Definition 2.11. Let (E, F) be a resistance form on a set X. R(E,F ) is called
the resistance metric on X associated with the resistance form (E, F) on X.
If no confusion can occur, we write R(E,F ) = R.
Let (E, F) be a resistance form on X and let R be the associated resistance
metric on X. Then by (RF4), for any u ∈ F and any p, q ∈ X,
R(p, q)E(u, u) ≥ |u(p) − u(q)|2 .
(2.1)
Hence every u ∈ F is uniformly 1/2-Hölder continuous with respect to R. So,
if Ω is the completion of X with respect to R, then any u ∈ F is naturally
extended to a continuous function on Ω. Using this extension, we may always
regard F as the collection of functions on Ω.
Theorem 2.12. Let (E, F) be a resistance form on X and let R be the associated resistance metric on X. If Ω is the completion of X with respect to R, then
(E, F) is a resistance form on Ω. Moreover, the resistance metric associated
with (E, F) on Ω is the natural extension of the resistance metric R associated
with (E, F) on X.
By the virtue of this theorem, if (E, F) is a resistance form on X and R is
the associated resistance metric, then (X, R) may be assumed to be complete.
6
Theorem 2.13. Let S = {(Vm , Hm )}m≥0 be a compatible sequence. Then
(ES , F(S)) is a resistance form on V∗ and the associated resistance metric coincides with RS . Moreover, if (Ω, R) is the completion of (V∗ , RS ), then (Ω, R)
is separable and R is the resistance metric associated with the resistance form
(ES , F(S)) on Ω.
By Theorem 2.13, from a compatible sequence, we can construct a resistance
form on a set which is complete and separable under the associated resistance
metric. The next theorem shows that the converse is also true.
Theorem 2.14. Let (E, F) is a resistance form on X and let R be the associated
resistance metric on X. Assume that (X, R) is separable. If {Vm }m≥0 is a
increasing sequence of finite subsets of X, then S = {(Vm , HVm )}m≥0 , where
HVm is defined in Proposition 2.10, is a compatible sequence and RS = R on
∪m≥0 Vm . In particular, if ∪m≥0 Vm is dense in X, then RS = R and (E, F) =
(ES , F(S)).
Next we will define the notion of harmonic functions.
Proposition 2.15. Let (E, F) be a resistance form on X and let V be a finite
subset of X. Then for any ρ ∈ (V ), there exists a unique u ∈ F such that
u|V = ρ and
E(u, u) = EHV (ρ, ρ) = min{E(v, v)|v ∈ F, v|V = ρ}.
Moreover, u is the unique element of F that satisfies, for any finite set U ⊆ X
containing V ,
HU u|U \V = 0
.
(2.2)
u|V
=ρ
Denoting u appearing in the above theorem by hV (ρ), we see that hV :
(V ) → F is linear.
Definition 2.16. Let (E, F) be a resistance form on X and let V be a finite
subset of X. We define HV = Im(hV ). An element of HV is called a V -harmonic
function with respect to (E, F). More precisely, if u = hV (ρ) for ρ ∈ (V ), then u
is called the V -harmonic function with boundary value ρ with respect to (E, F).
Also, for any p ∈ V , hV (χVp ) is denoted by ψpV .
It is easy to see that HV is spanned by {ψpV }p∈V . In fact, u = p∈V u(p)ψpV
for any u ∈ HV .
The second characterization of harmonic functions, (2.2), immediately implies the following proposition.
Proposition 2.17. Let (E, F) be a resistance form on X. If U and V are finite
subsets of U and V ⊆ U , then HV ⊆ HU .
We also obtain the following maximum principle.
7
Proposition 2.18. Let (E, F) be a resistance form on X and let V be a finite
subset of X. If u is a V -harmonic function with respect to (E, F), then
min u(p) = min u(x) ≤ max u(x) = max u(p).
p∈V
x∈X
x∈X
p∈V
Proposition 2.19. Let V = ∅ be a finite subset of X. Define FV = {u|u ∈
F, u|V ≡ 0}. Then E is an inner product on FV and (FV , E) is a Hilbert space.
(EV , FV ) may be regarded as a resistance form imposed Dirichlet boundary
condition on V .
Proof. If u ∈ FV and u is constant on X, then u = 0 on X. Hence FV
can be thought of as an closed subspace of (F/ ∼, E). By (RF2), (FV , E) is
complete.
It follows that HV ⊕ FV = F. In fact, defining PV : F → HV by PV (u) =
hV (u|V ), we see that, for any u ∈ F, u = PV u + (u − PV u), where PV u ∈ HV
and u − PV u ∈ FV . Although E is not an inner product on F, the following
lemma says that each of HV and FV may be thought of as the “orthogonal
complement” of the other with respect to E.
Lemma 2.20. Let V = ∅ be a finite subset of X.
(1) For any u ∈ HV and any v ∈ FV ,
E(u, v) = 0
(2) For any u, v ∈ F,
E(u, v) = E(u − PV u, v − PV v) + E(PV u, PV v)
= E(u − PV u, v − PV v) + EHV (u|V , v|V ).
where HV is the Laplacian on V associated with (E, F).
By this lemma, if u ∈ HV and v ∈ F, then E(u, v) = EHV (u|V , v|V ).
Proof. (1) Since PV (αu + v) = αu, we have E(αu + v, αu + v) ≥ E(αu, αu) for
any α ∈ R. Therefore E(u, v) = 0.
(2) By (1), E(u, v) = E(u − PV u, v − PV (v)) + E(PV u, PV v). Now, Proposition 2.15 implies that E(PV u, PV v) = EHV (u|V , v|V ).
3
Resistance between a point and a set
In this section, we study resistance between a point and a set and introduce the
notion of shorted resistance form.
Throughout this section, (E, F) is a resistance form on a set X and R is the
associated resistance metric.
8
Proposition 3.1. Let V = ∅ be a finite subset of X and let p ∈ X\V . Define
R(p, V ) = E(ψpV ∪p , ψpV ∪p )−1 . Then
−1
R(p, V ) = min{E(u, u)|u ∈ F, u(p) = 1, u|V ≡ 0}
.
Moreover, if H = HV ∪p , then
R(p, V ) = −(Hpp )−1 = (
Hpq )−1 .
(3.1)
(3.2)
q∈V
R(p, V ) is called the effective resistance between p and V . If p ∈ V , then we
set R(p, V ) = 0.
Proof. (3.1) is immediate by Proposition 2.15 and Definition 2.16. Also by
Proposition 2.15,
E(ψpV ∪p , ψpV ∪p ) = EH (χVp , χVp ).
This implies (3.2).
Next we state three useful lemmas. The first lemma is used to prove the
following two lemmas. It says that the effective resistance between two terminals
is no larger than the resistance of the resistor directly attached between them.
Lemma 3.2. Let U be a finite subset of X and let H = HU be the Laplacian
on U associated with (E, F). Define Rpq = (Hpq )−1 for any p = q ∈ U . Then
Rpq ≥ R(p, q).
Proof. Let W = {p, q}. Then by Proposition 2.15, there exists a W -harmonic
function ψ ∈ F such that ψ(p) = 1, ψ(q) = 0 and R(p, q) = E(ψ, ψ)−1 . Note
that Proposition 2.17 impliesthat ψ is also a U -harmonic function. Hence
2
E(ψ, ψ) = EH (ψ|U , ψ|U ) =
r,s∈U Hrs (ψ(r) − ψ(s)) /2 ≥ Hpq . Therefore,
Rpq ≥ R(p, q).
Lemma 3.3. Let V = ∅ be a finite subset of X. Then, for any p ∈ X,
(#V )−1 min R(p, q) ≤ R(p, V ) ≤ min R(p, q),
q∈V
q∈V
where #V is the number of elements of V .
Proof. Let A = {u|u ∈ F, u(p) = 1, u|V ≡ 0}. Also let Aq = {u|u ∈ F, u(p) =
1, u(q) = 0} for each q ∈ V . Since Aq ⊇ A, R(p, q)−1 = minu∈Aq E(u, u) ≤
minu∈A E(u, u) = R(p, V )−1 . Hence R(p, V ) ≤ minq∈V R(p, q).
Let H =HV ∪p and let Rpq = (Hpq )−1 for any q ∈ V . Then by (3.2),
R(p, V ) = ( q∈V (Rpq )−1 )−1 ≥ (#V )−1 minq∈V Rpq . Now the required inequality follows immediately by Lemma 3.2.
Lemma 3.4. Let V = ∅ be a finite subset of X. For any p ∈ V and q ∈ X\p,
0 ≤ ψpV (q) ≤
9
R(q, V )
.
R(p, q)
Proof. Since ψpV (q) = 0 for q ∈ V , we may assume that q ∈
/ V . Let H = HV ∪q .
By (2.2),
0 = (HψpV )(q) =
Hrq (ψp (r) − ψpV (q)) = Hpq − (
Hrq )ψpV (q).
r∈V
r∈V
Using (3.2) and Lemma 3.2, we obtain ψpV (q) = R(q, V )Hpq ≤ R(q, V )/R(p, q).
Now we consider an electrical network shorted on a finite set.
Definition 3.5. Let V = ∅ be a finite subset of X. Set
F V = {u|u ∈ F, u is constant on V .}
and define a quadratic form on F V , E V by E V (u, v) = E(u, v) for any u, v ∈ F V .
Let X V = (X\V ) ∪ b, where b is an element of V .
XV is the set where V is retracted to one point b. If no confusion can occur,
we denote b ∈ X V by V . Note that F V is identified with a collection of real
valued functions on X V .
Proposition 3.6. (1) (E V , F V ) is a resistance form on X V .
(2) Let U be a finite subset of X which contains V . Then u ∈ F V is a U harmonic function with respect to (E, F) if and only if u is a U V -harmonic
function with respect to (E V , F V ), where U V ⊂ X V is defined by U V = U \V ∪
{V }.
(3) Let RV be the resistance metric associated with (E V , F V ). Then for any
x, y ∈ X\V , RV (x, y) ≤ R(x, y). Also, if x ∈ X\V , RV (x, V ) = R(x, V ).
(E V , F V ) is called the V -shorted resistance form of (E, F).
Proof. (1) It is straightforward to show (RF1) through (RF5).
(2) Suppose that ρ ∈ (U ) is constant on V . Note that ρ is naturally identified
with an element in (U V ). Hence
min{E(v, v)|v ∈ F, v|U = ρ} = min{E V (v, v)|v ∈ FV , v|U V = ρ}.
This immediately implies (2).
(3)
R(x, y)−1 = min{E(u, u)|u ∈ F, u(p) = 1, u(q) = 0} ≤
min{E V (u, u)|u ∈ F V , u(p) = 1, u(q) = 0} = RV (x, y)−1 .
The rest is obvious by definitions of RV (x, V ) and R(x, V ).
10
4
Green function
In this section, we will define the Green function associated with a resistance
form with Dirichlet boundary condition and show that the Green function is the
reproducing kernel of the form. Then, the Green function will be shown to be
uniformly Lipschitz continuous with respect to the resistance metric. Also we
will see that harmonic functions are uniformly Lipschitz continuous with respect
to the resistance metric as well.
