LP -ESTIMATES AND POLYHARMONIC BOUNDARY VALUE PROBLEMS ON THE SIERPINSKI GASKET AND GAUSSIAN FREE FIELDS ON HIGH DIMENSIONAL SIERPINSKI
CARPET GRAPHS
A Dissertation Presented to the Faculty of the Graduate School
of Cornell University in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by Baris Evren Ugurcan
August 2014

© 2014 Baris Evren Ugurcan ALL RIGHTS RESERVED

LP -ESTIMATES AND POLYHARMONIC BOUNDARY VALUE PROBLEMS ON THE SIERPINSKI GASKET AND GAUSSIAN FREE FIELDS ON HIGH
DIMENSIONAL SIERPINSKI CARPET GRAPHS Baris Evren Ugurcan, Ph.D. Cornell University 2014
We define a suitable trace space on the set X halving the Sierpinski Gasket, then we prove Lp-estimates for p > 1 for the restriction operator on domLp∆(SG). We also construct a right inverse to the restriction operator, that is the extension operator, and provide similar Lp-estimates. Then, we consider the polyharmonic boundary value problem which involves finding a biharmonic function with prescribed values and Laplacian values on the bottom line (identified with the interval) and top vertex of the SG. After constructing a suitable orthogonal basis of piecewise biharmonic splines, we express the solution to the BV P in terms of the Haar expansion coefficients of the prescribed data and this basis. After constructing a Sobolev type space on SG, which is analogous to the H2-Sobolev space in classical analysis, we prove how smoothness of the prescribed data is reflected in the smoothness of the solution to the BV P . In the second part of the thesis, we focus on Gaussian Free Fields on High dimensions Sierpinski Carpet graphs. We assume that a “hard wall” is imposed at height zero so that the field stays positive everywhere. Our first result, in the second part of the thesis, is a large deviation type estimate which identifies the rate of exponential decay for P(Ω+VN ), namely the probability that the field stays positive. Then, in our second theorem we prove the leading-order asymptotics for the local sample mean of the free field above the hard wall on any transient Sierpinski carpet graph.

BIOGRAPHICAL SKETCH Baris Evren Ugurcan was born in Corum, Turkey in 1985. His interest in mathematics started from early ages. After obtaining a B.S. and M.S. degree from Bilkent University (Turkey), he went on to Cornell University to obtain his PhD in Mathematics under the direction of Professor Robert S. Strichartz. His research interests include analysis and stochastic processes on graphs and their natural scaling limits: fractals and operator theory. He has many research papers with several coauthors in this areas.
iii

Dedicated to my family. iv

ACKNOWLEDGEMENTS
First and foremost, I would like to thank my advisor Professor Robert S. Strichartz. I am grateful to him for introducing me to this field and supporting my research. His patience and full support made possible this thesis. Especially, his putting his full trust to his students made my PhD years a very productive and rewarding experience. As a reflection of his trust, in the future, I will try to achieve and prove even grater things and I am sure that his guidance and support will continue. I am also sure that our collaboration, which was established long before, will continue in the future. It is a great experience learning from him!
I also would like to thank my ”big sibling” and collaborator Joe P. Chen. He has always been a model-mathematician for me who never hesitated to share his precious experiences. Our collaboration proved fruitful and I expect it to continue in the future.
I would like to thank Professors Laurent Saloff-Coste and Camil Muscalu for serving on my committee and many fruitful discussions.
My thanks also goes to many mathematicians whose works and as a person has been inspiring for my career. Among them are: Professors Len Gross, Lionel Levine, Palle Jorgensen, Luke Rogers, Sasha Teplyaev and Ben Steinhurst.
It is my pleasure to thank my collaborators Professor Lionel Levine, Professor Yuval Peres, Mathav Murugan, Ben Li and Nick Ryder. I am so happy having the chance to work with these people!
I also would like to thank my family, to whom I dedicated this thesis, for their continuous support since my childhood.
v

TABLE OF CONTENTS

Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Introduction 1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Lp-Estimates on SG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Polyharmonic Boundary Value Problems on SG . . . . . . . . . . . 1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Gaussian Free Fields on High Dimensional Sierpinski Carpet Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2 4 5
7 8

2 Lp-Estimates on SG 2.1 Background and Notation . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Definition of the Trace Space . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 12 16 26

3 Polyharmonic Boundary Value Problems on SG 3.1 Construction of the Basic Functions . . . . . . . . . . . . . . . . . . . 3.2 Orthogonality of Basic Functions . . . . . . . . . . . . . . . . . . . . . 3.3 Solution of Polyharmonic BVP and Convergence Theorems . . . .

28 32 46 49

4 Gaussian Free Fields on High Dimensional Sierpinski Carpet Graphs 59 4.1 High Dimensional Sierpinski Carpet Graphs . . . . . . . . . . . . . 59 4.1.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2 Dirichlet Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2.1 Kusuoka-Zhou construction of Dirichlet forms . . . . . . . 65 4.2.2 Convergence of discrete Green forms . . . . . . . . . . . . . 66 4.2.3 Comparison of discrete Dirichlet & Green forms . . . . . . 67 4.2.4 The main lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3 Different Characterizations of Capacity . . . . . . . . . . . . . . . . . 70 4.4 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4.1 Lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4.2 Upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.5 Proof of Theorem 4.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.5.1 Lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.5.2 Upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

vi

A Chapter 1 Bibliography

95 97

vii

LIST OF TABLES
1.1 Comparison of relevant parameters on Zd and on the Sierpinski carpet graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

viii

LIST OF FIGURES

1.1 Half-Sierpinski Gasket (SG) . . . . . . . . . . . . . . . . . . . . . . . .

1.2 The 3rd-level approximation of, respectively, the outer Sierpinski

carpet graph G∞ and the inner Sierpinski carpet graph I∞, here

shown for the standard 2-dimensional Sierpinski carpet. Accord-

ing to the conventions in the text, when embedded in (R+)d, the

least vertex of G∞ is situated at the origin, while the least vertex

of

I∞

is

situated

at

(

1 2

,

⋯,

1 2

).

All

edges

have

Euclidean

distance

1.

3 9

3.1 We define [u](x) ∶= (u(x), ∆u(x), ∂nu(x)). . . . . . . . . . . . . . . 35 3.2 an denotes the function values on corresponding vertices. . . . . . 37
3.3 an denotes the values on corresponding vertices and the num-
bers in the triangles denote the level numbers. . . . . . . . . . . . . 38 3.4 Here D = (x0, a0), E = (x1, a1), A = (x2, a2), B = (x3, a3) where xi
denote the function and ai denote the Laplacian values. . . . . . . 39 3.5 In (xi, ai), xi stands for the function values and ai for the Lapla-
cian values. The numbers inside the triangles denote the level
number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.6 Here A = (x1, a1), B = (x1, a1) where xi denote the
function and ai denote the Laplacian values. Similarly, (0, 0, 0), (m, n, q), (−m, −n, −q) are the triples in the form (u(x), ∆u(x), ∂nu(x)) at the corresponding vertex x . . . . . . . . 42 3.7 Here x and y = x1 are the function values, the Laplacian is equal to a on F2K and F3K. Similarly, (0, a, m) is a triple in the form (u(x), ∆u(x), ∂nu(x)) at the corresponding vertex x . . . . . . . . 44 3.8 The numbers in the triangles denote the level number and xi are
the values of the function on the corresponding vertices. . . . . . 45
3.9 The Picture of Dn for the first three levels. . . . . . . . . . . . . . . . 50

4.1 The coarse-graining and conditioning scheme on the outer Sierpinski carpet graph G∞. Vertices indicated by filled dots are the representative interior points (CN ), while vertices covered by the

solid lines (the conditioning grid) are where the free field ϕ is conditioned upon (DN ). . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 The coarse graining and conditioning scheme upon translation.

As in Figure 4.1, the filled dots indicate the original representative interior points (CN ). Applying a translation by z−x0 for some z ∈ Vk (one of the hollow dots), one obtains the new representative interior points (C˜Nz , hollow dots) and conditioning grid (D˜Nz , solid lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

ix

CHAPTER 1 INTRODUCTION
1.1 General Introduction
In this introduction chapter, we basically give a summary of the relevant literature and results established in each chapter followed by a light introduction section. Our pace in the introduction chapter is lively and intuitive whereas in the following chapters we become more technical and give full detailed proofs of the results introduced before, followed by a thorough background section. We start our journey with a boundary value problem (BVP) on the Sierpinski Gasket (SG).
1.2 Lp-Estimates on SG
The first construction of Laplacian on the Sierpinski Gasket dates back to 1987 [G87] [K89]. The laplacian was constructed as a generator of a stochastic process. However, later on an analytic realization of the Laplacian as a renormalized limit of Graph Laplacians was established by Kigami [Ki1]. Kigami’s theory develops many tools and analytical objects which is specific to the fractals such as renormalized graph energies, normal derivatives and renormalized graph laplacians.
Although, there has been much work on analysis on fractals since then, the research on boundary value problems on bounded subsets of fractals has just taken off. We mention some recent work on this area: [OS] [LS] [GQS]. As there
1

is much research on function spaces on fractals [Str03], [Str99], [HK06], [HZ], [LRSU] it is possible to ask questions such as:
• (Extension operators) Given a bounded set Ω on SG and some function f on this subset lying in a certain function space (e.g. C(Ω))). Is it possible to extend this function such that the extension lies in dom(E), where E denotes the self-similar energy on SG?
• (Restriction Operators) Given a function f in a certain function space on SG (e.g. dom(E)) does its restriction on Ω lie in a certain function space on Ω (e.g. C(Ω)))? Of course, by the currently established theory, these questions and similar questions not only make sense on SG but also Kigami’s PCF (postcritically finite) fractals [Ki1]. Extending this theory to other examples such as the Sierpinski Carpet is certainly a challenge.
1.2.1 Results
In Chapter 2, we report our results obtained in [U]. In our work, building up the work in [LS], we extend the Lp estimates for the extension and restriction operators for all p > 1.
As in [LS], we work on the half-Sierpinski Gasket shown in Figure 1.1. For simplicity, using the same notation in Figure 1.1, we put X = {xm}, am ∶= u(xm), ηm ∶= ∂nu(xm). ym and Ym are also defined as shown in Figure 1.1. As we will be interested in extension and restriction operators, we define zm and Zm to be the reflections of ym and Ym across the symmetry line containing X.
2

Also, define the restriction map by Ru ∶= {(am, ηm)} where u is a function defined on a set containing X. We say that u ∈ domLp∆(SG) if u is continuous on SG and ∆u ∈ Lp(SG). We study the image of the restriction operator R and give
estimates on the norm.

Figure 1.1: Half-Sierpinski Gasket (SG)

Let p > 1 and q be its conjugate. We consider the following trace space

Tp ∶= {(am, ηm)

am = A1 + A2(3 5)m + a′m,

(5p

3)m pa′m

p,

m(p−1)
3 p ηm

p < ∞}

with the norm given by

(am, ηm)

p Tp

=

A1 p + A2 p +

(5p 3)m pa′m

p p

+

m(p−1)
3 p ηm

pp.

For harmonic functions on SG we have the following result.

Lemma 1.2.1. If h is a harmonic function, then h ∈ Tp with

Rh

p Tp

=

h(q0) p +

1 2p

h(q1) + h(q2) − 2h(q0) p +

1 2p+1

h(q1) − h(q2) p.