Throughout this section, we assume that (E, F) is a resistance form on a set
X, that R is the associated effective resistance on X and that B is a non-empty
finite subset of X.
x
Definition 4.1 (Green function). For any x ∈ X, define gB
= R(x, B)ψxB∪x .
x
Also define gB (x, y) = gB (y) for any x, y ∈ X. gB is called the Green function
of the resistance form (E, F) associated with the boundary B or the B-Green
function of (E, F).
x
By the above definition, gB
∈ F B ⊂ F. Also recalling Proposition 3.1, we
have E(ψxB∪x , ψxB∪x ) = R(x, B)−1 for any x ∈ X\B. These facts immediately
imply the following proposition.
Proposition 4.2. (1) gB (x, y) ≥ 0 for any x, y ∈ X. Also gB (x, y) = 0 if
x ∈ B or y ∈ B.
x
x
(2) E(gB
, gB
) = R(x, B) = gB (x, x) for any x ∈ X.
(3) gB (x, x) ≥ gB (x, y) for any x, y ∈ X.
In fact, gB is symmetric.
x
Proposition 4.3. For any u ∈ FB , E(gB
, u) = u(x). In particular, for any
y
x
x, y ∈ x, E(gB , gB ) = gB (x, y) = gB (y, x).
This fact shows that gB is the reproducing kernel of the form (E, FB ). Recall
that (FB , E) is a Hilbert space as we have shown in Proposition 2.19.
Also by this fact and Definition 4.1, we immediately have a relation between the effective resistance and the hitting time (when a stochastic process is
associated with the resistance form). See Appendix B for details.
Proof. Let V = B ∪ x and let D = HV ∈ LA(V ) be the Laplacian on V
x
associated with (E, F). Then since gB
is a V -harmonic function with respect to
(E, F),
x
x
x
E(gB
, u) = ED (gB
|V , u|V ) = −gB
(x)(Du)(x) = −gB (x, x)Dxx u(x).
(4.1)
x
Also by (3.2), R(x, B)−1 = −Dxx . This along with (4.1) implies that E(gB
, u) =
u(x).
Next we give an alternative expression of the Green function. Let V be
a finite subset of X containing B and let H = HV be the Laplacian on V
11
associated with (E, F). Then there exist linear maps T : (B) → (B), J :
(B) → (V \B) and X : (V \B) → (V \B) such that
T tJ
H=
.
J X
It is know that X is invertible. See [16, Lemma 2.1.5]. Set G = (−X)−1 .
Proposition
4.4. In the above situation, gB (p, q) = Gpq for any p, q ∈ V \B.
p
Also gB
= q∈V \B Gpq ψqV for any p ∈ V \B.
Proof. For any u ∈ HV ∩ FB and any p ∈ V \B,
p
p
p
u(p) = E(u, gB
) = EH (u, gB
) = −(u|V \B , X(gB
|V \B ))V \B ,
where (·, ·)V \B is the standard inner-product defined in Section 2. This imV \B
p
mediately implies that−X(gB
|V \B ) = χp . Hence gB (p, q) = Gpq for any
p
V
p, q ∈ V \B. Set f = q∈V \B Gpq ψq for p ∈ V \B. Then both gB
and f are
p
p
V -harmonic functions and gB |V = f |V . Therefore gB = f .
It is remarkable that the Green function is uniformly Lipschitz continuous
with respect to the resistance metric as follows.
Theorem 4.5. For any x, y, z ∈ X,
|gB (x, y) − gB (x, z)| ≤ R(y, z).
Corollary 4.6. Let V = ∅ be a finite subset of X and let D = HV ∈ LA(V ) be
the Laplacian on V associated with (E, F). If u is a V -harmonic function with
respect to (E, F), then, for any x, y ∈ X,
|u(x) − u(y)| ≤ (−tr(D))( max |u(p) − u(q)|)R(x, y),
p,q∈V
where tr(·) is the trace of matrices.
Proof. Set Vp = V \p for any p ∈ V . Then gVpp = R(p, Vp )ψpV . Hence Theorem 4.5 implies that
|ψpV (x) − ψpV (y)| ≤ R(p, Vp )−1 R(x, y)
for any x, y ∈ X. Since u =
p∈V
u(p)ψpV ,
|u(x) − u(y)| ≤
|u(p)|
R(x, y).
R(p, Vp )
p∈V
Replacing u by u − α, we obtain
|u(x) − u(y)| ≤
|u(p) − α|
R(x, y)
R(p, Vp )
p∈V
12
Note that R(p, Vp )−1 = E(ψpV , ψpV ) = −Dpp . If α = u(p∗ ) for some p∗ ∈ V , then
|u(p) − u(p∗ )|
−Dpp max |u(p) − u(q)|.
≤
p,q∈V
R(p, Vp )
p∈V
p∈V
The rest of this section is devoted to proving Theorem 4.5.
Lemma 4.7. (1) For any x, y ∈ X,
|gB (x, x) − gB (y, y)| ≤ R(x, y).
(2) For any x, y ∈ X,
0 ≤ gB (x, x) + gB (y, y) − 2gB (x, y) ≤ R(x, y).
Proof. (1) Consider the shorted resistance form (E B , F B ) on X B and the
shorted effective resistance RB . Since RB is a metric on X B , |RB (x, B) −
RB (y, B)| ≤ RB (x, y). By Proposition 3.6, |R(x, B) − R(y, B)| ≤ R(x, y). This
immediately implies the required inequality.
y
x
(2) Let h = gB
− gB
. Then E(h, h) = h(x) − h(y) = gB (x, x) + gB (y, y) −
2gB (x, y). Since E(h, h)R(x, y) ≥ |h(x)−h(y)|2 , it follows that 0 ≤ h(x)−h(y) ≤
R(x, y).
Lemma 4.8. For any x, y ∈ X,
|gB (x, x) − gB (x, y)| ≤ R(x, y).
Proof. By Lemma 4.7, 2|g(x, x) − g(x, y)| ≤ |g(x, x) + g(y, y) − 2g(x, y)| +
|g(x, x) − g(y, y)| ≤ 2R(x, y).
Lemma 4.9. For any x, y, z ∈ X,
gB (y, y)gB (z, x) ≤ gB (y, x)gB (y, z).
Proof. Fix y and z. Let u(x) = gB (y, y)gB (z, x) and let v(x) = gB (y, x)gB (y, z).
Then u(x) and v(x) are harmonic functions with respect to (X, E, F, B ∪ y ∪ z).
Since u(y) = v(y), u|B = vB ≡ 0 and u(z) = gB (y, y)gB (z, z) ≥ gB (y, z)2 =
v(z), the maximum principle implies that u(x) ≥ v(x) for any x ∈ X.
Proof of Theorem 4.5. By Lemma 4.9,
gB (y, y) ≥
gB (x, y)
(gB (y, y) − gB (y, z))
gB (y, y)
gB (x, y)gB (y, z)
≥ gB (x, y) −
≥ g(x, y) − g(x, z).
gB (y, y)
13
Exchanging y and z,
gB (z, z) − gB (z, y) ≥ gB (x, z) − gB (y, z).
Hence by Lemma 4.8,
|gB (x, y) − gB (x, z)| ≤ max{gB (z, z) − gB (z, y), gB (y, y) − gB (y, z)} ≤ R(y, z).
5
Green operators
By making use of the Green function gB (x, y), the associated Green operator
GB is formally given by
(GB f )(x) =
gB (x, y)f (y)µ(dy),
(5.1)
X
where µ is a measure on X In this section, we will define Green operators in
a rather universal way in Theorem 5.5. Our Green operators coincide with
the integral operator given by (5.1) in restricted situations. Through Green
operators, we will finally obtain the universal domain of Laplacians, DL , in
Definition 5.11.
As in the last section, (E, F) is a resistance form on a set X and R is the
associated resistance metric on X. Also we assume that (X, R) is separable in
this section.
Definition 5.1. For any p ∈ X and any u : X → R, we define
|u(x)|
||u||p, 12 = sup x∈X
1 + R(x, p)
.
(5.2)
and
C 12 (X, R) = {u|u : X → R, u is continuous on X, ||u||p, 12 < ∞}
In (5.2), we allow ∞ as a value of the supremum.
It is easy to see that the definition of C 12 (X, R) does not depend on p ∈ X.
In fact, || · ||p, 12 is a norm on C 12 (X, R) and
(1 + R(p, q))− 2 ||u||q, 12 ≤ ||u||p, 12 ≤ (1 + R(p, q)) 2 ||u||q, 12
1
1
for any q ∈ X and any u : X → R. Moreover, (C 12 (X, R), || · ||p, 12 ) is a Banach
space.
Proposition 5.2. F ⊂ C 12 (X, R). Moreover, let B = ∅ be a finite subset of
X. Then the natural inclusion map from (FB , E) to (C 12 (X, R), || · ||p, 12 ) is
continuous. In particular, if p ∈ B, then E(u, u) ≥ ||u||p, 12 .
14
Proof. By (2.1),
E(u, u) + |u(p)|
≥
|u(x)|
1 + R(x, p)
1 + R(x, p)
for any u ∈ F. Hence E(u, u) + |u(p)| ≥ ||u||p, 12 .
Hereafter, when no confusion can occur, we write || · || 12 or, simply, || · ||
instead of || · ||p, 12 .
1 (X, R) be the completion of F with respect to || · ||p, 1
Definition 5.3. Let C
2
2
1 (X, R), || · ||p, 1 ) is denoted by M (X, R).
in C 1 (X, R). The dual space of (C
2
2
2
For any ϕ ∈ M (X, R), ||ϕ||M (X,R) is the dual norm:
1 (X, R), ||u||p, 1 = 1}.
||ϕ||M (X,R) = sup{|ϕ(u)| : u ∈ C
2
2
For ease of notation, we sometimes use || · || in place of || · ||M (X,R) .
Using Lemma 2.9, we immediately see the following lemma.
1 (X, R), u ∨ v and u ∧ v belong to C
1 (X, R).
Lemma 5.4. For any u, v ∈ C
2
2
Let ϕ ∈ M (X, R). Then by Proposition 5.2, ϕ|FB : FB → R is continuous
with respect to the inner product E on FB . Since (FB , E) is a Hilbert space, the
dual space of (FB , E) can be identified with (FB , E) itself. Therefore, we have
the following theorem.
Theorem 5.5. Let B = ∅ be a finite subset of X. Then there exists a unique
continuous linear map GB : M (X, R) → FB such that
E(GB ϕ, u) = ϕ(u)
(5.3)
for any ϕ ∈ M (X, R) and any u ∈ FB . In particular,
x
(GB ϕ)(x) = ϕ(gB
).
(5.4)
for any ϕ ∈ M (X, R) and any x ∈ X. Moreover, for any ϕ ∈ M (X, R), GB ϕ
is uniformly Lipschitz continuous with respect to R on X.
Proof. Existence of GB ϕ satisfying (5.3) follows from the arguments above. By
Proposition 5.2,
E(GB ϕ, GB ϕ) = ϕ(GB ϕ) ≤ ||ϕ||||GB ϕ||p, 12 ≤ ||ϕ|| E(GB ϕ, GB ϕ).
This implies E(GB ϕ, GB ϕ) ≤ ||ϕ||. Hence GB is continuous. Making use
of the B-Green function, we obtain (5.4). By Theorem 4.5, |(GB ϕ)(x) −
y
x
− gB
)| ≤ ||ϕ||R(x, y).
(GB ϕ)(y)| ≤ |ϕ(gB
Definition 5.6. Let B = ∅ be a finite subset of X. Then GB is called the
Green operator associated with boundary B with respect to (E, F) or the BGreen operator with respect to (E, F).
15
The Green operator GB is, indeed, the universal version of the integral
operator given by (5.1). See Section 8 for details.
Proposition 5.7. Let B = ∅ be a finite subset of X. Then, for any ϕ ∈
M (X, R) and any u ∈ F,
E(GB ϕ, u) = ϕ(u) −
ϕ(ψpB )u(p).
(5.5)
p∈B
In particular, ker GB = { p∈B αp δp }, where δp is the Dirac’s delta at p :
δp (u) = u(p) for any u ∈ C 12 (X, R).
Proof. Let uB = PB u = p∈B u(p)ψpB . By Lemma 2.20 and (5.3)
E(GB ϕ, u) = E(GB ϕ, u − uB ) = ϕ(u − uB ).
This immediately implies (5.5).
p
Lemma 5.8. Let B = ∅ be a finite subset of X. Then GB (δp ) = gB
for any
p ∈ X.
P
Proof. Let u = GB (δp ). Then both u and gB
belong to FB . Also, for any
p
p
v ∈ FB , E(u, v) = δp (v) = v(p) = E(gB , u). Then u = gB
.
In the rest of this section, we will study relations between the Green operators with different boundaries.
Lemma 5.9. Let B1 and B2 be non-empty finite subsets of X satisfying B1 ⊆
B2 .
p
(1) {gB
, ψqB1 |p ∈ B2 \B1 , q ∈ B1 } is a base of HB2 . In fact, for any u ∈ HB2 ,
1
p
u=
E(u, ψpB2 )gB
+
u(q)ψqB1 .
1
p∈B2 \B1
(2) For any ϕ ∈ M (X, R),
GB2 ϕ = G B1 ϕ −
q∈B1
p
ϕ(ψpB2 )gB
= GB1 (ϕ −
1
p∈B2 \B1
ϕ(ψpB2 )δp )
p∈B2 \B1
Proof. (1) Set u1 = u − q∈B1 u(q)ψqB1 . Then u1 = u − PB1 u. Also since
PB1 u ∈ HB1 ⊆ HB2 , we see that u1 ∈ HB2 . Hence by Lemma 2.20, for any
v ∈ FB1 ,
E(u1 , v) = E(u1 , PB2 v) = E(u1 ,
v(p)ψpB2 ).
(5.6)
p∈B2 \B1
Note that PB2 v ∈ FB1 . Again by E(PB1 u, PB2 v) = 0. This
along with (5.6) pimplies E(u1 , v) = p∈B2 \B1 E(u, ψpB2 )v(p). Hence u1 = p∈B2 \B1 E(u, ψpB2 )gB
.
1
B2
(2) By Proposition 5.7, E(GB2 ϕ, u) = ϕ(u) − p∈B2 \B1 ϕ(ψp )u(p) for any
u ∈ FB1 . Hence E(GB1 ϕ − GB2 ϕ, u) = p∈B2 \B1 ϕ(ψpB2 )u(p) for any u ∈ FB1 .
p
Since GB2 ϕ ∈ FB2 ⊆ FB1 , we see that GB1 ϕ − GB2 ϕ = p∈B2 \B1 ϕ(ψpB2 )gB
.
1
Now combining this with Lemma 5.8, we obtain the desired equality.
16
The next theorem is the principal relation between the images of the Green
operators.
Theorem 5.10. (1) Let B1 and B2 be non-empty finite subsets of X satisfying
L
L
L
⊇ DB
, where DB,0
= Im(GB ) for any non-empty finite
B1 ⊆ B2 . Then DB
1 ,0
2 ,0
subset of X, B.
L
L
L
(2) DB,0
+ HB is independent of B: DB
+ HB1 = DB
+ HB2 for any
1 ,0
2 ,0
non-empty finite subsets of X, B1 and B2 .
Proof. (1) By Lemma 5.9-(2), GB2 ϕ ∈ Im(GB1 ) for any ϕ ∈ M (X, R).
L
L
(2) Define DB
= DB,0
+ HB for any non-empty finite subset of X, B. It is
L
L
enough to show that DB
= DB
if B1 ⊆ B2 . Note that HB1 ⊆ HB2 . Since
1
2
p
L
gB1 ∈ HB2 for any p ∈ B2 \B1 , Lemma 5.9-(2) implies that GB1 ϕ + u ∈ DB
for
2
L
L
any ϕ ∈ M (X, R) and any u ∈ HB1 . Therefore DB1 ⊆ DB2 .
On the other
5.8 andLemma 5.9, we see that, for any
hand, by Lemma
L
2
u ∈ HB2 , u = p∈B2 \B1 E(u, ϕB
)G
δ + q∈B1 u(q)ψqB1 ∈ DB
. Combining
B
p
1 p
1
L
this with (1), it follows that GB1 ϕ+u ∈ DB1 for any ϕ ∈ M (X, R) and u ∈ HB2 .
L
L
Therefore DB
⊇ DB
.
1
2
L
Definition 5.11. Define DL ⊂ F by DL = DB,0
+HB , where B is a non-empty
finite subset of X.
In the next section, we will define Laplacians on DL , which may be thought of
as the universal domain of Laplacians. In 7, we give an characterization of DL
with respect to discrete Laplacians {HV |V is a finite subset of X} associated
with (E, F).
If B = {p} for p ∈ X, then HB is the collection of constants on X. So, in
L
+ R.
such a case DL = Dp,0
Proposition 5.12. For any u ∈ DL , u is uniformly Lipschitz continuous with
respect to R on X.
Proof. Let p ∈ X and let B = {p}. Then, for any u ∈ DL , there exist ϕ ∈
M (X, R) and a constant c such that u = GB ϕ + c. As GB ϕ is uniformly
Lipschitz continuous by Theorem 5.5, so is u.
Define
CL (X, R) = {u|u is uniformly Lipschitz continuous with respect to R on X}.
(5.7)
Knowing the above proposition, we might expect that CL (X, R) ⊆ F and that
DL = CL (X, R) ∩ F. Both conjecture are not true however even if (X, R) is
compact. See Corollary 9.14.
17
6
Laplacians
In this section, we continue to assume that (E, F) is a resistance form on a set
X, that R is the associated resistance metric on X and that (X, R) is separable.
To define Laplacians, we need to know more about the space M (X, R). If
(X, R) is locally compact and F is dense in C 12 (X, R), then the Riesz theorem
(see [20] for example) implies that
M (X, R) = {µ|µ = µ+ − µ− , where µ± are Borel regular measures on X
that satisfy
1 + R(x, p)µ± (dx) < ∞}.
X
Although (X, R) may not be locally compact in general, we can still divide
ϕ ∈ M (X, R) into the positive part ϕ+ and the negative part ϕ− by similar
arguments to those in the proof of the Riesz theorem.
Definition 6.1. (1) Let u and v be real valued function on X. We write
+
u ≤ v if and only if u(x) ≤ v(x) for any x ∈ X. Define C
1 (X, R) = {u|u ∈
2
C 1 (X, R), u ≥ 0}.
2
+
(2) We say ϕ ∈ M (X, R) is non-negative if ϕ(u) ≥ 0 for any u ∈ C
1 (X, R).
Define M + (X, R) = {ϕ|ϕ ∈ M (X, R), ϕ is non-negative.}
2
Theorem 6.2. For any ϕ ∈ M (X, R), there exists a unique pair (ϕ+ , ϕ− ) ∈
M + (X, R)2 satisfying ϕ = ϕ+ − ϕ− and
ϕ+ (u) + ϕ− (u) =
sup
1 (X,R)
h∈C
|ϕ(h)|
(6.1)
2
0≤|h|≤u
+
for any u ∈ C
1 (X, R).
2
By analogy with the Riesz theorem, ϕ+ + ϕ− corresponds to the “total
variation” of ϕ.
1 (X, R) and any ν ∈ M + (X, R).
Remark. We see ν(|f |) ≥ |ν(f )| for any f ∈ C
2
Therefore, (6.1) implies ||ϕ|| = ||ϕ+ + ϕ− || ≥ ||ϕ± ||.
+
+
Proof. First, for u ∈ C
1 (X, R), define Uu = {h|h ∈ C 1 (X, R), 0 ≤ h ≤ u} and
2
2
ϕ+ (u) = sup ϕ(h).
h∈Uu
+
Since ϕ(0) = 0, ϕ+ (u) ≥ 0. Let u and v belong to C
1 (X, R). Then,
2
ϕ+ (u + v) = sup ϕ(h) ≥
h∈Uu+v
sup
h1 ∈Uu ,h2 ∈Uv
18
ϕ(h1 + h2 ) = ϕ+ (u) + ϕ+ (v).
(6.2)
On the other hand, for any h ∈ Uu+v , define h1 and h2 by h1 = h ∧ u and
h2 = h − h1 . By Lemma 5.4, h1 ∈ Uu and h2 ∈ Uv . This immediately implies
that equality holds in (6.2). So we have obtained
ϕ+ (u + v) = ϕ+ (u) + ϕ+ (v)
(6.3)
+
for any u, v ∈ C
1 (X, R). Also it is easy to see that ϕ+ (αu) = αϕ+ (u) for any
2
+
1 (X, R). Note
u ∈ C 1 (X, R) and α ≥ 0. Next we define ϕ+ (u) for any u ∈ C
2
2
1 (X, R), there exist u1 and u2 such that u1 , u2 ∈ C
+
that, for any u ∈ C
1 (X, R)
2
2
and u = u1 −u2 . In fact, we may let u1 = u∨0 and u2 = u1 −u. So, if u = u1 −u2
+
for u1 , u2 ∈ C
1 (X, R), then we set ϕ+ (u) = ϕ+ (u1 ) − ϕ+ (u2 ). By (6.3), ϕ+ (u)
2
+
is well-defined: if u = u1 − u2 = v1 − v2 for u1 , u2 , v1 , v2 ∈ C
1 (X, R), then
2
ϕ+ (u1 )−ϕ+ (u2 ) = ϕ+ (v1 )−ϕ+ (v2 ). It is routine to show that ϕ+ ∈ M + (X, R).
Defining ϕ− = ϕ+ − ϕ, we see that ϕ− = (−ϕ)+ . Obviously ϕ = ϕ+ − ϕ−
+
and ϕ± ∈ M + (X, R). Next we show (6.1). Let u ∈ C
Note that
1 (X, R).
2
1 (X, R)
|h1 − h2 | ≤ u for any h1 , h2 ∈ Uu . On the other hand, for any h ∈ C
2
with 0 ≤ |h| ≤ u, define h+ = h ∨ 0 and h− = (−h) ∨ 0. Then h± ∈ Uu and
h = h+ − h− . Therefore, by the fact that ϕ− = (−ϕ)+ , we have
ϕ+ (u) + ϕ− (u) =
sup
h1 ,h2 ∈Uu
ϕ(h1 − h2 ) =
sup
1 (X,R)
h∈C
|ϕ(h)|.
(6.4)
2
|h|≤u
Hence (6.1) holds.
The remaining part is the uniqueness. Let ϕ± be the ones defined above.