(1.1)

We define the following norm on domLp∆(SG):

u =domLp ∆(SG)

u

p L∞(SG)

+

∆u pp.

In the following result, we give an estimate on Ru Tp in the case u vanishes on the analytical boundary of SG, namely u V0 = 0.
3

Theorem 1.2.2. Let p > 1, u ∈ domLp∆(SG) and u V0 = 0. Then, Ru ∈ Tp and Ru Tp ≤ C ∆u Lp(SG).

(1.2)

Putting together Lemma 1.2.1 and Theorem 1.2.2 we obtain the following estimate on Ru Tp in the general case.
Theorem 1.2.3. (Lp- Trace Theorem) The restriction operator R ∶ domLp∆(SG) → Tp is bounded and
Ru Tp ≤ C1 u L∞(SG) + C2 ∆u Lp(SG).

Of course, the curious question is whether we can find a right inverse to the restriction operator. Answering the question, in the following Theorem we give the existence of a right inverse to the restriction operator R.
Theorem 1.2.4. (Lp-extension Theorem) Let p > 1. There exists a bounded linear extension map E ∶ Tp → domLp∆(SG) with R ○ E = Id.

1.3 Polyharmonic Boundary Value Problems on SG
In this introductory section and in the corresponding chapter (Chapter 3) we denote Sierpinski Gasket by SG or K. L denotes the bottom line of the SG naturally identified with the unit interval [0, 1]. If v1 denotes the top vertex of the SG we put SG = SG ∖ L ∪ v1. Ψ denotes the function Ψ = 1 on the relevant domain. ψ denotes the skew-symmetric mother wavelet on [0, 1] i.e. 1 on [0, 1 2] and −1 on (−1 2, 1]. If w is the word in 0, 1’s telling us the address of a dyadic interval, ψw denotes the scaled down copy of the mother wavelet ψ onto this dyadic interval.
4

We solve the following boundary value problem on the Sierpinski Gasket
∆2u = 0 ∆u L = f2 ∆u(v1) = c′
u L = f1 u(v1) = c.
We present a solution to the above BVP in terms of the coefficients of the Haar expansion of f1 and f2 on L i.e. on the unit interval. We take our analysis further and also investigate which Sobolev spaces f1 and f2 should belong to so that the solution u will belong to the space on the SG which is analogous to the H2 Sobolev space in the classical case with norm given by
(u, u) = (u L)2dx + E(u, u) + (∆u)2dµ < ∞. SG
1.4 Results
We construct a basis {h0, h2, h3, h1ψ, h1w, h2ψ, h2w} on SG and express the solution to the BVP as an infinite series in terms of this basis. We provide necessary and sufficient conditions, namely we prove the following main results: Theorem 1.4.1. Suppose the Haar series of f1, f2 on L is given by
5

M
f1 = cΨ(f1)Ψ +

cw(f1)ψw

m=0 w =m

M
f2 = cΨ(f2)Ψ +

cw(f2)ψw.

m=0 w =m

Then the unique biharmonic function satisfying

(1.3) (1.4)

is given by

∆2u = 0 ∆u L = f2 ∆u(v1) = c′
u L = f1 u(v1) = c

u(x) = (c′ − cΨ(f2))h3(x) + cΨ(f1)Ψ + (a − cψ(f1))h0(x) + cΨ(f2)h2

(1.5)

M
+

M
cw(f1)h1w(x) +

cw(f2)h2w(x)

m=0 w =m

m=0 w =m

The (⋅, ⋅)-norm of u, in the case c′ = 0, cΨ(f2) = 0, is given by

M
(u, u) = cΨ(f1) 2 +

cw(f1) 2 + (c − cψ(f1))2E0

m=0 w =m

M
+ E1 m=0 w =m

10 3

mM
cw(f1) 2 + E2 m=0 w =m

2 15

m
cw(f2) 2+

M
+ L1 m=0 w =m

2 3

m
cw(f2) 2

(1.6)

where E0 = E (h0), E1 = E (h1ψ), E2 = E (h2ψ), L1 = (h2ψ, h2ψ)4.

6

Theorem 1.4.2. A biharmonic function u on SG has

(u, u) = (u L)2dx + E(u, u) + (∆u)2dµ < ∞ SG

if and only if u L

= f1

is in Hs1(L) for s1

=

log

10 3

log 4

>

0 and ∆u L

= f2

is in Hs2(L) for

s2

=

log

2 3

log 4

<

0.

In

this

case,

(1.3),

(1.5)

and

(1.6)

hold

for

M

=

∞

and

(1.4)

converges

in

the Hs2 norm to f2.

Theorem 1.4.3. Let u be a function in SG with (u, u) < ∞. Then, u has boundary

values with u L

= f1 in Hs1(L) for s1

=

log

10 3

log 4

>

0 and ∆u L

= f2 in Hs2(L) for

s2

=

log

2 3

log 4

< 0.

1.5 Gaussian Free Fields on High Dimensional Sierpinski Car-
pet Graphs
The discrete GFF on a graph is a centered Gaussian process whose covariance matrix is given by the Green’s function of the corresponding Laplacian operator on the graph. As a simple example, if we have just one point, we can assign a random variable to this point in which case we have a single random variable. If we consider 2 points now, we can assign 2 independent Random variables, in which case we have a 2-tuple of independent Gaussian variables. We can go on for sure, but we can do something a little more complicated: we can take a graph G and put a centered Gaussian random variables to each vertex x ∈ G that we denote by φx. Rather than having independent Gaussian random variables we can put correlations namely cov(φx, φy) = G(x, y) where G is the Green’s function corresponding to the Laplacian operator on the graph.
Another way to go about is to directly give the Dirichlet energy. Recall that
7

in a general context the Dirichlet energy is defined by E(f, f ) = (∆f, f ). Let G = (V, E) be a finite or transient infinite graph, and let

EG(f )

=

1 2

[f (x)
x,y∈V

−

f (y)]2

(f ∶ V → R)

x∼y

be the Dirichlet energy on G. A Gaussian free field on G is a collection of Gaus-

sian random variables ϕ = {ϕx}x∈V with mean zero and covariance given by the

Green’s function G(x, y) (x, y ∈ V ) for simple random walk on G. Formally, the

law of ϕ has density proportional to exp

−

1 2

EG

(⋅)

with respect to the Lebesgue

measure on RV .

1.5.1 Results
In this section, we summarize our results obtained in [CU] (joint with Joe P. Chen).
Now, assume that a ”hard wall” is imposed at height zero so that the field stays positive everywhere. We prove the leading-order asymptotics for the local sample mean of the free field above the hard wall on any transient Sierpinski carpet graph. Therefore, we extend the results of Bolthausen, Deuschel, and Zeitouni [BDZ95] for the free field on Zd, d ≥ 3, to the fractal setting. In our proofs, we heavily use the theory of transient regular Dirichlet forms together with coarse graining, and conditioning arguments introduced in the previous literature. Thus, our results stands as a fine blend of analytic, in particular potential theoretic, and probabilistic techniques.
In what follows, F is a transient generalized Sierpinski carpet, namely a Sierpinski Carpet on which Brownian motion is transient. Recall that the existence
8

Zd, d ≥ 3

[−

L 2

,

L 2

]d

∩

Zd

L

≍ Ld

≍ L2

≍ L2−d ∈ (0, 1)

Infinite graph Approximating subgraph (”box”)
Side length of box Volume of box
Expected escape time of random walk from box Resistance across opposite faces of box

Transient GSC graph
GN = (VN , ∼)
N F
≍ mNF ≍ tNF ≍ ρNF = (tF mF )N ∈ (0, 1)

Table 1.1: Comparison of relevant parameters on Zd and on the Sierpinski carpet graph.

of a Dirichlet form was proved in [BB99] [KusuokaZhou] and [BBKT] as a scaling limit of corresponding Dirichlet forms on Sierpinski Carpet graphs.
Rather than writing out the definitions in full length, we give an impressionist sketch through Table 1.1 and Figure 1.2 for the reader’s convenience.

Figure 1.2: The 3rd-level approximation of, respectively, the outer Sierpinski carpet graph G∞ and the inner Sierpinski carpet graph I∞,

here shown for the standard 2-dimensional Sierpinski carpet.

According to the conventions in the text, when embedded in

(R+)d, the least vertex of G∞ is situated at the origin, while the

least

vertex

of

I∞

is

situated

at

(

1 2

,

⋯,

1 2

).

All edges have Eu-

clidean distance 1.

9

We introduce some notations before we give a summary of our results.

As shown in Figure 1.2, G∞ denotes the outer Sierpinski Carpet graph and

GG∞ ∶ V∞ × V∞ → R the Green’s function for simple random walk thereon. We denote the measure space on the Sierpinski Carpet F by (F, ν), where ν is the

constant multiple of the dh(F )-dimensional Hausdorff measure on F . We also

consider the unbounded version of the Sierpinski CarpetF∞ ∶= ⋃∞N=0

N F

F

and

ν∞,

accordingly, the σ-finite self-similar Borel probability measure on F∞, assigning

mass mNF to

N F

F

.

We also recall the notion of the (0-order) capacity of the compact carpet F with respect to a Dirichlet form (E, F ) ∈ E on L2(F∞, ν∞), given by

Cap E

(F

)

∶=

inf{E(f, f )

∶

f

∈

F

∩

Cc(F∞), f

≥

1

a.e.

on

F },

Let P be the law of the Gaussian free field on G∞ with covariance GG∞, and let Ω+VN denote the entropic repulsion event {ϕx ≥ 0 for all x ∈ VN }. Our first main result identifies the rate of exponential decay for P(Ω+VN ).

Theorem 1.5.1. There exists positive constants C1 and C2 such that

−C1

≤

lim
N →∞

log P(Ω+VN ) ρ−FN log(tNF )

≤

lim
N →∞

log P(Ω+VN ) ρ−FN log(tNF )

≤

−C2

(1.7)

We elaborate on where the constants come from in Chapter 4, actually we

give explicitly what the constants are. The constants C1 and C2 depend on

Cap E

(F

)

and

the

Green’s

function

GG∞ .

Two other sources which needs to be

considered for explicit computation of constants are: comparing the Dirichlet

forms on G∞ and on I∞ and comparing the (maximal or minimal) cluster point of the sequence of renormalized Dirichlet forms on I∞ with an element of E.

10

Although the upper and lower bounds in Theorem 4.1.1 are different, we

are still able to give a precise description of entropic repulsion on G∞. In the

following

Theorem,

we

prove

that

conditional

on

Ω+VN

,

the

local

sample √

mean

of

the free field on VN is pushed to a height which is proportional to N , and as

N → ∞, the rescaled height converges in probability to a constant.

Theorem 1.5.2. For any > 0 and η > 0,

⎛

lim
N →∞

sup
x∈VN

P⎝

VN, (x)⊂VN

ϕ¯N, (x) log(tNF )

−

2G ≥ η

Ω+VN

⎞ ⎠

=

0,

(1.8)

where ϕ¯N, (x) ∶=

1 VN, (x)

ϕz and VN, (x) ∶=
z∈VN, (x)

z ∈ VN

∶ max 1≤i≤d

zi − [

N F

xi]

≤

⋅

N F

.