Assume that there exist ν± ∈ M + (X, R) satisfying that ϕ = ν+ − ν− and
ν+ (u) + ν− (u) = suph∈C 1 (X,R),0≤|h|≤u |ϕ(h)|. Then ν+ (u) + ν− (u) = ϕ+ (u) −
2
ϕ− (u). Since ϕ+ − ϕ− = ν+ − ν− , it follows that ν+ = ϕ+ and ν− = ϕ− .
Definition 6.3. Let {Vm }m≥0 be a family of finite subsets of X. We say that
{Vm }m≥0 is an admissible sequence of (X, R) if and only if Vm ⊆ Vm+1 for any
m ≥ 0 and V∗ is dense in (X, R), where V∗ is defined by V∗ = ∪m≥0 Vm .
Lemma 6.4. Let {Vm }m≥0 be an admissible sequence of (X, R). Then for any
p ∈ V∗ and any ϕ ∈ M (X, R), ϕ(ψpVm ) converges as m → ∞. Moreover the
limit limm→∞ ϕ(ψpVm ) does not depend on the choice of {Vm }m≥0 : if {Um }mge0
is an admissible sequence of (X, R) and p ∈ U∗ ∩ V∗ , then limm→∞ ϕ(ψpUm ) =
limm→∞ ϕ(ψpVm ).
Proof. By Theorem 6.2, we may assume that ϕ is non-negative without loss of
V
generality. Note that both ψpVm and ψp m+1 are Vm+1 -harmonic functions. Hence
V
the maximum principle implies that ψpVm ≥ ψp m+1 ≥ 0. Hence {ϕ(ψpVm )}m≥0 is
monotonically decreasing and uniformly bounded. Hence it converges as m →
∞. Let a = limm→∞ ϕ(ψpVm ) and let b = limm→∞ ϕ(ψpUm ). Assume that a > b.
19
Then there exists m ≥ 0 such that ϕ(ψpUm ) < a. Note that ϕ(pVp m ) ≥ a and
hence Vm = Um . So, V = (Vm ∪ Um )\Vm is not empty. Now by the maximum
principle and Lemma 3.4,
sup |ψpVk (x) − ψpVk ∪V (x)| ≤ max ψpVk (q) ≤ max
q∈V
x∈X
q∈V
R(q, Vk )
→0
R(p, q)
as k → ∞. Therefore ||ψpVk − ψpVk ∪V ||p, 12 → 0 as k → ∞. This implies the
following contradiction:
a = lim ϕ(ψpVk ) = lim ϕ(ψpVk ∪V ) ≤ ϕ(ψpVm ∪Um ) ≤ ϕ(ψpUm ) < a.
k→∞
k→∞
Hence a = b.
Definition 6.5. Let p ∈ X and let {Vm }m≥0 be an admissible sequence of
(X, R) with p ∈ V0 . For any ϕ ∈ M (X, R), define ϕ(p) = limm→∞ ϕ(ψpVm ).
By Lemma 6.4, ϕ(p) does not depend on a choice of {Vm }m≥0 .
Next we define Neumann derivatives of u ∈ DL at p ∈ X.
Theorem 6.6. Let p ∈ X and let {Vm }m≥0 be an admissible sequence of (X, R)
with p ∈ V0 . Then, for any u ∈ DL , −(HVm u)(p) converges as m → ∞, where
HVm is the Laplacian on Vm associated with (E, F). Moreover, define
(du)p = − lim (HVm u)(p).
m→∞
Then (du)p is independent of a choice of {Vm }m≥0 . (du)p is called the Neumann
derivative of u at p. In particular, if B = ∅ is a finite subset of X and u =
f − GB ϕ, where f ∈ HB and ϕ ∈ M (X, R), then, for any x ∈ X,
−(HB f )(x) − ϕ(x) + ϕ(ψxB ) if x ∈ B,
(du)x =
−ϕ(x)
if x ∈
/ B,
where HB is the Laplacian on B associated with (E, F).
In Proposition 8.6, we will see that (du)p is “usually” equal to zero for p ∈
/ B.
Proof. First note that −(HVm u)(p) = E(u, ψpVm ) by Lemma 2.20-(2). Let B = ∅
be a finite subset of X. Choose {Vm }m≥0 so that V0 = B, that Vm ⊆ Vm+1
and that ∪m≥0 Vm is dense in (X, R). Suppose u = f − GB ϕ for f ∈ HB and
ϕ ∈ M (X, R). Then, for x ∈ B, Lemma 2.20 along with (5.5) implies
E(u, ψxm ) = E(f, ψxB ) − E(GB ϕ, ψxm ) = −(HB f )(x) − ϕ(ψxm ) + ϕ(ψxB ),
where ψpm = ψpVm . By Lemma 6.4, limm→∞ E(u, ψxm ) = −(HB f )(x) − ϕ(x) +
ϕ(ψxB ). When x ∈
/ B, we assume that x ∈ V1 . Then Lemma 2.20 along with
(5.5) implies
E(u, ψxm ) = E(f, PB ψxm ) − ϕ(ψxm ) +
ϕ(ψqB )ψxm (q).
q∈B
20
Using Lemma 3.4, we see that ψxm (q) → 0 as m → ∞ for any q ∈ B. Hence by
Lemma 6.4, we obtain E(u, ψxm ) → −ϕ(x) as m → ∞.
Choosing B = {p} and using Lemma 6.4, we also verify that (du)p is independent of a choice of {Vm }m≥0 .
Now we define “Laplacians”. First we consider a Laplacian with boundary
condition on a finite set B.
Definition 6.7. Let B be a non-empty finite subset of X.
(1) Define MBN A (X, R) = {ϕ|ϕ ∈ M (X, R), ϕ(p) = 0 for
any p ∈ B}.
(2) Define LB : DL → MBN A (X, R) by LB u = ϕ − p∈B ϕ(p)δp , where u =
uB − GB ϕ for uB ∈ HB and ϕ ∈ M (X, R). LB is called the B-Laplacian on X
associated with the resistance form (E, F).
Using Proposition 5.7, we see that LB is well-defined.
Theorem 6.8. For a non-empty finite set B, define G∗B = GB |MBN A (X,R) . Then
L
L . Moreover,
G∗B : MBN A (X, R) → DB,0
is bijective and (G∗B )−1 = −LB |DB,0
(1) For any u ∈ F and any v ∈ DL ,
E(u, v) =
u(p)(dv)p − (LB v)(u).
(6.5)
p∈B
(2) For any f ∈ (B) and any ϕ ∈ MBN A (X, R),
LB u = ϕ
and
u|B = f
(6.6)
if and only if
u=
f (p)ψpB − G∗B ϕ.
(6.7)
p∈B
(6.5) and (6.6) are counterparts of the Gauss-Green formula and the solution
to the Dirichlet problem for Poisson’s equation, respectively. (6.7) is equivalent
to
x
u(x) =
f (p)ψpB (x) − ϕ(gB
)
p∈B
for any x ∈ X.
NA
NA
Proof. Define prNA
B : M(X, R) → MB (X, R) by prB (ϕ) = ϕ −
p∈B ϕ(p)δp .
NA
L
Then Proposition 5.7 implies that ker GB = ker prB . Hence Im(G∗B ) = DB,0
∗
∗
and ker GB = {0}. Therefore GB is bijective. By the definition of LB , it follows
that (G∗B )−1 = −LB . This fact immediately implies that (6.6) is equivalent to
(6.7).
21
To show (6.5), assume that v = vB − GB ϕ for vB ∈ HB and ϕ ∈ M (X, R).
Then by (5.5) and Theorem 6.6,
E(u, v) = E(u, vB ) − E(u, GB ϕ)
=−
u(p)(HB vB )(p)−(ϕ(u)−
u(p)ϕ(ψpV )) =
u(p)(dv)p −(LB v)(u).
p∈B
p∈B
p∈V
Corollary 6.9. Let B be a non-empty finite subset of X. Then, for v ∈ DL ,
define Lv ∈ M (X, R) by
Lv = LB v −
(dv)p δp .
(6.8)
p∈B
Then L is independent of a choice of B and
E(u, v) = −(Lv)(u)
(6.9)
for any u ∈ F.
L is called the Neumann Laplacian (or the N-Laplacian for short) on X
associated with (E, F). Comparing (6.9) with (6.5), we might regard L as LB
with B = ∅.
Proof. Let B1 and B2 be non-empty finite subsets of X. Then, by (6.5),
u(p)(dv)p − (LB2 v)(u) =
u(p)(dv)p − (LB1 v)(u)
p∈B2
p∈B1
1 (X, R). Therefore,
for any u ∈ F. Note that F is dense in C
2
(dv)p δp − LB2 v =
p∈B2
(dv)p δp − LB1 v.
p∈B1
Hence L is independent of B. (6.9) is obvious by (6.5).
The following proposition express the basic property of L, that it is Fredholm
with index zero.
Proposition 6.10. Let ϕ ∈ M (X, R). There exists v ∈ DL such that Lv = ϕ if
and only if ϕ(1) = 0. Moreover, if Lv = ϕ, then L(v + c) = ϕ for any constant
c ∈ R. In particular, ker L = {u|u is constant on X} and Im(L) = {ϕ|ϕ ∈
M (X, R), ϕ(1) = 0}.
Proof. Since E(v, 1) = −(Lv)(1) = 0, Lv = ϕ implies ϕ(1) = 0. Conversely,
suppose ϕ(1) = 0. Let v = −Gp ϕ for p ∈ X. Then by (5.5), E(u, v) =
−ϕ(u) + ϕ(1)u(p) = −ϕ(u). Hence Lv = ϕ. The rest is obvious.
22
By (6.5) and (6.8), we immediately deduce the following relations between
Laplacians.
Lemma 6.11. (1) Let B be non-empty finite subset of X. Then, for any
v ∈ DL , Lv = LB v if and only if (dv)p = 0 for any p ∈ B.
(2) Let B1 , B2 be non-empty finite subsets of X satisfying B1 ⊆ B2 . Then, for
any v ∈ DL , LB1 v = LB2 v if and only if (dv)p = 0 for any p ∈ B2 \B1 .
Lemma 6.11-(1) may be thought of as a special case of Lemma 6.11-(2) with
B2 = B and B1 = ∅.
7
Characterization of the domain of the Laplacian
As in the previous sections, (E, F) is a resistance form on X, R is the resistance
metric on X associated with (E, F). (X, R) is assumed to be separable.
Definition 7.1. (1) Cb (X, R) is the collection of continuous and bounded
b (X, R) is the
functions on X equipped with the supremum norm || · ||∞ . Also C
completion of F ∩ Cb (X, R) with respect to the norm || · ||∞
(2) For any ϕ ∈ M (X, R), the total variation of ϕ, ||ϕ||T V , is defined by
||ϕ||T V = ϕ+ (1) + ϕ− (1),
where ϕ± is defined in Theorem 6.2.
By Theorem 6.2, the total variation of ϕ is given by
||ϕ||T V =
sup
1 (X,R)
h∈C
|ϕ(h)|.
(7.1)
2
0≤|h|≤1
Theorem 7.2. For any u : X → R, define
||u||D = sup{
|(HV u)(p)| : V is a non-empty finite subset of X}.
p∈V
Then ||u||D = ||Lu||T V < +∞ for any u ∈ DL
. Moreover, let {Vm }m≥0 be an
admissible sequence of (X, R). Then limm→∞ p∈Vm |(HVm u)(p)| = ||u||D for
any u ∈ DL .