11

CHAPTER 2 LP -ESTIMATES ON SG

2.1 Background and Notation

In Chapters 2 and 3 of this thesis, we mainly work on the Sierpinski Gasket (SG). We can think of it as approximated by a sequence of graphs. In this section, we introduce some known facts and background material taken from [Ki1] and [Str]. We begin this section with the concrete definition of SG.

Definition 2.1.1. Let {q0, q1, q2} denote the vertices of an equilateral triangle where q0 is the top vertex, q1 is the lower left and q2 the lower right. Consider three functions Fi ∶ R2 → R2, i = 0, 1, 2, defined by

Fix

=

1 2

(x

−

qi)

+

qi.

SG is the unique nonempty compact set which satisfies

SG = ∪2i=0Fi(SG).

Definition 2.1.2. We define a word w = w1w2...wm of length w = m on the alpha-

bet {0, 1, 2}, that is each wi ∈ {0, 1, 2}. We put Fw = Fw1 ○ Fw2 ○ ... ○ Fwm and call

Fw(SG) a cell of level m. The standard self-similar measure µ is characterized by

µw ∶= µ(FwSG) =

1 3

w.

Definition 2.1.3. We define the (analytical) boundary of SG to be V0 = {q0, q1, q2}. We

define Vm = ∪2i=0FiVm−1 and V∗ = ∪∞i=0Vm. Let Γ0 be the complete graph on V0. We

construct

a

graph

Γm

with

vertices

Vm

by

defining

the

edge

relation

y

∼
m

x

if

there

is

a

cell of level m containing both x and y (i.e. there exist a word w of length m such that

x = Fwqi and y = Fwqj for some distinct i, j ∈ {0, 1, 2})

12

We call V∗ ∖ V0 the set of junction points and observe that for each point x ∈ V∗ ∖ V0 , x = Fwqi = Fw′qj for distinct words w, w′ with the same length.

Definition 2.1.4. Given a real-valued function u defined on V∗, we define the renormal-

ized

graph

energy

on

Γm

by

Em

=

r−mEm(u),

where

r

=

3 5

and

Em(u)

=

∑y ∼x(u(x) m

−

u(y))2.

For instance, E0(u) = (u(q0) − u(q1))2 + (u(q1) − u(q2))2 + (u(q2) − u(q0))2. It is easy to show that Em(u) is an increasing sequence so limm→∞ Em(u) exists. We

define

E (u)

=

lim
m→∞

Em(u)

to be the energy of a function u. We say u ∈ domE if and only if E(u) < ∞. Even

though the definition of E(u) only involves V∗, we can regard u as function defined on

SG because function of finite energy admits a unique continuous extension on SG. We

can define the associated bilinear form

E(u, v)

=

1 4

(E

(u

+ v)

−

E(u −

v))

for u, v ∈ domE.

In addition, we define dom0E = {u ∈ domE ∶ u V0 ≡ 0}.

Definition 2.1.5. Let ζ be a positive continuous measure. We can define a Laplacian ∆ζ weakly via bilinear energy form. Let u ∈ domE and f be continuous on SG. Then we say u ∈ dom∆ζ with ∆ζu = f if

E(u, v) = − f vdζ for all v ∈ dom0E. SG
More generally, if we only assume f ∈ L2(dζ) and the above equality holds, then we say u ∈ domL2∆ζ and ∆ζu = f . We can also define a graph Laplacian ∆m on Γm by

∆mu(x) = (u(y) − u(x)), x ∈ Vm ∖ V0. y∼x m

13

Definition 2.1.6. The normal derivative of a function u at a boundary point q ∈ V0 is given by

∂nu(q)

=

lim
m→∞

1 rm

(u(q)
y∼q

−

u(y)),

where

r

=

3 ,
5

if

the

limit

exists.

m

It is known that the normal derivative exists if u ∈ dom∆ζ. We can also define the

normal derivative on junction points. Let x = Fwqi be a boundary point of the m-cell

Fw(SG). We define ∂nu(x) with respect to the cell Fw(SG) to be r− w ∂n(u ○ Fw)(qi).

Notice that if x = Fwqi = Fw′qi′ is a junction point, then the normal derivative defined

at x with respect to Fw(SG) and Fw′(SG) may not be equal. However, we have

∂nu(Fwqi) + ∂nu(Fw′qi′) = 0 if u ∈ domL2∆ζ.

We also define the effective resistance between x, y ∈ SG to be R(x, y), where

R(x, y) =

min (E(u, u)

u(x) = 1, u(y) = 0)

−1
.

u

We can obtain an analogue of Gauss-Green formula in a fractal setting [Ki1, Str].

Theorem 2.1.7. Suppose u ∈ domL2∆ζ for some measure ζ.Then ∂nu(x) exists for all

x ∈ V0 and we have

for all v ∈ domE.

E(u, v) = − (∆ζu)vdζ + v∂nu SG V0

We can also get a localized version of this formula.

EFwK(u, v) = −

(∆ζu)vdζ +

v∂nu,

Fw K

∂ Fw K

where K denotes the Sierpinski gasket.

14

(2.1)

Definition 2.1.8. Let u0 be a function initially defined on V0, with u0(q0) = a, u(q1) = b, u(q2) = c. There exists a unique function u1, called the harmonic extension of u0, which extends u0 to V1 such that E(u1) is minimized. We put p1 = F0(q1), p2 = F0(q2), p3 = F1(q2). The values of u1 on V1 ∖ V0 = {p1, p2, p3} is given by

u(p1)

=

1 c
5

+

2 a
5

+

2 b
5

u(p2)

=

2 c
5

+

2 a
5

+

1 b
5

u(p3)

=

2 c
5

+

1 a
5

+

2 b.
5

Thereby, u0 can be extended to all cells in V∗ by applying the above harmonic extension algorithm recursively. Observe that with this harmonic structure SG has a 3dimensional space of harmonic functions which is exactly the cardinality of V0.

The space S(H0, Vm) of piecewise harmonic splines of level m is defined to be the space of continuous functions such that u ○ Fw is harmonic for all w = m.

We see that S(H0, Vm) is contained in dom(E) and is finite dimensional of dimension Vm . These functions are obtained by specifying values of u on Vm then extending harmonically, by using the above algorithm, to all higher levels. We have E(u) = Em(u) for these functions.

ψxm denotes the piecewise harmonic spline in S(H0, Vm) satisfying ψxm(y) = δxy for y ∈ Vm. In the sequel, we drop the superscript and just write ψx when it is clear from the subscript index.

15

2.2 Definition of the Trace Space

Recall that we have am = u(xm) and ηm = ∂nu(xm). Let p > 1 and q be its conju-

gate

i.e.

1 p

+

1 q

=

1.

We

recall

the

definition

of

the

following

trace

space

Tp ∶= {(am, ηm)

am = A1 + A2(3 5)m + a′m,

(5p

3)m pa′m

p,

m(p−1)
3 p ηm

p < ∞}

with the norm given by

(am, ηm)

p Tp

=

A1 p + A2 p +

(5p 3)m pa′m

p p

+

m(p−1)
3 p ηm

pp.

Remark 2.2.1. Observe that these norms converge to their counterparts in [LS], namely to 5ma′m ∞ and 3mηm Lip as p → ∞.

We define the following norm on domLp∆(SG):

u =domLp ∆(SG)

u

p L∞(SG)

+

∆u pp.

The following sequence of key lemmas are the Lp-versions of the corresponding Lemmas in [LS].
Lemma 2.2.2. Let am be a sequence and r > 1. Then, am = A + a′m with rm pa′m p < ∞ iff rm p(am+1 − am) p < ∞. Furthermore, we have
rm pa′m p ≤ C rm p(am+1 − am) p
Proof. The first statement implies the second statement. Observe that am+1 = A + a′m+1 and am = A + a′m, which gives
rm p(am+1 − am) p = rm p(a′m+1 − a′m) p ≤ C1 r(m+1) pa′m+1 p + rm pa′m p < ∞

16

In order to get the other direction, we first show that am is a Cauchy sequence. For m > n we have
m−1
am − an = (ak+1 − ak)rk pr−k p
k=n
applying Holder’s inequality yields

am − an ≤

m−1

1p

ak+1 − ak prk

k=n

m−1 1 k=n (rq p)k

1q
≤C

1 rqn p

which implies that am is a Cauchy sequence. So, am → A for some A. We can

write

∞∞
am − A = (ak − ak+1) = (am+k − am+k+1).

k=m

k=0

Multiplying by rm p yields

∞
rm p(am − A) = r(m+k) pr−k p(am+k − am+k+1).
k=0
Finally, by Minkowski’s inequality we obtain

∞
rm p(am − A) p ≤ r−k p (ak+m − ak+m+1)r(k+m) p p
k=0 ∞
≤ r−k p (am − am+1)rm p p.
k=0
Hence, the result.

Lemma 2.2.3. Let am be a sequence. We have (5p 3)m p(5am+2 − 8am+1 + 3am) p < ∞. 17

if and only if am = A1 + A2(3 5)m + a′m with (5p 3)m pa′m p < ∞. Also, the following estimate holds
(5p 3)m pa′m p ≤ C (5p 3)m p(5am+2 − 8am+1 + 3am) p

Proof. If we do the calculation we obtain (5p 3)m p(5am+2 − 8am+1 + 3am) p = (5p 3)m p(5a′m+2 − 8a′m+1 + 3a′m) p ≤ D1 (5p 3)(m+2) pam+2 p + D2 (5p 3)(m+1) pam+1 p + D3 (5p 3)m pam p < ∞
which shows that the second part implies the first part.

For

the

converse

implication,

we

use

the

substitution

dm

=

5m 3m

(am+1

−

am).

We

can write

∞ m=1

5p 3

m∞
(5am+2 − 8am+1 + 3am)p = 3p 3(p−1)m(dm+1 − dm)p < ∞. m=1

Observe that dm satisfies the assumptions of Lemma 3.1.2 with r = 3p−1. Hence,

dm = D + d′m with

∞∞

3(p−1)m d′m p ≤ C

3(p−1)m(dm+1 − dm)p.

m=1

m=1

Now, we apply the Lemma 3.1.2 for a second time for em = am + (5 2)(3 5)mD,

which yields

∞∞
3(p−1)m d′m p =

m=1

m=1

5p 3

m
(em+1 − em)p.

We obtain em = E + e′m where

∞ m=1

5p 3

m∞
e′m p ≤ C m=1

5p 3

m
(em+1 − em)p.

Putting everything together yields am = E − (5 2)(3 5)mD + e′m with

∞ m=1

5p 3

m∞
e′m p ≤ C m=1

5p 3

m
(5am+2 − 8am+1 + 3am)p.

Hence, the result.

18

Lemma 2.2.4. Let r < 1 be a constant and am be a sequence. Then

m
r p am

p < ∞ if and

only if

r

m p

(am+1

−

am)

p

< ∞.

Furthermore we have

m
r p am

p ≤ C1 a1

+ C2

r

m p

(am+1

−

am)

p.

Proof. It is easy to see that the first statement implies the second one. For the
m
converse, writing am as a telescoping series and multiplying by r p yields

rm

pam

=

rm

pa1

+

m−1

m
rp

(ak+1

−

ak)

=

m
rp

a1

+

m−1
(am−k+1

−

am−k

)r

m−k p

k
rp

.

k=1 k=1

(2.2)

We apply the Minkowski’s inequality to obtain

m−1
(am−k+1

−

am−k )r

m−k p

r

k p

k=1

∞
≤ rk p
p k=1

(am−k+1

−

am−k )r

m−k p

1k<m

p

∞
≤ rk p (am+1 − am)rm p p.
k=1

Applying Minkowski’s inequality one more time to (2.2) and using the above

estimate yields

rm pam p ≤ rm pa1 p + C2 (am+1 − am)rm p p.