Lemma 7.3. For any u ∈ DL and any non-empty finite set V ⊆ X,
|(HV u)(p)| ≤ ||Lu||T V
p∈∈V
Proof. Define α(p) = 1 if (HV u)(p) ≥ 0 and α(p) = −1 if |(HV u)(p) <
V
V
0.
Set
f
=
p∈V α(p)ψp . Since |(HV u)(p)| = |E(u, ψp )|, it follows that
p∈V |(HV u)(p)| = |E(u, f )| = |(Lu)(f )|. By the maximum principle (Proposition 2.18), |f | ≤ 1. Hence by the definition of || · ||T V , we obtain the desired
inequality.
23
1 (X, R) ∩
Proof of Theorem 7.2. Let u ∈ DL . For any > 0, there exists f ∈ C
2
Cb (X, R) such that ||f ||∞ ≤ 1 and
||Lu||T V − /3 ≤ |(Lu)(f )| ≤ ||Lu||T V .
(7.2)
1 (X, R). Hence we may choose h ∈ F such that
Note that F is dense in C
2
||Lu||||f − h|| ≤ /3. Set g = (h ∧ 1) ∨ (−1). Then, by Lemma 2.9, g ∈
F ∩ Cb (X, R) and ||g||∞ ≤ 1. Moreover, since ||f ||∞ ≤ 1, it follows that
|f (x) − h(x)| ≥ |f (x) − g(x)| for any x ∈ X. This implies |(Lu)(f − g)| ≤
||Lu||||f − g|| ≤ ||Lu||||f − h|| ≤ /3. By (7.2),
||Lu||T V − 2/3 ≤ |(Lu)(g)| ≤ ||Lu||T V .
(7.3)
Now let {Vm }m≥0 be an admissible sequence of (X, R). Then, for sufficiently
large m,
|(Lu)(g) − (Lu)(gm )| = |E(u, g) − E(u, gm )| ≤ /3,
where gm = PVm g = p∈Vm g(p)ψpVm . Hence by (7.3),||Lu||T V − ≤ |E(u, gm )|.
On the other hand, the fact that ||g||∞ ≤ 1 along with Lemma 7.3 implies
|E(u, gm )| = |
−g(p)(HVm u)(p)| ≤
|(HVm u)(p)| ≤ ||Lu||T V . (7.4)
p∈Vm
Therefore, ||Lu||T V − ≤
p∈Vm
|(HVm u)(p)| ≤ ||Lu||T V .
It is noteworthy that the sequence p∈Vm |(HVm )u(p)| in Theorem 7.2 is
monotonically nondecreasing by the following lemma.
p∈Vm
Lemma 7.4. Let U and V be non-empty finite subsets of X with V ⊆ U . Then
for any u : X → R,
(HV u)(p) =
ψpV (q)(HU u)(q),
(7.5)
q∈U
where HV and HU are
and U , respectively, associated with
Laplacians on V (E, F). In particular, p∈V |(HV u)(p)| ≤ p∈U |(HU u)(p)|.
Proof. Divide HU into four parts as in Proposition 2.6:
T tJ
,
HU =
J X
where T : (V ) → (V ), J : (V ) → (U \V ) and X : (U \V ) → (U \V ). Since
(V, HV ) ≤ (U, HU ), Proposition 2.6 implies that HV = T − tJX −1 J. For any
u ∈ lU , set u0 = u|V and u1 = u|U \V . Then,
HV u1 = (T − tJX −1 J)u1 = (T u0 + tJu1 ) − tJX −1 (Ju0 + Xu1 )
= (HU u)|V − tJX −1 (HU u)|U \V .
24
(7.6)
Now, ψpV is the V -harmonic function with boundary value χVp . Hence ψpV |U \V =
−X −1 JχVp . Therefore (−tJX −1 )pq = ψpV (q) for any p ∈ V and any q ∈ U \V .
Combining this with (7.6), we immediately verify (7.5) and hence
|(HV u)(p)| ≤
ψpV (q)|(HU u)(q)|.
q∈U
The rest of the statement follows by summing this for all p ∈ V .
Theorem 7.5. Suppose that (X, R) is bounded. Then u ∈ DL if and only if
u ∈ Cb (X, R) and ||u||D < +∞. Moreover, DL is a Banach space with respect
to the norm || · ||∞ + || · ||D .
Proof. Suppose u ∈ Cb (X, R) and ||u||D < +∞. Let {Vm }m≥0 be a sequence of
finite subsets of X satisfying that Vm ⊆ Vm+1 for any m ≥ 0 and that ∪m≥0 Vm
b (X, R) = C
1 (X, R) because (X, R) is
is dense in (X, R). Note that F ⊆ C
2
bounded. For any v ∈ F,
|EHVm (v|Vm , u|Vm )| ≤ ||v||∞
|(HVm u)(p)|.
p∈Vm
Therefore, u ∈ F and |E(v, u)| ≤ ||v||∞ ||u||D for any v ∈ F. Since F is dense in
b (X, R), E(·, u) can be extended to be a bounded linear functional on C
b (X, R).
C
Hence there exists ϕ ∈ M (X, R) such that E(v, u) = ϕ(v) for any v ∈ F. This
implies that u ∈ DL and ϕ = −Lu.
Obviously || · ||∞ + || · ||D is a norm on DL . We also see that DL is complete
under this norm.
8
Realization of Green operator and Laplacian
Let (E, F) be a resistance form on a set X and let R be the associated resistance
metric on R. We assume that (X, R) is separable and locally compact and
that, for any f ∈ C0 (X, R), there exists a sequence {fn }n≥0 ⊂ F ∩ Cb (X, R)
such that ||f − fn ||∞ → 0 as n → ∞, where C0 (X, R) is the collection of
continuous functions with compact support. Also in this section, µ is a σ-finite
Radon measure on (X, R): µ is a σ-finite Borel regular measure on (X, R) and
µ(K) < +∞ for any compact subset K ⊆ X. Under those assumptions, we
obtain
Proposition 8.1. F ∩ C0 (X, R) is dense in C0 (X, R).
Proof. Let f ∈ C0 (X, R) and let K be the support of f . Define f+ = f ∨ 0
and f− = (−f ) ∨ 0. Then f± ∈ C0 (X, R) and f = f+ − f− . Now, for any
≥ 0, there exists g ∈ F ∩ Cb (X, R) such that ||f+ − g||∞ < /2. Set h =
(g − /2) ∨ 0. The Markov property of (E, F) implies that h ∈ F. Also if
x∈
/ K, then |f+ (x) − g(x)| = |g(x)| < /2. Hence h ∈ C0 (X, R). Also we see
that||f+ − h|| < . The same discussion implies that ||f− − u|| < for some
u ∈ F ∩ C0 (X, R). Therefore, F ∩ C0 (X, R) is dense in C0 (X, R).
25
Definition 8.2. If (X, R) is not bounded, then, for p ∈ X, define µp, 12 by
µp, 12 (A) = A 1 + R(x, p)µ(dx) for any Borel set A ⊆ X. If (X, R) is bounded
then we set µp, 12 = µ for any p ∈ X.
1 (X, R) =
Note that if (X, R) is bounded, then C 12 (X, R) = Cb (X, R) and C
2
Cb (X, R). Hereafter, if (X, R) is bounded, C 12 (X, R) is regarded as equipped
with the supremum norm. Accordingly, we modify the definition of the norm
|| · ||M (X,R) as follows:
b (X, R), ||u||∞ = 1}.
||ϕ||M (X,R) = sup{|ϕ(u)| : u ∈ C
1 (X, R), define
Proposition 8.3. For any f ∈ L1 (X, µp, 12 ) and any u ∈ C
2
ϕf (u) = X f (x)u(x)µ(dx). Then ϕf ∈ M (X, R) and ||ϕf ||M (X,R) = ||f ||1 ,
where ||f ||1 is the L1 -norm with respect to µp, 12 .
1 (X, R). The
Proof. It is easy to see that |ϕf (u)| ≤ ||u||||f ||1 for any u ∈ C
2
equality ||f ||1 = ||ϕf ||M (X,R) is shown by routine arguments using the facts that
1 (X, R) contains C0 (X, R). (Note that C0 (X, R) ⊆
µ is Borel regular and that C
2
1 (X, R) if F ∩ C0 (X, R) is dense in C0 (X, R) with respect to the supremum
C
2
norm.)
Define Φ : L1 (X, µp, 12 ) → M (X, R) by Φ(f ) = ϕf . Then, by the above
theorem, Φ is an isometric embedding from L1 (X, µp, 12 ) to M (X, R). Hereafter,
through Φ, we regard L1 (X, µp, 12 ) as a subset of M (X, R).
Theorem 8.4. Let B = ∅ be a finite subset of X. If GB,µ = GB |L1 (X,µp, 1 ) ,
2
then
(GB,µ f )(x) =
gB (x, y)f (y)µ(dy)
(8.1)
X
for any f ∈ L1 (X, µp, 12 ).
Proof. By (5.4),
x
(GB,µ f )(x) = (GB ϕf )(x) = ϕf (gB
)
for any x ∈ X. This immediately implies (8.1).
L
Definition 8.5. Let B = ∅ be a finite subset of X. Define DB,µ,0
= Im(GB,µ )
L
L
and DB,µ = DB,µ,0 ⊕ HB .
Proposition 8.6. Assume that µ is non-atomic: µ(p) = 0 for any p ∈ X.
L
(1) Let B = ∅ be a finite subset of X. Then (du)p = 0 for any u ∈ DB,µ
and
any p ∈ X\B.
(2) Let B1 and B2 be non-empty finite subsets of X with B1 ⊆ B2 . Then
L
L
DB
= {u|u ∈ DB
, (du)p = 0 for any p ∈ B2 \B1 }.
1 ,µ
2 ,µ
L
L
In particular, DB
⊆ DB
.
1 ,µ
2 ,µ
26
Proof. (1) By Theorem 6.6, (du)x = −ϕf (x) = −f (x)µ(x) = 0 for any
x ∈ X\B.
L
L
L
(2) Lemma 5.9-(2) implies that GB1 ,µ f ∈ DB
. Hence DB
⊆ DB
. More2 ,µ
1 ,µ
2 ,µ
L
over, by (1), (du)x = 0 for any u ∈ DB1 and any x ∈ B2 \B1 .
Conversely let u = uB2 − GB2 ,µ f for uB2 ∈ HB2 and f ∈ L1 (X, µp, 12 ).
Assume that (du)p = 0 for any p ∈ B2 \B1 . So using Theorem 6.6, we obtain
(du)p = −(HB2 uB2 )(p) + ϕf (ψpB2 ) = 0
(8.2)
for any p ∈ B2 \B1 . By Lemma 5.9,
p
u = uB2 − GB1 ,µ f +
ϕf (ψpB2 )gB
1
=
p∈B2 \B1
(E(u, ψpB2 )
p
+ ϕf (ψpB2 ))gB
+
1
p∈B2 \B1
u(q)ψqB1 − GB1 ,µ f.
(8.3)
q∈B1
Since E(u, ψpB2 ) = −(HB2 uB2 )(p) by Lemma 2.20, (8.3) along with (8.2) implies
L
u = q∈B1 u(q)ψqB1 − GB1 ,µ f . Hence u ∈ DB
.