Hence, the result.

Lemma 2.2.5. Let ηm be a sequence. Then,

m(p−1)
3p

(3ηm+2

−

16ηm+1

+ 5ηm)

p

<

∞

if

and only if ηm = 5mA + ηm′ with

m(p−1)
3 p ηm′

p < ∞.

Furthermore, we have

m(p−1)
3 p ηm′

p ≤ C1 η2 − 5η1

+ C2

3

m(p−1) p

(3ηm+2

−

16ηm+1

+

5ηm)

p.

Proof. The second statement obviously implies the first one. For the converse, let em = 3m(ηm+1 − 5ηm). We have

∞
3m(p−1)
m=1

3ηm+2

− 16ηm+1

+ 5ηm

p

=

∞ m=1

1 3m

em+1

− em

p.

19

Applying Lemma 2.2.4 to em yields

∞ m=1

1 3m

em

p

≤

C1 e1

+ C2

3

m(p−1) p

(3ηm+2

−

16ηm+1

+

5ηm)

p

p
< ∞.

We put 5mdm = ηm. Then, we can write

∞ m=1

1 3m

em

p

=

∞
3m(p−1)
m=1

ηm+1

− 5ηm

p

=

∞
5p
m=1

3p−15p

m dm+1 − dm p < ∞

Now, we apply 3.1.2 to the sequence dm to obtain dm = D + d′m with

∞ 3p−15p m d′m p ≤ C ∞ 3p−15p m dm+1 − dm p

m=1

m=1

where C = C(p) is a constant. By definition of dm we have ηm = 5mD + 5md′m. Putting everything together along with the definition ηm′ ∶= 5md′m yields

∞
3m(p−1) ηm′ p ≤

C1 e1

+ C2

3

m(p−1) p

(3ηm+2

−

16ηm+1

+

5ηm)

p

p
.

m=1

Now, having established the key Lemmas, we are ready to prove the first main result of this chapter. The proof is rather technical so we give some motivation. Recall that am = u(xm) and ηm = ∂nu(xm). The main tool lurking in the background turns out to be the Green’s formula. Recall that given any function u on SG for which ∆u exists, we can write as
u(x) = G(x, y)∆u(y)dy + h(x) SG
where G(x, y) is the Green’s function and h is the harmonic function having the same values as u on V0. By using the results in Chapter 1 of Appendix A, we relate the linear combinations 5am+2 − 8am+1 + 3am and 3ηm+2 − 16ηm+1 + 5ηm to integrals which we can bound using the sequence of Lemmas we proved in this
20

section. We also note that the definition of Ψm and certain estimates on its size was given in (A.3) and (A.4).

Theorem 2.2.6. Let p > 1, u ∈ domLp∆(SG) and u V0 = 0. Then, Ru ∈ Tp and

Ru Tp ≤ C ∆u Lp(SG).

(2.3)

Proof. We follow the ideas in [LS]. We start with u ∈ domLp∆(SG) with Ru = {(am, ηm)}. Applying the Green’s formula in Theorem A.0.3 to 5am+2 − 8am+1 + 3am and the equation for G(xm, y) in Lemma A.5, we obtain

5am+2

− 8am+1

+

3am

=

3m 5m

Gm∆udy.
SG

where Gm is defined to be

1 50

(9Ψm+2(3,

1,

1)

−

20Ψm+1(1,

0,

0)

+

25Ψm

(1,

−1,

−1))

.

(2.4)

Arguing in a similar manner to [LS], we deduce that Gm is supported only on Dm = Ym ∪ Ym+1 ∪ Ym+2 ∪ Zm ∪ Zm+1 ∪ Zm+2. So, we have

5am+2

−

8am+1

+

3am

=

3m 5m

Gm∆udy.
Dm

(2.5)

We observe that

ψxm q

≤

ψxm

and ∫SG ψxm dy =

.2
3m+1

Therefore, for C

=

C(a, b, c) we obtain

SG

Ψm(a, b, c)

q

≤

C .
3m

Having this estimate at our disposal, we apply Holder’s inequality to (2.5) and

obtain

5am+2 − 8am+1 + 3am p ≤ C

3 5

mp

∆u

p1 3Lp(Dm) pm

q

.

21

Pluggging in 1 − 1 q = 1 p and rearranging yields

5p 3

mp p
(5am+2 − 8am+1 + 3am) ≤ C

∆u

.p
Lp(Dm)

Recall that Dm = Ym ∪ Ym+1 ∪ Ym+2 ∪ Zm ∪ Zm+1 ∪ Zm+2 which implies

m+2

∆u

=p
Lp(Dm)

∆u

.p
Lp (Yk ∪Zk )

k=m

We also have

∞

∆u

p Lp(SG)

=

∆u

.p
Lp (Yk ∪Zk )

k=1

(2.6) (2.7) (2.8)

Therefore, using (2.6) we can write

5p 3

mp
(5am+2 − 8am+1 + 3am)

p
≤C
p

∆u

p Lp

(SG)

.

We apply Lemma 2.2.3 to obtain am = A1 + A2(3 5)m + a′m where A1 = limm→∞ am and A2 = limm→∞(5 3)m(am − A1). By the same lemma, we also have
the following estimate

5p 3

mp
a′m

≤
p

5p 3

mp
(5am+2 − 8am+1 + 3am)

.
p

Combining the above inequalities, we immediately get

5p 3

mp
a′m

≤C
p

∆u Lp(SG).

(2.9)

We want to show that A1 = 0 and A2 ≤ C ∆u L2(SG). The Green’s formula

for am reads

am = u(xm) = G(xm, y)∆u(y)dy. SG

By Lemma A.5, we know that

G(xm,

y)

=

2 15

3 5

m

m k=1

ψk(1,

2,

2)(y)

+

1 6

3 5

m
ψm(1, −1, −1)(y).

22

Then, by using Holder’s inequality and ψk q ≤ ψk we can write

5 3

m
am ≤

5m 3

G(xm, y) ∆u(y) dy
SG

m

≤ D1

ψk(1, 2, 2) ∆u dy + D2

ψm(1, −1, 1) ∆u dy

k=1 SG

SG

m
≤ D1 ( ψk q)1 q ∆u p + D2 ( ψm q)1 q ∆u p

k=1

≤

∆u

m1

p k=1

3k

q

+ D2

∆u

1 p 3m q

≤ C1

∆u

p + C2

∆u

1 p 3m q .

This estimate finally implies that

A1 = 0 and A2 ≤ C ∆u p

as asserted.

(2.10)

Now, we want to bound the normal derivative ηm.

We use Lemma A.0.6 to calculate 3ηm+2 − 16ηm+1 + 5ηm and obtain

3ηm+2 − 16ηm+1 + 5ηm =

Φm∆udy − (φm+2 − 16φm+1 + 5φm).

Dm

(2.11)

where and

φm =

ψm∆udy

Zm

Φm

=

1 10

(−3Ψm+2(5,

1,

−1)

+

10Ψm+1(8,

1, −1)

−

25Ψm(1, −1,

1)).

Observe that Zm ⊂ Dm, which will be important for the estimates.

Applying Holder’s inequality to (2.11) and φm and taking pth power yields

3ηm+2 − 16ηm+1 + 5ηm p ≤

1
pm

3q

∆u

.p
Lp(Dm)

23

Since p q = p − 1 we obtain

3m(p−1) 3ηm+2 − 16ηm+1 + 5ηm p ≤

∆u

.p
Lp(Dm)

Now, we use again (2.7) and (2.8) to conclude

3

m(p−1) p

(3ηm+2

−

16ηm+1

+

5ηm)

p≤C

∆u

Lp(SG).

Then, the sequence ηm satisfies the assumption of Lemma 2.2.5 which gives us ηm = 5mA + ηm′ with

m(p−1)
3 p ηm′

p ≤ C1 η2 − 5η1

+ C2

3

m(p−1) p

(3ηm+2

−

16ηm+1

+

5ηm)

p.

We recall from Lemma A.0.6 that

ηm

=

∂u(xm)

=

3 5

−

1 2

1 3m

m
3k
k=1

ψk(0, −1, 1)∆udy
SG

ψm(1, −1, 1)∆udy − φm.
SG

Applying Holder to the above equation yields

ηm

≤ C1

∆u

1 p 3m

m
3k
k=1

1 3k

q

+ C2

∆u

1 p 3m q

≤ C1

∆u

1 (31 p)m+1 − 31 p p 3m 31 p − 1

+ C2

∆u

1 p 3m q

≤ C1

∆u

1 p 3m

(31

p)m+1

+

C2

∆u

1 p 3m q

≤ C3

∆u

1 3m

q

+

C2

∆u

1 p 3m q

≤C

∆u

1 p 3m q

24

where C1, C2, C3 = C3(p) are constants. By the above inequality, we obtain A = 0

and ηm = ηm′ . We also get the bound η2 − 5η1 ≤ C ∆u Lp(SG). Putting everything

together yields

m(p−1)
3 p ηm′

p≤C

∆u

Lp(SG).

(2.12)

Since am = A2(3 5)m + a′m, by using (2.9), (2.10) and (2.12) we obtain

Ru

p p

=

A1 p +

A2 p +

5p 3

mp
a′m

p
+
p

m(p−1)
3 p ηm′

p p

≤

C

∆u

p Lp

(S

G)

.

Hence, the result.

Lemma 2.2.7. If h is a harmonic function, then h ∈ Tp with

Rh

p Tp

=

h(q0) p +

1 2p

h(q1) + h(q2) − 2h(q0) p +

1 2p+1

h(q1) − h(q2) p.

(2.13)

Proof. Writing h as a linear combination of the constant, skew-symmetric and symmetric harmonic function yields h(xm) = A1 + A2(3 5)m and ∂nh(xm) = A3 3m, where

A1 = h(q0)

A2

=

1 2

(h(q1

)

+

h(q2)

−

2h(q0))

A3

=

1 2

(h(q1

)

−

h(q2)).

Theorem 2.2.8. (Lp- Trace Theorem) The restriction operator R ∶ domLp∆(SG) → Tp is bounded and
Ru Tp ≤ C1 u L∞(SG) + C2 ∆u Lp(SG).
Proof. Let u ∈ domLp∆(SG) and h be the harmonic function with h V0 = u V0. We put w = u − h and observe that ∆u = ∆w. We have
Ru Tp ≤ Rw Tp + Rh Tp
25

Since w = 0 on V0, by (2.3) we have

Rw Tp ≤ C2 ∆u Lp(SG).

By (2.13) we can write

Rh

p Tp

=

u(q0) p +

1 2p

u(q1) + u(q2) − 2u(q0) p +

1 2p+1

u(q1) − u(q2) p

≤

u

p L∞(SG)

+

1 2p

(4

u

L∞(SG))p

+

1 2p+1

(2

u

L∞(SG))p ≤ C1

u

p L∞(SG)

where C1 = C1(p), C2 = C2(p) are constants. Hence, the result.

2.3 Extension Theorem

We quote the following result from [LS]:

Theorem 2.3.1. ( [LS]*7.3 & 7.4 ) Given any sequences am and ηm, there exists a piecewise biharmonic function u on SG and sequences Cm′ and Cm such that Ru = {(am, ηm)}, ∆u = Cm′ on Ym, ∆u = Cm on Zm and the normal matching conditions hold at xm, ym and zm. We have

Cm′ = 5m

3 8

(5am+1 − 8am + 3am−1) − 3m

3 8

(3ηm+1 − 16ηm + 5ηm−1)

Cm = 5m

3 8

(5am+1 − 8am + 3am−1) + 3m

3 8

(3ηm+1 − 16ηm + 5ηm−1)

(2.14) (2.15)

The extension operator that maps two sequences {(am, ηm)} to the function u in the above theorem is denoted by E. The following lemma is a standard result that we include without proof.
Lemma 2.3.2. Let α, β be complex numbers and p ≥ 1 we have
α + β p ≤ 2p−1( α p + β p).

26

We prove the following theorem.
Theorem 2.3.3. (Lp-extension Theorem) Let p > 1. There exists a bounded linear extension map E ∶ Tp → domLp∆(SG) with R ○ E = Id.

Proof. Let {(am, ηm)} ∈ Tp and u = E{(am, ηm)} where E is as defined above. We

have am = A1 + A2(3 5)m + a′m with

(5p 3)m pa′m p < ∞ and

m(p−1)
3 p ηm

p < ∞.

It

follows that a′m → 0 and ηm → 0. By the same argument for the T∞ case in [LS],

u is continuous at q0 thus continuous everywhere by construction. We want to show that ∆u ∈ Lp(SG). By using the definitions of Cm′ , Cm and the previous

Lemma we have

Cm′ p ≤ 5m

3 8

(5a′m+1 − 8a′m + 3a′m−1) − 3m

3 8

p
(3ηm+1 − 16ηm + 5ηm−1)

≤ 2p−1 5m

3 8

p
(5a′m+1 − 8a′m + 3a′m−1) + 2p−1 3m

3 8

p
(3ηm+1 − 16ηm + 5ηm−1)

≤ K15(m+1)p a′m+1 p + K25mp a′m p + K35(m−1)p a′m−1 p

+ H13(m+1)p ηm+1 p + H23(m)p ηm p + H33(m−1)p ηm−1 p

for some constants Ki = Ki(p), Hi = Hi(p). Then, we can write

∞ m=1

Cm′ p 3m

∞
≤ 3K m=1

5p 3

m∞

a′m p + 3H

3m(p−1) ηm p

m=1

≤ D1 ∞ 2 m=1

5p 3

m

a′m

p

+

D2 2

∞
3m(p−1)
m=1

ηm

p

= D1 2

(5p

3)m pa′m

p p

+

D2 2

m(p−1)
3 p ηm

p p

<

∞

A similar estimate also holds for ∑∞m=1

Cm 3m

p

,

hence

we

obtain

∆u

∞

p Lp(SG)

=

m=1

Cm′ p + Cm p 3m

∞
≤ D1 m=1

5p 3

m∞

a′m p + D2

3m(p−1) ηm p < ∞

m=1

where D1 = D1(p), D2 = D2(p) are constants. Hence, the result.

27

CHAPTER 3 POLYHARMONIC BOUNDARY VALUE PROBLEMS ON SG

Recall that K denotes SG and L the bottom line of SG identified with [0, 1].

Similarly L1 denotes [0, 1 2] and L2 denotes (1 2, 1]. Let Ψ = 1 denote the con-

stant function on the relevant domain. The Haar functions give us an orthonor-

mal basis on L2[0, 1]. The Haar basis we use consists of the functions Ψ and ψn,k

for n = 0, 1, 2, . . . and k = 0, 1, . . . , 2n−1 which are defined as follows

ψn,k(t) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

2n 2 −2n 2 0

t

∈

[

k 2n

,

k+1 2n

2

)

t

∈

(

k+1 2n

2,

k+1 2n

]

otherwise .

Observe that ψn,k are nothing but the dilated and translated versions of the mother wavelet

ψ(t) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

1 −1 0

t ∈ [0, 1 2) t ∈ (1 2, 1]
otherwise .

We have ψ = ψ0,0. For, f ∈ L2[0, 1], we have the Haar coefficients of f as

cΨ(f ) =< f, Ψ > cn,k(f ) =< f, ψn,k > where < ⋅, ⋅ > denotes the usual inner product. We know that the Haar series

∞ 2n−1

cΨ(f )Ψ +

cn,k(f )ψn,k.

n=0 k=0

28

(3.1)

converges to f in the L2-norm. As in [OS] we use a different notation to have

the representation of f comparable to the analysis on SG. As described in sec-

tion

2.1

w

denotes

a

word

in

0

and

1’s.

Observe

that

the

contractions

F0(t)

=

1 2

t

and

F2(t)

=

1 2

t

+

1 2

are

the

restrictions

of

the

two

of

the

three

contraction

map-

pings defining SG. For details we refer the reader to Section 2.1. It follows that

Fw[0, 1] for a word with w = m is a dyadic interval in the form [k 2m, (k+1) 2m].

Using this notation we have that

ψw = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩02m 2ψ ○ Fw−1

on Fw[0, 1]. otherwise .

Similar to the previous case, we put

cΨ(f ) =< f, Ψ > cw(f ) =< f, ψw > .
That is, f has the representation

∞
f = cΨ(f )Ψ +

cw(f )ψw.

m=0 w =m

(3.2)

Following [OS] we define the following Sobolev-type function spaces and norms on L. For f as in (3.2) we put

∞

f

2 Hs

=

cΨ(f ) 2 +

2smcw(f ) 2

m=0 w =m

29

(3.3)

and

Hs(L) = {f ∶ f Hs < ∞}.

(3.4)

Observe that putting s = 0 yields H0(L) = L2(L) in which case f H0 = f 2 since Haar functions are orthonormal in L2(L). For s < 0, the elements of H−s are distributions on L. Hs and H−s are dual spaces of each other with respect to
the usual pairing [J4, J5].

In this Chapter, we denote the 3 boundary vertices of SG by v1, v2, v3. Recall that we put SG = SG ∖ L ∪ {v1} We want to solve the following BVP on K:

∆2u = 0 ∆u L = f2 ∆u(v1) = c′
u L = f1 u(v1) = c
Next we write the Haar expansion of both f1 and f2.

∞
f1 = cΨ(f1)Ψ +

cw(f1)ψw

m=0 w =m

∞
f2 = cΨ(f2)Ψ +

cw(f2)ψw.

m=0 w =m

(3.5) (3.6)

we want to obtain the following representation for u.

u(x) = (c′ − cΨ(f2))h3(x) + cΨ(f1)Ψ + (a − cψ(f1))h0(x) + cΨ(f2)h2

∞
+

∞
cw(f1)h1w(x) +

cw(f2)h2w(x)

m=0 w =m

m=0 w =m

30

where h0, h1ψ are the harmonic functions as constructed in [OS] and the biharmonic functions h2, h3 and h2ψ are constructed, in the following section (Section 3.1), so that they satisfy

h0(v1) = 1 h0 L = 0 ∆h0 L = 0 ∆h2 = 1 h2(v1) = 0 h2 L = 0
∆h3(v1) = 1 h3(v1) = 0 h3 L = 0 ∆h3 L = 0 h1ψ(v1) = 0 h1ψ L = ψ ∆h1ψ L = 0 h2ψ(v1) = 0 h2ψ L = 0 ∆h2ψ L = ψ.

Also, we define the miniaturized versions of h1ψ and h2ψ, as h1w and h2w.

h1w = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩02m 2h1ψ ○ Fw−1

on FwK otherwise .

h2w = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩025mm2 h2ψ ○ Fw−1

on FwK otherwise .

Remark

3.0.4.

The

appearance

of

the

factor

1 5m

in

the

definition

of

h2w

is

because

Fw−1

scales the Laplacian by 5m and we want to bring it down to 1.

We note that h1w and h2w satisfy

h1w(v1) = 0 h1w L = ψw ∆h1w L = 0 h2w(v1) = 0 h2w L = 0 ∆h2w L = ψw.
31

3.1 Construction of the Basic Functions

In the following proposition below, we give the rules of how the normal deriva-

tives of a biharmonic function is distorted when miniaturized. For the definition

of normal derivatives on SG we refer the reader to Section 2.1. We already know

that

∆n(φ

○

Fi)

=

(

1 5

)n(∆nφ)

○

Fi

(3.7)

By iteration for w = k we obtain

∆n

(φ

○

Fw

)

=

(

1 5

)nk

(∆n

φ)

○

Fw

.

For n = 2, namely in the bilaplacian case, we have

∆2(φ

○

Fw)

=

(

1 5

)2k(∆2φ)

○

Fw.

Remark 3.1.1. Similarly we have ∆(φ ○ Fw−1) = 5n(∆φ) ○ Fw−1.

Lemma 3.1.2. Let gi denote the harmonic function with gi(vj) = δij and let w = k.

Then we have

∂ngi ○ Fw(vi) = 2

5 3

k

∂ngi ○ Fw(vj) = −

5 3

k

Proposition 3.1.3. Let u be a biharmonic function and w = k. Then, we have

∂n(u ○ Fw−1)(Fwv1) = +

5 3

k
(2u(v1) − u(v2) − u(v3))

5 3

k

∆u(v1)

7 45

+

∆u(v2)

4 45

+

∆u(v3)

4 45

.

∂n(u ○ Fw−1)(Fwv2) = +

5 3

k
(2u(v2) − u(v1) − u(v3))

5 3

k

∆u(v1

)

4 45

+

∆u(v2)

7 45

+

∆u(v3)

4 45

32

∂n(u ○ Fw−1)(Fwv3) = +

5 3

k
(2u(v3) − u(v1) − u(v2))

5 3

k

∆u(v1)

4 45

+

∆u(v2)

4 45

+

∆u(v3)

7 45

.

Proof. We use the localized symmetric Gauss-Green formula on A = FwK. We have

(u ○ Fw−1)∆(v ○ Fw−1) − (v ○ Fw−1)∆(u ○ Fw−1)
AA
= ((u ○ Fw−1)(Fwq)∂n(v ○ Fw−1)(Fwq) − (v ○ Fw−1)(Fwq)∂n(u ○ Fw−1)(Fwq)).
q∈{v1,v2,v3}
We first put v = g1, since g1 is harmonic we obtain

∂n(u ○ Fw−1)(Fwv1) =

u(q)∂n(v ○ Fw−1)(Fwq) + (v ○ Fw−1)∆(u ○ Fw−1).

q∈{v1,v2,v3}

A

Observe that since u is biharmonic, ∆u = h with h = ∆u(v1)g1 + ∆u(v2)g2 +

∆u(v3)g3. So we obtain, by using Lemma 3.1.2,

∂n(u ○ Fw−1)(Fwv1) =

5 3

k
(2u(v1) − u(v2) − u(v3))

+ 5k ∆u(v1)

(g1 ○ Fw)2 dµ + ∆u(v2)

(g2 ○ Fw)(g1 ○ Fw)dµ

Fw K

Fw K

+ ∆u(v3)

(g3 ○ Fw)(g1 ○ Fw)dµ .