1 ,µ
Proposition 8.7. Suppose µ is non-atomic. Let B be a non-empty finite subset
L
→ L1 (X, µp, 12 ) by ∆B,µ u = f for u = uB − GB,µ f
of X. Define ∆B,µ : DB,µ
1
where uB ∈ HB and f ∈ L (X, µp, 12 ). Then,
L
(1) ∆B,µ u = LB u for any u ∈ DB,µ
. In particular,
E(v, u) =
v(p)(du)p −
v∆B,µ udµ
(8.4)
X
p∈B
for any v ∈ F.
(2) For any f ∈ L1 (X, µp, 12 ) and any h ∈ (B),
∆B,µ u = f
and
if and only if
u(x) =
u|B = h.
h(p)ψpB (x) −
gB (x, y)f (y)µ(dy)
X
p∈B
L
for any x ∈ X. In particular, GB,µ is invertible and ∆B,µ |DB,µ,0
= −(GB,µ )−1 .
(3) ||u||D = p∈B |(du)p | + X |∆B,µ f |dµ.
(4) Let B1 and B2 be non-empty finite subsets of X with B1 ⊆ B2 . Then
∆B1 ,µ = ∆B2 ,µ |DBL ,µ .
1
Proof. Theorem 6.8 immediately implies (1) and (2).
(3) By Theorem 7.2, (7.1) and (8.4) imply
||u||D =
sup
h(p)(du)p −
h∆B,µ udµ.
1 (X,R) p∈B
h∈C
2
0≤|h|≤1
X
27
Hence we see that ||u||D ≤
p∈B |(du)p | + X |∆B,µ u|dµ. Now by Proposi 1 (X, R) contains C0 (X, R). This shows the equality.
tion 8.1, C
2
(4) Combining Proposition 8.6-(2) and (8.4), we obtain
v∆B1 ,µ udµ =
v∆B2 ,µ udµ
X
X
L
for any u ∈ DB
and and v ∈ F. Since C0 (X, R) ∩ F is dense in C0 (X, R), it
1 ,µ
follows that ∆B1 ,µ u = ∆B2 ,µ u.
By (8.4), it follows that ∆B,µ u is the unique element in L1 (X, µp, 12 ) that
satisfies
E(u, v) = − (∆B,µ u)vdµ
X
for any v ∈ FB .
Definition 8.8. Suppose µ is non-atomic. Define DµL by
L
DµL = {u|u ∈ DB,µ
, (du)p = 0 for any p ∈ B},
where B is a non-empty finite subset of X.
By Proposition 8.6, DµL is independent of a choice of B. In fact,
DµL =
L
DB,µ
.
B:B=∅
B is a finite set
Proposition 8.9. Suppose µ is non-atomic. Lu ∈ L1 (X, µp, 12 ) for any u ∈ DµL .
Let ∆µ = L|DµL . Then ∆µ = ∆B,µ |DµL for any non-empty finite set B ⊆ X and
∆µ u is the unique element in L1 (X, µp,µ ) that satisfies
E(u, v) = − (∆µ u)vdµ
(8.5)
X
for any v ∈ F. Moreover ||u||D =
X
|∆µ u|dµ for any u ∈ DµL .
By Proposition
6.10, we see that ker ∆µ = constants and Im(∆µ ) = {f |f ∈
L1 (X, µ), X f dµ = 0}.
Proof. Let B be a non-empty finite subset of X. For any u ∈ DµL , since (du)p = 0
for any p ∈ X, (8.4) implies that
E(u, v) = −
v∆B,µ udµ
(8.6)
X
for any v ∈ F. On the other hand, E(u, v) = −(Lu)(v). Therefore, Lu =
∆B,µ u.
28
Next, we identify ∆B,µ and ∆µ with the non-negative self-adjoint operators
on L2 (X, µ) associated with (E, FB ∩ L2 (X, µ)) and (E, F ∩ L2 (X, µ)). Using
the results in [16, Section 2.4] (in particular, Theorem 2.4.1 and 2.4.2), we
immediately have the following theorem.
Theorem 8.10. Suppose µ is non-atomic.
(1) Let B be a non-empty finite subset of X. Then (E, FB ∩ L2 (X\B, µ)) is a
regular Dirichlet form on L2 (X\B, µ).
(2) (E, F ∩ L2 (X, µ)) is a regular Dirichlet form on L2 (X, µ).
Remark. Ordinarily, one assumes that the space is locally compact for a Dirichlet form. In the above theorem, however, (X, R) may not be locally compact in
general.
If µ is non-atomic, then L2 (X\B, µ) may be identified with L2 (X, µ). Hence
we regard (E, FB ) as a Dirichlet form on L2 (X, µ) hereafter.
The next proposition gives direct relations between the non-negative selfadjoint operators associated with the Dirichlet forms and the Laplacians ∆B,µ
and ∆µ .
Proposition 8.11. Assume that µ is non-atomic.
(1) Let B be a non-empty finite subset of X and let ΓB,µ be the non-negative
self-adjoint operator on L2 (X, µ) associated with the Dirichlet form (E, FB ).
Then L2 (X, µ) ∩ GB,µ (L2 (X, µ) ∩ L1 (X, µp, 12 )) ⊆ Dom(ΓB,µ ) and ΓB,µ u =
−∆B,µ u for any u ∈ L2 (X, µ) ∩ GB,µ (L2 (X, µ) ∩ L1 (X, µp, 12 )).
(2) Let Γµ be the non-negative self-adjoint operator on L2 (X, µ) associated
with the Dirichlet form (E, F). If u ∈ DµL ∩ L2 (X, µ) and ∆µ u ∈ L2 (X, µ), then
u ∈ Dom(Γµ ) and Γµ u = −∆µ u.
Proof. (1) By the definition of the non-negative self-adjoint operator associated
with a closed form (see [16, Appendix B.1] for example), u ∈ Dom(ΓB,µ ) and
ΓB,µ = f if and only if
E(u, v) =
f vdµ
X
for any v ∈ F ∩ L (X, µ). This along with (8.6) immediately implies the desired
statement.
(2) Using (8.5), we immediately see the desired statement by similar arguments
as in (1).
2
We have a simpler statement under a restricted situation.
Lemma 8.12. Suppose that µ is non-atomic. If X (1 + R(x, p))µ(dx) < ∞,
then F ⊂ L2 (X, µ) ⊂ L1 (X, µp, 12 ).
Proof. Let u ∈ F. Then |u(x) − u(p)|2 ≤ R(x, p)E(u, u). This implies that
|u(x)|2 ≤ 2R(x, p)E(u, u)+2|u(p)|2 . Hence u ∈ L2 (X, µ). Next let u ∈ L2 (X, µ).
Then
1 + R(x, p)u(x)µ(dx) ≤
(1 + R(x, p))µ(dx)
|u(x)|2 µ(dx) < ∞.
X
X
X
29
Note that if X (1 + R(x, p))µ(dx) < ∞, then ΓB,µ and Γµ have compact
resolvents. (See [16, Theorem 2.4.2] for a proof.)
Theorem 8.13. Assume that µ is non-atomic. Suppose X (1+R(p, x))µ(dx) <
∞.
L
(1) Let B be a non-empty finite subset of X. Then Dom(ΓB,µ ) ⊂ DB,µ,0
and
−1
is a compact
ΓB,µ u = −∆B,µ u for any u ∈ Dom(ΓB,µ ). Moreover, (ΓB,µ )
operator and (ΓB,µ )−1 = GB,µ |L2 (X,µ)
(2) Dom(Γµ ) ⊆ DµL and Γµ u = −∆B,µ u for any u ∈ Dom(Γµ ).
Proof. (1) Since ΓB,µ has compact resolvent and 0 is not an eigenvalue of ΓB,µ ,
ΓB,µ is invertible and (ΓB,µ )−1 is a compact operator. In particular, for any
f ∈ L2 (X, µ), there exists a unique u ∈ Dom(ΓB,µ ) such that ΓB,µ u = f . For
any v ∈ F,
E(u, v) =
f v = E(GB,µ f, v).
X
Hence u = GB,µ f . Therefore Dom(ΓB,µ ) = GB,µ (L2 (X, µ)). By Proposition 8.11, it follows that ΓB,µ u = −∆B,µ u for any u ∈ Dom(ΓB,µ ).
(2) Let u ∈ Dom(Γµ ). Then
vΓµ udµ
E(u, v) =
X
for any v ∈ F. By Lemma 8.12, Γµ u ∈ L1 (X, µp, 12 ). Since X Γµ udµ = E(1, u) =
0, we see that Γµ u ∈ Im(∆µ ). Therefore there exists h ∈ DµL such that −∆µ h =
Γµ u. By (8.5), E(h, v) = E(u, v) for any u ∈ F. Therefore u − h is a constant
and hence u ∈ DµL . Finally, by Proposition 8.11-(2), Γµ u = −∆µ u.
9
P. c. f. self-similar sets
In this section, we apply the results in the previous sections to self-similar
resistance forms (coming from harmonic structures) on post critically finite selfsimilar structures. In particular, we show relations between the domain of
resistance forms, F, the domain of Laplacian in generalized sense, DL , and
uniformly Lipschitz continuous functions.
First we give a quick review of the theory of analysis on post critically finite
self-similar sets. See [16, Chapter 3].
Definition 9.1. Let K be a compact metrizable topological space and let S be
a finite set. Also, let Fi , for i ∈ S, be a continuous injection from K to itself.
Then, (K, S, {Fi }i∈S ) is called a self-similar structure if there exists a continuous
surjection π : Σ → K such that Fi ◦π = π◦i for every i ∈ S, where Σ = S N is the
one-sided shift space and i : Σ → Σ is defined by i(w1 w2 w3 · · · ) = iw1 w2 w3 · · ·
for each w1 w2 w3 · · · ∈ Σ.
30
Note that if (K, S, {Fi }i∈S ) is a self-similar structure, then K is self-similar
in the following sense:
K=
Fi (K).
(9.1)
i∈S
Notation. Wm = S m is the collection of words with length m. For w =
w1 w2 · · · wm ∈ Wm , we define Fw : K → K by Fw = Fw1 ◦ · · · ◦ Fwm and
Kw = Fw (K). In particular, W0 = {∅} and F∅ is the identity map. Also we
define W∗ = ∪m≥0 Wm .
Definition 9.2. Let (K, S, {Fi }i∈S ) be a self-similar structure. We define the
critical set C ⊂ Σ and the post critical set P ⊂ Σ by
C = π −1 ( (Ki ∩ Kj )) and P =
σ n (C),
n≥1
i=j
where σ is the shift map from Σ to itself defined by σ(ω1 ω2 · · · ) = ω2 ω3 · · · . A
self-similar structure is called post critically finite (p. c. f. for short) if and only
if #(P) is finite.
Now, we fix a connected p. c. f. self-similar structure (K, S, {Fi }i∈S ).
Definition 9.3. Let V0 = π(P). For m ≥ 1. Also set
Vm =
Fw (π(P)) and V∗ =
Vm .
w∈Wm
m≥0
It is easy to see that Vm ⊂ Vm+1 and that K is the closure of V∗ .
Next we explain how to construct Laplacians on a p. c. f. self-similar set. First
we define a Laplacian on a finite set.
Proposition 9.4. Let D ∈ LA(V0 ) and let r = (r1 , r2 , · · · , rN ), where ri > 0
for i ∈ S. Define a symmetric bilinear form E (m) on (Vm ) by E (m) (u, v) =
−1
ED (u ◦ Fw , v ◦ Fw ), where rw = rw1 · · · rwm for w = w1 w2 · · · wm ∈
w∈Wm rw
Wm . Then E (m) ∈ DF(Vm ). We denote the Laplacian on Vm corresponding
E (m) by Hm .