Fw K

Since

∫K

gi gj dµ

=

4 45

and

∫K

gi2dµ

=

7 45

,

plugging

in

yields

∂n(u ○ Fw−1)(Fwv1) = +

5 3

k
(2u(v1) − u(v2) − u(v3))

5 3

k

∆u(v1)

7 45

+

∆u(v2)

4 45

+

∆u(v3)

4 45

.

Similar calculation with v = g2, g3 yields

33

∂n(u ○ Fw−1)(Fwv2) = +

5 3

k
(2u(v2) − u(v1) − u(v3))

5 3

k

∆u(v1

)

4 45

+

∆u(v2)

7 45

+

∆u(v3)

4 45

and

∂n(u ○ Fw−1)(Fwv3) = +

5 3

k
(2u(v3) − u(v1) − u(v2))

5 3

k

∆u(v1)

4 45

+

∆u(v2)

4 45

+

∆u(v3)

7 45

.

As a Corollary of Lemma 3.1.3, we have:

Corollary 3.1.4. Assume that ∆u = C on a cell of level m with boundary p0, p1, p2. Then, the outward normal derivative of u reads

∂nu(pj) =

5 3

k

(2u(pj )

−

u(pj+1)

−

u(pj−1))

+

C 3m+1

.

As shown in Figure 3.1, we start with the prescribed data a, b, c, m, n, q and want to obtain an equation relating these with x, y, z. In order to do so, we use the two basis given in [SU] for the space of biharmonic functions. Again, by the Figure 3.1, we can write
u = xf01 + af02 + mf03 + yf11 + bf12 + nf13 u = xf0(11) + af0(21) + mf0(31) + zg0(11) + cg0(12) + qg0(13)

34

Figure 3.1: We define [u](x) ∶= (u(x), ∆u(x), ∂nu(x)).

We know by [SU] that the two basis are related by the equations
3
f0(k1) = f0k + bklf1l
l=1 3
g0(1k) = dklf1l
l=1
Here, dkl = Jkl and bkl = − ∑3m=1 Jml∂nf0k(vm) where J is the inverse matrix of the inner products of the easy basis for H0 and ∂nf0k(vm) = −Hmk where {Hmk} denotes the Dirichlet form E0 on V0 × V0. Namely, we have

H

⎛ = ⎜⎜⎜⎜⎜⎜⎝

−2 1 1

1 −2
1

1 1 −2

⎞ ⎟⎟⎟⎟⎟⎟⎠

J

⎛ = ⎜⎜⎜⎜⎜⎜⎝

11 −4 −4

−4 11 −4

−4 −4
11

⎞ ⎟⎟⎟⎟⎟⎟⎠ .

We obtain bii = −30, bij = 15 and dii = 11, dij = −4. The conversion equations between two basis give rise to the equation

35

15 a − 4 c + 15 m − 4 q − 30 x + 11 z = y −30 a + 11 c + 15 m − 4 q + 15 x − 4 z = b
15 a − 4 c − 30 m + 11 q + 15 x − 4 z = n
which has the solution

x = r2,

y

=

−

135 2

a

−

11 4

b

+

105 4

c

−

15

r1

+

45 4

r2

+

225 4

r3,

z

=

−

15 2

a

−

1 4

b

+

11 4

c

−

r1

+

15 4

r2

+

15 4

r3,

m = r3,

n = 45 a + b − 15 c + 15 r1 − 45 r3,

q = r1

(3.8) (3.9) (3.10) (3.11) (3.12) (3.13)

for three free parameters r1, r2, r3. We will use the equations 3.8 to construct the functions we need.
Lemma 3.1.5. There exists a biharmonic function h0 satisfying h0(v1) = 1 h0 L = 0 ∆h0 L = 0.
Proof. Since a harmonic function is fully determined by its values on the boundary, we prescribe certain values on the boundary of the triangle then extend harmonically. Of course, one needs to check the normal derivative matching conditions. As shown in Figure 3.2 we construct a highly symmetric harmonic

36

Figure 3.2: an denotes the function values on corresponding vertices.

function, this construction is inspired by [OS]. The construction gives rise to the structure shown in Figure 3.3.

The matching conditions for the normal derivative yields

2an+1

− an+1

−

an

=

−5 3

(2an+1

−

2an+1)

which we can write as

an+1

−

an

=

10 3

(an+2

−

an+1).

(3.14) (3.15)

We apply the transformation ∆n = an+1 −an and observe that ∑Nn=0 ∆n = aN+1 − a0 and ∑Nn=0 ∆n+1 = aN+2 − a1. Observe that we can write (3.15) as

3 10

∆n

=

∆n+1.

Summing both sides from 0 to N yields

3 10

N n=0

∆n

=

N
∆n+1.
n=0

37

Figure 3.3: an denotes the values on corresponding vertices and the numbers in the triangles denote the level numbers.
By our previous observations we obtain the recurrence

aN +2

=

3 10 aN+1

+

a1

−

3 10 a0.

Remarks 3.1.6.

1.

Observe

that

for

the

initial

condition

a0

=

1, a1

=

3 10

one

gets

an =

3 10

n. In this case, L ∶= limn→∞ an = 0.

2.

We

can

play

with

a0

and

a1

in

order

to

get

L

=

a1

−

3 10

a0

≠

0.

Proposition 3.1.7. There exists a function h3(x) biharmonic on SG which satisfies the

38

following properties

h3(v1) = 0, ∆h3(v1) = 1
h3 L = 0 ∆h3 L = 0.

Figure 3.4: Here D = (x0, a0), E = (x1, a1), A = (x2, a2), B = (x3, a3) where xi denote the function and ai denote the Laplacian values.
Proof. We use the fact from [SU] that a biharmonic function on a triangle is fully determined by its function and Laplacian values on the boundary of the triangle. We construct a symmetric biharmonic function as shown in Figure 3.4 which gives rise to the structure shown in Figure 3.5. In this case, by Proposition 3.1.3, the matching equations for the normal derivatives read, using the same notation as in Figure 3.5,
39

Figure 3.5: In (xi, ai), xi stands for the function values and ai for the Laplacian values. The numbers inside the triangles denote the level
number.

2xn+1

−

xn+1

−

xn

+

7 45 an+1

+

4 45 an+1

+

4 45 an

=

−5 3

(2xn+1

−

2xn+2

+

7 45 an+1

+

4 45 an+2

+

4 45

an+2)

which we can rearrange as

xn+1

−

xn

+

11 45 an+1

+

4 45

=

10 3

(xn+2

−

xn+1)

−

5 3

7an+1 + 8an+2 45

.

(3.16)

We can apply the transformation ∆n = xn+1 − xn and write (3.16) as

∆n+1

=

3 10 ∆n

+

λn

where

λn

=

11an+1+4an 150

+

.7an+1+8an+2 90

40

(3.17)

Now, we sum both sides of (3.17) from 0 to N to get

Remarks 3.1.8.

xN +2

=

3 10 xN+1

+ x1

−

3 10 x0

N
+ λn n=0

(3.18)

1. Observe that when a0 and a1 are chosen so that L < 1 then we

have that ∑∞n=0 λn < ∞ in which case the equation (3.18) has a solution which

gives a biharmonic function with non-zero Laplacian.

In this general construction, in order to get h3, we simply set a0 = 1 and a1 = 3 10 in which case an = (3 10)n. Observe that with these, we have ∑∞n=0 λn < ∞. We simply set x0 = 1 and x1 = 3 10. Of course, we need to subtract off a harmonic function to get the correct boundary values, which does not effect the value of the Laplacian.
Proposition 3.1.9. There exists a biharmonic function, h2ψ on SG satisfying the following properties
h2ψ(v1) = 0, ∆h2ψ(v1) = 0 ∂nh2ψ(v1) = 0
h2ψ L = 0 ∆h2ψ L = ψ.

Proof. We use the construction in Lemma 3.1.7. Observe that we can construct a biharmonic function with [u](v1) = (0, 0, 0), [u](v2) = (m, n, q), [u](v3) = (−m, −n, −q). We simply put this function on F1K as shown in Figure 3.6. Then, apply the construction in a skew-symmetric manner both to F2K and F3K by
41

Figure 3.6: Here A = (x1, a1), B = (x1, a1) where xi denote the function and ai denote the Laplacian values. Similarly, (0, 0, 0), (m, n, q), (−m, −n, −q) are the triples in the form (u(x), ∆u(x), ∂nu(x)) at the corresponding vertex x
choosing the parameters in a suitable way. We again subtract off a harmonic function to get the correct boundary values.
Remark 3.1.10. Observe that when we have a recurrence relation of the following form xn+1 = rxn + bn
with r < 1 and bn < C then limn→∞ xn < ∞. This follows from the fact that xN = ∑Ni=0−1 birN−1−i. Since, xN ≤ C ∑Ni=0−1 r i < ∞. Lemma 3.1.11. There exists a biharmonic function u on K with the following property
(u(v1), ∆u(v1), ∂nu(v1)) = (1 3, 1, 1) (u(v2), ∆u(v2), ∂nu(v2)) = (0, 1, 0) (u(v3), ∆u(v3), ∂nu(v3)) = (0, 1, 0).
42

If we put u1(x) = (u ○ F1−1)(x), x ∈ F1K then we have (u1(F1(v1)), ∆u1(F1(v1)), ∂nu1(F1(v1))) = (1 3, 5, 5 3) (u1(F1(v2)), ∆u1(F1(v2)), ∂nu1(F1(v2))) = (0, 5, 0) (u1(F1(v3)), ∆u1(F1(v3)), ∂nu1(F1(v3))) = (0, 5, 0)
Proof. We obtain u by plugging in a = c = r3 = r1 = 0, b = n = 1, r2 = 1 3 in the equations (3.8). For the second part, we scale the Laplacian accordingly and change the normal derivatives by using Proposition 3.1.3. Proposition 3.1.12. There exists a function h2(x) biharmonic on SG which satisfies the following properties
h2(v1) = 0 ∆h2 = 1 h2 L = 0.
Proof. We first define a function R. We define R to be equal to u1, as in Lemma 3.1.11, on F1K. After that keeping the derivative to be 5, we continue in a symmetric way all the way down to the bottom line L, as shown in Figure 3.7.
As shown in Figure 3.7 we miniaturize this function to the top of the SG (we denote the normal derivative by m and Laplacian value by a) and continue all the way to the bottom in a symmetric way. This gives rise to the structure shown in Figure 3.8.
As seen in Figure 3.7, we continue to the bottom the Laplacian in a symmetric way. By using Corollary 3.1.4, the first compatibility equation for normal derivatives reads:
43

Figure 3.7: Here x and y = x1 are the function values, the Laplacian is equal to a on F2K and F3K. Similarly, (0, a, m) is a triple in the form (u(x), ∆u(x), ∂nu(x)) at the corresponding vertex x

x

=

a 2.52.3

+

5 3

2m .
2

Now, we write the matching condition for the general point xn+1 in figure 3.8

to obtain

5 3

n+2
(2xn+1

− xn+1

−

xn) +

a 3n+3

+

5 3

n+3
(2xn+1

− 2xn+2) +

a 3n+4

=

0.

In order to solve this recursion we put ∆n = xn − xn−1. Then the equation becomes

∆n+2

=

3 10 ∆n+1

+

2a .
3.5n+3

We observe that ∑Nn=−01 ∆n+1 = xN since we have the initial conditions x1 = x, x0 = 0, ∆1 = x. Putting these together yields the equation

44

Figure 3.8: The numbers in the triangles denote the level number and xi are the values of the function on the corresponding vertices.

xN +1

=

3 10 xN

+

N −1 n=0

2a 3.5n+3

+

x.