Definition 9.5. (D, r) is called a harmonic structure if and only if the sequence
{(Vm , Hm )}m≥0 is a compatible sequence. Furthermore, a harmonic structure
(D, r) is called regular if and only if 0 < ri < 1 for any i ∈ S.
It is know that (D, r) is a harmonic structure if and only if (V0 , D) ≤
(V1 , H1 ). See [16, Proposition 3.1.3] for details. Hereafter we fix a regular
harmonic structure (D, r) on (K, S, {Fi }i∈S ).
If (D, r) is a harmonic structure, then by Theorem 2.13, we have a resistance
form (E, F) on V∗ associated with the compatible sequence {(Vm , Hm )}m≥0 . Let
R be the resistance metric on V∗ corresponding to (E, F). Since (D, r) is assumed
to be regular, we have the following fact.
31
Theorem 9.6. Let (Ω, R) be the completion of (V∗ , R). Then Ω is naturally
identified with K. Through this identification, R gives the same topology to K
as the original distance of K. In particular, (K, R) is compact and F is a dense
subset of C(K, R) with respect to the supremum norm.
See [16, Section 3.3] for the proof of the above theorem.
By this theorem, we may naturally think of C(K, R) as a subset of (V∗ ).
1 (K, R) = C(K, R). Hence M (K, R) is
Also, if follows that C 12 (K, R) = C
2
the dual space of C(K, R) (i.e. measures on (K, R)). The following fact is a
immediate corollary of Theorems 6.6 and 7.5. Recall that DL ⊂ F ∩ CL (K, R),
where CL (K, R) is the collection of uniformly Lipschitz continuous functions on
K with respect to R.
Proposition 9.7. For any u ∈ (V∗ ), ||u||D = limm→∞ p∈Vm |(Hm u)(p)|.
Also u ∈ DL if and only if ||u||D < ∞. Moreover, for any u ∈ DL and any
p ∈ K, E(u, ψpVm ∪p ) = −(HVm ∪p u)(p) converges as m → ∞. The limit is
denoted by (du)p .
The Green function gB coincides with the one defined in [16, Appendix A.2]
when B is a finite subset of V∗ . T. Watanabe has studied the case where B
is a general finite subset of K in [23]. He has obtained the Green function,
harmonic functions and Laplacians and extended the results in [16]. He has
also considered the case where the harmonic structure is not regular.
Let µ be a Borel regular measure on K with µ(K) = 1. Also we assume that
µ is non-atomic. Then µp, 12 = µ and we may apply all the results in Section 8.
DµB defined in [16, Appendix A.2] is equal to GB,µ (C(K, R))⊕HB in our context,
L
L
while DB,µ
= GB,µ (L1 (K, R)). So, the Laplacian ∆B,µ in this paper maps DB,µ
1
to L (K, µ) and −∆B,µ |GB,µ (L2 (K,µ)) is the non-negative self adjoint operator
on L2 (K, µ) associated with the Dirichlet form (E, FB ) on L2 (K, µ).
We will study relations between F, DL and CL (K, R) in the rest of this
section.
Definition 9.8. (1) Define ψpm = ψpVm for p ∈ Vm . Set i(p) = m if p ∈
i(p)
Vm \Vm−1 for any p ∈ V∗ . (We think of V−1 as ∅.) Then, define ψp = ψp .
(2) Define um = p∈Vm u(p)ψpm for any u ∈ (V∗ ) and any m ≥ 0.
Any function u ∈ (V∗ ) has an expansion with respect to the basis {ψp }p∈V∗ .
Proposition 9.9. Let u ∈ (V∗ ). If αp (u) = ui(p) (p) − ui(p)−1 (p) for any p ∈
V∗ , then
u(x) =
αp (u)ψp (x)
(9.2)
p∈V∗
for any x ∈ V∗ . (In the definition of αp (u), we set u−1 = 0 when i(p) = 0.)
Note that the converse
of the above proposition is also true: for given
{αp }p∈V∗ , letting u =
p∈V∗ αp ψp , we see that there exists u ∈ (V∗ ) such
that αp (u) = αp for any p ∈ V∗ .
32
Definition 9.10. Let u ∈ (V∗ ) and let w ∈ W∗ . Define αw (u) ∈ (V1 \V0 )
by (αw (u))q = αFw (q) (u) for any q ∈ V1 \V0 . Also define Hw u ∈ (V1 \V0 ) by
(Hw u)q = (H|w|+1 u)(Fw (q)) for any q ∈ V1 \V0 . Let aw (u) = u|w|+1 ◦ Fw − u|w| ◦
Fw .
It is easy to see that aw (u) = q∈V1 \V0 αFw (q) (u)ψq . In particular, αw (u) =
aw (u)|V1 \V0 .
Lemma 9.11. For any u ∈ (V∗ ) and any w ∈ W∗ ,
αw (u) = rw X −1 Hw u.
Proof. Let m = |w|. Then, for any q ∈ V1 \V0 ,
(Hm+1 u)(Fw (q)) = (rw )−1 (H1 (u ◦ Fw ))(q) = (rw )−1 (H1 (um+1 ◦ Fw )(q).
Since (H1 um ◦ Fw )(q) = 0 for any q ∈ V1 \V0 , we have Hw u = (rw )−1 H1 aw (u) =
(rw )−1 Xαw (u).
Theorem 9.12. Let u ∈ (V∗ ).
(1) u ∈ F if and only if
rw (|Hw u|V1 \V0 )2 < ∞.
(9.3)
m≥0 w∈Wm
(2) u ∈ C(K) if
m≥0
( max rw |Hw u|V1 \V0 ) < ∞.
w∈Wm
(9.4)
(3) If u ∈ DL , then
m≥0
|Hw u|V0 \V1 ) < ∞.
(9.5)
( max |Hw u|V1 \V0 ) < ∞.
(9.6)
sup (
w∈Wm
(4) u ∈ CL (K, R) if
m≥0
w∈Wm
This theorem will be proven at the end of this section. Meanwhile, applying
the theorem to a special class of functions, we show relations between F, DL
and CL (K, R).
Corollary 9.13. Let (βi )i∈S ∈ (S) and let c ∈ (V1 \V0 ). Assume that c = 0.
Define u ∈ (V∗ ) by
for any p ∈ V0
α (u) = rw βw c for any w ∈ W∗ ,
αp (u) = 0
w
33
(9.7)
where βw = bw1 · · ·bwm for
w = w1 w2 · · · wm ∈ W∗ .
(1) u ∈ F if and only if i∈S ri |βi |2 < 1.
(2) If maxi∈S ri |βi | <
1, then u ∈ C(K).
(3) If u ∈ DL , then i∈S |βi | ≤ 1.
(4) If maxi∈S |βi | < 1, then u ∈ CL (K, R).
Proof. By Lemma 9.11, Hw u = βw Xc. Hence the equations (9.3) - (9.6) is
immediately translated into the corresponding conditions on β.
By Proposition 5.12, it follows that DL ⊆ F ∩ CL (K, R). The next corollary
says that the converse is not true.
Corollary
9.14. (1) F ∩ CL (K, R) ∩ (DL )c = ∅.
(2)
i∈S ri = 1 if and only if CL (K, R) ⊂ F
Remark. By [15, Theorem 3.2], it follows that i∈S ri ≥ 1. (We can prove this
fact by using (9.8) below as well. Let f be a nontrivial V0 -harmonic
function.
Then f ∈ CL (K, R) ∩ F and Em (f, f ) = E(f, f ) > 0.By (9.8), i∈S ri ≥ 1.)
Also by [15, Theorem 3.2] (or [16, Theorem 4.1.2]), i∈S ri = 1 if and only if
the Hausdorff dimension of (K, R) is one.
Proof. (1) Let u be given by (9.7). Set βi = N −1/2 for all i ∈ S,
where N is
the number of elements in S. Since 0 < ri < 1 for any i ∈ S, i∈S ri < N .
L c
This fact along with
.
Corollary 9.13 implies that u ∈ F ∩ CL (K, R)
∩ (D )−1/2
(2) First assume i∈S ri > 1. Let u be given by (9.7) with βi = ( i∈S ri )
.
Then by Corollary 9.13, u ∈ F c ∩ CL (K, R).
Next we show the converse. Define L(f ) = supx,y∈K |f (x) − f (y)|/R(x, y)
for f ∈ CL (K, R). Note that there exists c > 0 such that
E0 (h, h) ≤ c max |h(p) − h(q)|2
p,q∈V0
for any h ∈ (V0 ). Using this and Theorem A.1, we see that
Em (f, f ) =
w∈Wm
1
E0 (f ◦ Fw , f ◦ Fw )
rw
1
max |f (Fw (p)) − f (Fw (q))|2
rw p,q∈V0
w∈Wm
1
m
≤ cL(f )
max R(Fw (p), Fw (q))2 ≤ c
L(f )
ri
rw p,q∈V0
≤c
w∈Wm
(9.8)
i∈S
for any m ≥ 0 and any f ∈ CL (K, R), where c
= c maxp,q∈V0 R(p, q)2 . If
i∈S ri = 1, then (9.8) implies that Em (f, f ) is bounded and hence f ∈ F.
Example 9.15. Let K = [0, 1]. Define R(x, y) = |x − y| for any x, y ∈ K.
Choose r1 , r2 > 0 with r1 + r2 = 1. Set F1 (x) = r1 x and F2 (x) = r2 x +
r1 . Then K = F1 (K) ∪ F2 (K) and F1 (K) ∩ F2 (K) = {r1 }. We see that
34
−1 1
and
1 −1
let r = (r1 , r2 ). Then (D, r) is a regular harmonic structure. If (E, F) is the
resistance form corresponding this harmonic structure, then F = H 1 ([0, 1]) =
{f |f ∈ L2 ([0, 1], dx)}, where f is the derivative of f in the generalized sense,
1
and E(f, g) = 0 f (x)g (x)dx. Also the corresponding effective resistance is R.
In this case,
(K, {1, 2}, {F1 , F2 }) is a p. c. f. self-similar structure. Let D =
DL = {f |f is bounded variation},
CL (K, R) = {f |f is bounded}
and CL (K, R) ⊂ F.
The rest of this section is devoted to proving Theorem 9.12.
Proof of Theorem
9.12.
(1) By [16, Proposition 3.2.19] and Lemma 9.11, u ∈ F
if and only if m≥0 w∈Wm rw (Hw u, −X −1 Hw u)V1 \V0 . Since −X −1 is positive
definite, this is equivalent
to (9.3).
(2) Let um =
p∈Vm \Vm−1 αp ψp for m ≥ 0, where V−1 = ∅. Then u =
u
.
Note
that
um ∈ C(K). By Lemma 9.11 and (9.4), it follows that
m
m≥0
u
is
uniformly
convergent on K as m → ∞.
m
m≥0
(3) If u ∈ DL , Theorem 7.2 implies that supm≥0 p∈Vm |(Hm u)(p)| < ∞.
Therefore, we obtain
(9.5)
(4) Let um = p∈Vm \Vm−1 αp (u)ψp for m ≥ 0. (u0 = p∈V0 u(p)ψp .) Then
u = m≥0 um . Define m maxw∈Wm−1 |Hw u|. By (9.6), m≥0 m < ∞. Next
we consider the Lipschitz constant of um .
Case 1: Suppose that x, y ∈ Kw for some w ∈ Wm−1 . Set x1 = (Fw )−1 (x)
and y1 = (Fw )−1 (y). Note that ψq is uniformly Lipschitz continuous for any
q ∈ V1 \V0 . By Theorem A.1, we see
|um (x) − um (y)| ≤
|αFw (q) (u)||ψq (x1 ) − ψq (y1 )| ≤ c1 |αw (u)|R(x1 , y1 )
q∈V1 \V0
≤ c2 |αw (u)|(rw )−1 R(x, y) ≤ c3 m R(x, y),
where c1 , c2 and c3 are independent of x, y, w and m.
. Then, for
Case 2: Suppose that w, w
∈ Wm−1 , w = w
, x ∈ Kw and y ∈ Kw
any z ∈ Fw (V0 ), the result of Case 1 along with Lemma 3.3 implies that
|um (x)| ≤ c3 m R(x, z) ≤ c4 m R(x, Fw (V0 )),
where c4 = c3 #(V0 ). In the same manner, |um (y)| ≤ c4 m R(y, Fw (V0 )). Note
that R(x, Fw (V0 )) + R(y, Fw (V0 )) = RV (x, y) ≤ R(x, y), where V = Fw (V0 ) ∪
Fw (V0 ). Hence |um (x) − um (y)| ≤ |um (x)| + |um (y)| ≤ c4 m R(x, y).
The above two cases
c4 m R(x, y) for any
implies that |um (x) − um (y)| ≤ x, y ∈ K. Since u = m≥0 um , we see |u(x) − u(y)| ≤ c4 m≥0 m R(x, y) for
any x, y ∈ K.
35
Appendix A
Assume the same situation as in the last section: L = (K, S, {Fi }i∈S ) is a
p. c. f. self-similar structure, (D, r) is a regular harmonic structure on L, and
(E, F) and R are the corresponding resistance form and the resistance metric
respectively.
In this appendix, we show that Fi is asymptotically a similitude with a
contraction ratio ri under R. Precisely we have the following theorem.
Theorem A.1. There exists c1 such that
c1 rw R(x, y) ≤ R(Fw (x), Fw (y)) ≤ rw R(x, y)
for any w ∈ W∗ and any x, y ∈ K.
The upper estimate of R(Fw (x), Fw (y)) can be found in [16, Lemma 3.3.5].
So what really matters here is the lower estimate. We will do this in several
steps.
First, we prove the following result on resistance forms. It shows that if
one moves a resistor, the effective resistance between the new terminals of the
resistor become smaller than before.
Theorem A.2. Let (E, F) be a resistance form on X and let R be the resistance
metric associated with (E, F). Let x, x1 , . . . , xn , y, y1 , · · · , yn ∈ X and assume
that x = y. For r1 , . . . , rn > 0, define
n
1
(u(xi ) − u(yi ))(v(xi ) − v(yi ))
r
i=1 i
n
1
E2 (u, v) = E(u, v) +
(u(x) − u(y))(v(x) − v(y))
r
i=1 i
E1 (u, v) = E(u, v) +
for any u, v ∈ F. Then (E1 , F) and (E2 , F) are resistance forms on X. Moreover, if R1 and R2 are the resistance metrics associated with E1 and E2 , respectively, then
R1 (x, y) ≥ R2 (x, y) =
1 −1
1
.
+
R(x, y) i=1 ri
n
Proof. We may assume that n = 1, because the general case easily follows by induction. Without loss of generality, we may also suppose that X = {x, y, x1 , y1 }.
We write p1 = x, p2 = y, p3 = x1 and p4 = y1 . (X is assumed to contain exactly
four points; otherwise the following discussion is much easier.) Let
E(u, v) =
1≤i<j≤4
1
(u(pi ) − u(pj ))(v(pi ) − v(pj )).
rij
We denote a = (r14 )−1 , b = (r13 )−1 , c = (r24 )−1 , d = (r24 )−1 and e = (r34 )−1 .
Writing h = (r1 )−1 and letting H2 (h) = R2 (x, y)−1 , we see that H2 (h) =
36
R(x, y)−1 + h. On the other hand, by an elementary calculation, it follows that
H1 (h) =
1
(a + b)(c + d)(h + e) + (ac(b + d) + db(a + c))
,
+
r12
(a + c)(b + d) + (h + e)(a + b + c + d)
where H1 (h) = R1 (x, y)−1 . Therefore,
∂H1
(ad − bc)2
=
≤ 1.
∂h
((a + c)(b + d) + (h + e)(a + b + c + d))2
Since H1 (0) = H2 (0) = R(x, y)−1 , we obtain H1 (h) ≥ H2 (h) for any h > 0.
This immediately implies the desired result.
Define
Λ(r) = {τ = τ1 τ2 · · · τm |rτ1 τ2 ···τm−1 > r ≥ rτ1 τ2 ···τm }
for 0 < r < 1. It is known that Λ(r) is a partition of Σ. (See [16, Chapter 1].)
We write Λ = Λ(rw ), Λw = {τ |τ ∈ Λ, τ = w, Kw ∩ Kτ = ∅} and Λ
w = {τ |τ ∈
Λ, Kw ∩ Kτ = ∅}. Then we have the following facts.
Lemma A.3 ([16, Lemma 4.2.3]). There exists B > 0 such that #(Λw ) ≤ B
for any w ∈ W∗ .
Lemma A.4 ([16, (3.3.1)]). For any u, v ∈ F,
E(u, v) =
1
E(u ◦ Fτ , u ◦ Fτ ).
rτ
τ ∈Λ
Next define VΛ = ∪τ ∈Λ Fτ (V0 ), V = VΛ ∪{Fw (x), Fw (y)} and U = V0 ∪{x, y}.
Then V = ∪τ ∈Λw ∪Λw Fτ (V0 ) ∪ Fw (U ). By Proposition 2.10, there exist HV ∈
LA(V ) such that RHV = R|V ×V . In the same way, we have HU ∈ LA(U ).
Then Lemma A.4 implies the following fact. Note that D = HV0 .
Lemma A.5. For any u, v ∈ (V ),
EHV (u, v) =
1
EH (u ◦ Fw , v ◦ Fw ) +
rw U
τ ∈Λw ∪Λw
1
ED (u ◦ Fτ , v ◦ Fτ ).
rτ
Note that R(Fw (x), Fw (y)) = RHV (Fw (x), Fw (y)).
Proof of Theorem A.1. Let V0 = {p1 , · · · , pl }. If I = {(i, j)|i < j, Dpi pj = 0}
and rij = (Dpi pj )−1 for (i, j) ∈ I, it follows that
ED (u, v) =
(i,j)∈I
1
(u(pi ) − u(pj ))(v(pi ) − v(pj )).
rij
We consider the V \Fw (U )-shorted resistance form of (EHV , (V )) and denote it
by (E , F ), where F = (Fw (U ) ∪ {b}). (The one point b represents V \Fw (U ).)
37
Since the points in Fτ (V
0 ) for τ ∈ Λw contracts to the single point b, the part of
−1
EHV coming from Λw , τ ∈Λ w (rτ ) ED (u ◦ Fτ , v ◦ Fτ ), vanishes in the shorted
form E . The part coming from Λw becomes
m
1
(u(xk ) − u(yk ))(v(xk ) − v(yk )),
rk
k=1
where (xk , yk ) ∈ Fw (V0 ) × (Fw (V0 ) ∪ {b}), m ≤ B#(I) and rk
= rτ rij for some
τ ∈ Λw and some (i, j) ∈ I. Hence,
1
1
EHu (u ◦ Fw , v ◦ Fw ) +
(u(xk ) − u(yk ))(v(xk ) − v(yk )).
rw
rk
m
E (u, v) =
k=1
Denote E by E1
and define E2
by
E2
(u, v) =
1
EH (u ◦ Fw , v ◦ Fw ) +
rw U
m
1
(u(Fw (x)) − u(Fw (y)))(v(Fw (x)) − v(Fw (y)).
rk
k=1
The effective resistance between Fw (x) and Fw (y) with respect to the form
(rw )−1 EHU (u ◦ Fw , v ◦ Fw ) is rw R(x, y). By this fact along with Theorem A.2,
R(Fw (x), Fw (y)) ≥ R1
(Fw (x), Fw (y)) ≥
R2
(Fw (x), Fw (y)) =
1 −1
1
,
+
rw R(x, y)
rk
m
k=1
where Ri
and R2
corresponds to the effective resistance with respect to Ei
for
i = 1, 2. Note that if c = (mini∈S ri )(min(i,j)∈I rij ), then rk
≥ crw for any k.
Hence,
R2
(Fw (x), Fw (y)) ≥
m −1
1
1
+
≥ rw min{R(x, y), A},
rw R(x, y) crw
2
where A = c(B#(I))−1 . Now if rw R(x, y) ≤ A, then R(x, y) ≥ rw R(x, y)/2.
Otherwise, let d = supp,q∈K R(p, q). Then A =≥ R(x, y)A/d. This implies
R(x, y) ≥ rw R(x, y)A(2d)−1 . Combining these, we see that R(Fw (x), Fw (y)) ≥
c1 rw R(x, y), where c1 = A(2d)−1 .
Appendix B
In this appendix, we show a relation between the hitting time and the effective
resistance by using the definition of the Green function, Definition 4.1. This
relation is an extension of Theorem 4.27 and Corollary 4.28 in [1], which was
originally given in [7]. Let (E, F) be a resistance form on X and let R be the
38
associated resistance metric on X. Also we assume that (E, F ∩ L2 (X, µ)) is
a regular Dirichlet form on L2 (X, µ), where µ is assume to be a σ-finite Borel
regular measure on X. Let ({Xt }t>0 , {Px }x∈X ) be the Hunt process associated
with the Dirichlet form (E, F ∩ L2 (X, µ)). See [9] about the relation between
Dirichlet forms and Hunt processes. For a subset A ⊆ X, define TA = inf{t|t >
0, Xt ∈ A}. Then we see that
Lemma B.1. For any finite subset B ⊂ X and any x ∈ X,
Ex (TB ) =
gB (x, y)µ(dy).
X
This lemma along with the definition of the Green function, Definition 4.1,
immediately imply the following theorem.
Theorem B.2. (1) For any x, y ∈ X,
R(x, y) = Ex (Ty ) + Ey (Tx ).
(2) For any finite subset B ⊂ X and any x ∈ X,
R(x, B) ≥ Ex (TB ).
Note that Ex (TB ) = R(x, B) X ψxB∪x (y)µ(dy).
The above theorem holds for non-regular harmonic structures on p. c. f. selfsimilar sets as well. More precisely, let (K, S, {Fi }i∈S ) be a p. c. f. self-similar
structure and let (D, r) be a harmonic structure on it. If (D, r) is not regular
(i.e. ri ≥ 1 for some i ∈ S), then Ω is identified with a proper subset of K,
where Ω is defined in Theorem 9.6. Hence in such a case, R is not a distance
on K. However, it has been shown in [23] that the B-harmonic functions are
naturally extended to continuous functions on K if B is a finite subset of Ω.
Therefore, Theorem B.2 remains true if x, y and B belong to Ω.
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