By plugging in a = 5, m = 0 we get the recurrence relation

xN +1

=

3 10 xN

+

N −1 n=0

2×5 3.5n+3

+

2 .
15

with

initial

conditions

x0

=

0, x1

=

2 15

,

∆1

=

2 15

.

We

have

limN→∞ xN

=

5 21

.

We consider K

=

R 5

which satisfies K(v1) =

1 15

,

∆K

=

1, K L

=

1 21

.

In order

to get h2

we subtract off

the harmonic function

Y

with

Y (v1)

=

1 15

and Y

L

=

1 21

.

Hence, the result.

45

3.2 Orthogonality of Basic Functions
So far, we have the basic functions: A ∶= {h0, h1ψ, h1w, h2ψ, h2w}. We want to show that this family of functions is orthogonal with respect to the inner product
(f, g) = f Lg Ldx + E(f, g) + ∆f ∆gdµ. SG
In the sequel we will put (f, g)2 = f Lg Ldx (f, g)3 = E(f, g) (f, g)4 = ∆f ∆gdµ.
SG
That is, (f, g) = (f, g)2 + (f, g)3 + (f, g)4. We will show that the family, {h0, h1ψ, h1w, h2ψ, h2w}, is orthogonal with respect to each (⋅, ⋅)i. Lemma 3.2.1. Let A ⊂ SG. If f is symmetric and g is skew-symmetric on A then EA(f, g) = 0.
Proof. It is a standard fact that for each f, g there exits a signed measure (energy measure) νf,g bilinear in f and g such that
EA(f, g) = 1Aνf,g = νf,g(A). Then, if f is symmetric and g is skew-symmetric it follows from bilinearity that EA(f, g) = νf,g(A) = 0. Remark 3.2.2. Observe that the measure νf,g does not have any atoms and therefore does not charge points. This follows from the fact that νf,f does not have any atoms because
νf,f (FwK) ≤ r w E (f, f ).
46

Together with the polarization identity the above inequality implies

νf,g(FwK) ≤ r w (E(f + g) + E(f − g))

which implies that νf,g is atomless.

Proposition 3.2.3. [OS] Let v ∈ dom(E) then, the Gauss-Green formula holds

E(h0, v)

=

−7 3

v Ldx + v(v1)∂nh(v1).
L

That is, if ∫L v Ldx = 0 then we simply have E(h0, v) = v(v1)∂nh(v1).

Lemma 3.2.4.

(h0, h1ψ)2 = 0 (h0, h1ψ)3 = 0 (h0, h1ψ)4 = 0 (h0, h2ψ)2 = 0 (h0, h2ψ)3 = 0 (h0, h2ψ)4 = 0 (h0, h1w)2 = 0 (h0, h1w)3 = 0 (h0, h1w)4 = 0 (h0, h2w)2 = 0 (h0, h2w)3 = 0 (h0, h2w)4 = 0

Proof. We put B ∶= A ∖ h0. Recall that h0 is the harmonic function on SG with h0(v1) = 1, h0 L = 0. Given that h0 is harmonic we readily have (h0, f )4 = 0 for any f ∈ B. Since h0 L = 0 we obtain (h0, f )2 = 0 for all f ∈ B.
By Lemma 3.2.1 we have (h0, f )3 = 0 for f ∈ {h1ψ, h2ψ, h1w, h2w}.

Lemma 3.2.5.

(h1ψ, h2ψ)2 = 0 (h1ψ, h2ψ)3 = 0 (h1ψ, h2ψ)4 = 0 (h1ψ, h1w)2 = 0 (h1ψ, h1w)3 = 0 (h1ψ, h1w)4 = 0 (h1ψ, h2w)2 = 0 (h1ψ, h2w)3 = 0 (h1ψ, h2w)4 = 0
47

Proof. This time we put B = A ∖ h1ψ.

Since h2ψ L = 0 we have (h1ψ, h2ψ)2 = 0. By, skew symmetry we easily obtain (h1ψ, h1w)2 = 0.

Similarly h2w L = 0 implies that (h1ψ, h2w)2 = 0.

We want to show (h1ψ, h2ψ)3 = E(h1ψ, h2ψ) = 0. We can write

E (h1ψ,

h2ψ )

=

3 i=1

1 3

E

((h1ψ

○

Fi,

h2ψ

○

Fi).

In the above summation we name the three parts as I, II and III. We denote by n1 and −n2 the normal derivatives ∂nh1ψ(F1v2) and ∂nh1ψ(F1v3) with respect to the cell F1K. We put h2ψ(F1v2) = x, h2ψ(F1v3) = −x and recall that h2ψ(v1) = 0, h2ψ L1 = 0, h2ψ L2 = 0. Applying local Gauss-Green to the cells F1K, F2K, F3K and noting the cancellations by Lemma 3.2.3 yields
I = n1x + n2x II = −n1x III = −n2x.
Therefore, I + II + III = 0.

By the skew symmetry of h1w, h2w and symmetry of h1ψ on FwK we have that (h1w, h1ψ)3 = (h2w, h1ψ)3 = 0.
Since h1ψ is harmonic on SG we have that (h1ψ, h2ψ)4 = (h1ψ, h1w)4 = (h1ψ, h2w)4 = 0.

48

Lemma 3.2.6.

(h2ψ, h1w)2 = 0 (h2ψ, h1w)3 = 0 (h2ψ, h1w)4 = 0 (h2ψ, h2w)2 = 0 (h2ψ, h2w)3 = 0 (h2ψ, h2w)4 = 0

Proof. Since h2ψ L = 0 we obtain (h2ψ, h1w)2 = 0, (h2ψ, h2w)2 = 0. By skew symmetry of h1w and h2w and symmetry of h2ψ on the cell FwK we have (h2ψ, h1w)3 = 0, (h2ψ, h2w)3 = 0 and (h2ψ, h2w)4 = 0. Since, h1w is harmonic we readily obtain (h2ψ, h1w)4 = 0.
Lemma 3.2.7.
(h1w, h2w)2 = 0 (h1w, h2w)3 = 0 (h1w, h2w)4 = 0 (h1w′, h2w)2 = 0 (h1w′, h2w)3 = 0 (h1w′, h2w)4 = 0.

Proof. The first row is simply the scaled down versions of the previous Lemmas, hence they are all zero. The second row follows from definitions and skewsymmetry.

3.3 Solution of Polyharmonic BVP and Convergence Theorems
Definition 3.3.1. Let DM denote the closed subset of SG which is the union of all the cells FwF0(SG) for w < M , here we take wj = 1, 2. Observe that ∂DM consists of the points q0 and Fwq0 for w = M . See Figure 3.9.
We put Vn = Dn+1 ∖ Dn. Observe that SG = ∪∞i=0Vi.
Lemma 3.3.2. There exists a unique biharmonic function on DM when we prescribe values and laplacian values on ∂DM .
49

Figure 3.9: The Picture of Dn for the first three levels.
Proof. For the harmonic case, we know from [OS] that the prescribed values on ∂DM determines a unique harmonic function on DM . We obtain the biharmonic case by an iterated application of the harmonic case. Suppose there are two biharmonic functions u1 and u2 with the same values and Laplacian values on DM . It follows that h1 = ∆u1 and h2 = ∆u2 are two harmonic functions on DM but observe that h1 and h2 has the same values on ∂DM . It follows from the harmonic case that ∆u1 = ∆u2 which implies ∆(u1 − u2) = 0. That is to say, u1 − u2 is harmonic on DM with (u1 − u2) ∂DM = 0. By another application of the harmonic case, we obtain u1 = u2.
Theorem 3.3.3. Suppose the Haar series of f1, f2 on L is given by

M
f1 = cΨ(f1)Ψ +

cw(f1)ψw

m=0 w =m

M
f2 = cΨ(f2)Ψ +

cw(f2)ψw.

m=0 w =m

50

(3.19) (3.20)

Then the unique biharmonic function satisfying

is given by

∆2u = 0 ∆u L = f2 ∆u(v1) = c′
u L = f1 u(v1) = c

u(x) = (c′ − cΨ(f2))h3(x) + cΨ(f1)Ψ + (a − cψ(f1))h0(x) + cΨ(f2)h2

(3.21)

M
+

M
cw(f1)h1w(x) +

cw(f2)h2w(x)

m=0 w =m

m=0 w =m

The (⋅, ⋅)-norm of u, in the case c′ = 0, cΨ(f2) = 0, is given by

M
(u, u) = cΨ(f1) 2 +

cw(f1) 2 + (c − cψ(f1))2E0

m=0 w =m

M
+ E1 m=0 w =m

10 3

mM
cw(f1) 2 + E2 m=0 w =m

2 15

m
cw(f2) 2+

M
+ L1 m=0 w =m

2 3

m
cw(f2) 2

(3.22)

where E0 = E (h0), E1 = E (h1ψ), E2 = E (h2ψ), L1 = (h2ψ, h2ψ)4. Corollary 3.3.4. By using the notation in Theorem 3.3.3 we have

(u, u)2 ≤ (u, u)3 + 2 cΨ(f1) .

51

Proof. By Theorem 3.3.3 we have

M
(u, u)2 = cΨ(f1) 2 +

cw(f1) 2

m=0 w =m

M
(u, u)3 = (c − cψ(f1))2E0 + E1 m=0 w =m

10 3

m
cw(f1) 2

M
+ E2 m=0 w =m

2 15

m
cw(f2) 2.

Since,

E0

=

7 3

>

1,

it

follows

that

(u, u)2 ≤ (u, u)3 + 2cΨ(f1).

(3.23)

Hence, the result.

Remark 3.3.5. In this remark, we explain how we consider the trace to the boundary, as in ∆u L = f2, when f ∈ Hs2 for s2 < 0. We know that the elements of Hs2 are distributions on the unit interval lying in the dual of H−s2. So, we denote the pairing between two spaces by < ⋅, ⋅ >. A natural interpretation is the following:

Consider the Haar series expansion of f2, which does not really converge to a function so we cut it at M and put

M
SM f2 = cΨ(f2)Ψ +

cw(f2)ψw.

m=0 w =m

We will denote convergence in Hs2 norm by ⇉. Using this notation, we have that

SM f2 ⇉ f2.

We define the following function, uM ∶ SG → R,

M
uM = C + Ah3 + Dh2 + Bh0 +

M
cM1,wh1w(x) +

cM2,w h2w (x).

m=0 w =m

m=0 w =m

(3.24)

We choose the coefficients A, B, C, D, c1,w, c2,w in such a way that uM and u have the

same values and Laplacian values on ∂DM (Definition 3.3.1). By Lemma 3.3.2, there is

52

a unique biharmonic function with prescribed values and Laplacian values on ∂DM so it follows that, restricted to DM , u and uM are the same.
At this point we make the important observation that actually cM1,w = cM1,w+1 and cM2,w = cM2,w+1 for w ≤ M . Namely, as we go to higher levels we do not “overwrite” the existing coefficients. For instance, in the simplest case, if we apply this procedure to the BVP in Theorem 3.3.3, we simply get:

C = cΨ(f1) D = cΨ(f2) A = c′ − cΨ(f2) B = a − cψ(f1) cM1,w = cw(f1) cM2,w = cw(f2).

This happens because h1w and h2w , by definition, do not have support on ∂DM for w ≥ M + 1. That is to say, the coefficients in uM is fully determined by the basis functions which has support on ∂DM , namely by Ψ, h0, h3, h1w and h2w for w ≤ M .
Now we are ready to define. We put
∆u L = f2 iff ∆uM L = SM f2.

Observe that with this definition ∆uM L ⇉ f2

(3.25)

53

We observe that (3.25) implies that

< ∆uM L, g >⇉< f2, g >

as M → ∞, where g ∈ H−s2. So that, ∆u L becomes an element in the dual of H−s2.

Theorem 3.3.6. A biharmonic function u on SG has

(u, u) = (u L)2dx + E(u, u) + (∆u)2dµ < ∞ SG

if and only if u L

= f1 is in Hs1(L) for s1

=

log

10 3

log 4

>

0 and ∆u L

=

f2 is in Hs2(L)

for s2

=

log

2 3

log 4

< 0.

In this case, (3.19), (3.21) and (3.22) hold for M

= ∞ and (3.20)

converges in the Hs2 norm to f2.

Proof. We first show the “if” part. As M → ∞, the contribution to the energy comes from h1w and h2w. If we put E(h1ψ, h1ψ) = E1 we obtain

E (h1w, h1w) =

10 m 3 E1.

So, for finite M the total contribution of energy coming from h1w’s is

m=M
E1
m=0 w =m

10 3

m
cw(f1) 2.

Therefore, the Hs1-norm is finite as M

→ ∞ when s1

=

log

10 3

log 4

> 0.

Similarly, putting E(h2ψ, h2ψ) = E2 we obtain

E (h2w, h2w) =

2m 15 E2.

In this case, the total contribution coming from h2w’s is

m=M
E2
m=0 w =m

2 15

m
cw(f2) 2

54

in

which

case

H s2 -norm

stays

finite

as

M

→∞

for

s2

=

log

2 15

log 4

< 0.

We also investigate the contribution to the total norm coming from h2w’s as (h2w, h2w)4 = (∆h2w)2dµ.

We put

(∆h2ψ)2dµ = L1.

Therefore, the total contribution ends up being

m=M
L1
m=0 w =m

2 3

m
cw(f2) 2.

In this case, Hs2-norm stays finite as M

→ ∞ for s2 =

log

2 3

log 4

< 0.

Since,

log

2 15

log 4

<

log

2 3

log 4

<

0, we pick the larger one because of the containment for Hs-spaces.

Now, we turn to the “only if” part. Let u be a biharmonic function on SG, first we suppose that c′ = cψ(f2) = 0 and then derive the general case as a Corollary. We define uM as exactly in (3.24). So that our observations, after (3.24), in Remark 3.3.5 hold.
We note that (uM , uM ) is equal to (3.22). We want to show that (u − uM , u − uM ) → 0 as M → ∞. We can find constants C1 C2 and C3 such that E(h1ψ) = C1E (h1ψ D1), E (h2ψ) = C2E (h2ψ D1), (h2ψ, h2ψ)4 = C3(h2ψ D1, h2ψ D1)4. By scaling and orthogonality we can use this to conclude the existance of a constant C > 1 such that
(uM , uM )3 ≤ C(uM DM , uM DM )3 (uM , uM )4 ≤ C(uM DM , uM DM )4.

55

By the above observations we have u DM = uM DM so we can write

(uM , uM )3 ≤ C(uM DM , uM DM )3 = C(u DM , u DM )3 ≤ C(u, u) (uM , uM )4 ≤ C(uM DM , uM DM )4 = C(u DM , u DM )4 ≤ C(u, u)

(3.26)

Therefore, by (3.22), it follows that (uM , uM )3 and (uM , uM )3 are bounded and increasing.

Now we want to show that (u − uM , u − uM )3 → 0 as M → ∞. Observe that

(u − uM , u − uM )3 = ((u − uM ) DM , (u − uM ) DM )3 + ((u − uM ) ,DMc (u − uM ) DMc )3 Since (u − uM ) DM = 0, we get ((u − uM ) DM , (u − uM ) DM )3 = 0. We need to show
((u − uM ) DMc , (u − uM ) DMc )3 → 0 as M → ∞. By triangle inequality we have

((u − uM ) DMc , (u − uM ) )DMc 3 ≤ (u DMc , u )DMc 3 + (uM DMc , uM DMc )3.

(3.27)

As (u, u)3 < ∞ by assumption, we readily obtain (u DMc , u )DMc 3 → 0 as M → ∞.

By using the notation in Definition 3.3.1, observe that

∞

(uM DMc , uM )DMc 3 =

(uM Vn , uM Vn )3.

n=M

We define the following double sequence
N
SNk = (uk Ve , uk Ve )3 e=1
which is, by (3.26), uniformly bounded and increasing in both k and N . We have
M
SNk − SMk = (uk Ve , uk Ve )3.
e=N

56

Taking k = N and letting M = ∞ yields

∞

SNN

−

Sk ∞

=

(uk Ve , uk Ve )3 = (uM DMc , uM DMc )3.

e=N

By the fact that the sequence SNk is monotone and uniformly bounded both

limN →∞

SNN

and

limN →∞

SN ∞

exists

and

equal.

Therefore

we

obtain

lim (uM
M →∞

DMc , uM

)DMc 3

=

0.

By (3.27), we have (u − uM , u − uM )3 → 0 as M → ∞. A similar argument shows that (u − uM , u − uM )4 → 0 as M → ∞.

We have that (3.21) and (3.22) hold for uM . Also, in our case (3.23) reads
(uM , uM )2 ≤ (uM , uM )3 + 2C. That is to say, by (3.26), (uM , uM )2 is also uniformly bounded. We obtain (u − uM , u − uM )2 → 0 as M → ∞.

In conclusion, we get (u − um, u − um) → 0 as M → ∞. It follows that we can take the limit as M → ∞ in (3.21) and (3.22) and this also implies the convergence of uM L to f1 in Hs1 and ∆uM L to f2 in Hs2.

In order to get the general case, as above, we solve the case c′ = cψ(f2) = 0 and then simply replace u by u + (c′ − cψ(f2))h3 + cψ(f2)h2. Observe that adding
on a constant does not change the regularity of f2.

Theorem 3.3.7. Let u be a function in SG with (u, u) < ∞. Then, u has boundary

values with u L

= f1 in Hs1(L) for s1

=

log

10 3

log 4

>

0 and ∆u L

= f2 in Hs2(L) for

s2

=

log

2 3

log 4

< 0.

Proof. We proceed as in the proof of Theorem 3.3.6. We form the functions uM in a similar way this time being equal to u on ∂DM . We have uM (v1) = u(v1)
57

and all the other norms (uM , uM )i for i = 2, 3, 4 are bounded by (u, u). We take the limit as M → ∞ to obtain a biharmonic function h with h L = f1 = u L and ∆h L = f2 = ∆u L. Hence, the result.
58

CHAPTER 4 GAUSSIAN FREE FIELDS ON HIGH DIMENSIONAL SIERPINSKI
CARPET GRAPHS
This Chapter is mostly drawn from [CU] which is joint work with Joe P. Chen.
4.1 High Dimensional Sierpinski Carpet Graphs

Construction of the fractal

Let F0 ∶= [0, 1]d be the unit cube in Rd, d ≥ 2, and fix an F ∈ N, F ≥ 3. For N ∈ Z,

let QN be the collection of closed cubes of side

−N F

with

vertices

in

−N F

Zd.

For

A ⊂ Rd, let QN (A) = {Q ∈ QN ∶ int(Q) ∩ A ≠ ∅}. Denote by ΨQ the orientation-

preserving affine map which maps F0 to Q ∈ QN .

We now introduce a decreasing sequence (FN )N of closed subsets of F0 as

follows. Fix mF ∈ N, 1 ≤ mF <

d F

,

and

let

F1

be

the

union

of

mF

distinct

elements

of Q1(F0). Then by induction we put

FN+1 = ⋃ ΨQ(F1) = ⋃ ΨQ(FN ) , N ≥ 1.

Q∈QN (FN )

Q∈Q1(F1)

It is a standard argument to show that F = ⋂∞N=0 FN is the unique fixed point

of the iterated function system of contractions {ΨQ}Q∈Q1(F1). Moreover, F has

Hausdorff dimension dh(F ) = log mF log F .

We say that F is a generalized Sierpinski carpet (GSC) if and only if F1 satisfies the following four conditions:

1. (Symmetry) F1 is preserved under the isometries of the unit cube. 59

2. (Connectedness) F1 is connected.
3. (Non-diagonality) Let m ≥ 1 and B ⊂ F0 be a cube of side length 2 −Fm, which is the union of 2d distinct elements of Qm. Then if int(F1 ∩ B) is non-empty, it is connected.
4. (Borders included) F1 contains the segment {(x1, 0, ⋯, 0) ∈ Rd ∶ x1 ∈ [0, 1]}.

For alternative ways of stating the non-diagonality condition 3, see [KajinoND]. Throughout the article, we shall refer to F and mF as, respectively, the length scale factor and the mass scale factor of the carpet F .

The stochastic analysis on the Sierpinski carpet is built upon the measure

space (F, ν), where ν is the self-similar Borel probability measure which assigns

mass m−FN to each ΨQ(F ), Q ∈ QN (FN ). Note that ν is a constant multiple of the dh(F )-dimensional Hausdorff measure on F . We will also consider the un-

bounded carpet F∞ ∶= ⋃∞N=0

N F

F

,

and

let

ν∞

be

the

σ-finite

self-similar

Borel

prob-

ability measure on F∞, assigning mass mNF to

N F

F

.

We introduce two other important scale factors associated with Sierpinski
carpets. Let DN be the network of diagonal crosswires obtained by connecting each vertex of a cube Q ∈ QN to the vertex at the center of the cube via a wire of unit resistance. Denote by RDN the resistance across two opposite faces of DN . It was shown in [BB90Resistance, McGillivray] that there exist ρF ∈ (0, ∞) and positive constants C(d) and C′(d) such that

CρNF ≤ RDN ≤ C′ρNF .

The constant ρF is henceforth referred to as the resistance scale factor of the carpet F . As of this writing, there’s no known exact formula for ρF : the best estimate,

60

obtained via a resistance shorting and a cutting argument, is [BB99]*Proposition

5.1

2 F

mF ≤ ρF ≤ 21−d

F.

Next, let tF = mF ρF , which stands for the time scale factor of the carpet F . The

significance of tF is due to the fact that the expected time for a d-dimensional

Brownian motion to traverse from one face of

N F

FN

to

the

opposite

face

scales

with tNF .

It is often convenient to introduce, respectively, the Hausdorff, walk, and spectral dimensions of F :

dh(F )

=

log mF log F

,

dw(F )

=

log tF log F

,

ds(F

)

=

2

log mF log tF

.

Under the strict inequality mF <

d F

,

one

has

1

≤

ds(F

)

<

dh(F

)

<

d

and

dw(F

)

>

2.

The latter inequality implies that diffusion on F (resp. F∞) is sub-Gaussian, in

contrast with Gaussian diffusion which has walk dimension 2.

For each generalized Sierpinski carpet F , we consider two associated graphs. See Figure 1.2.

Let VN =

N F

FN

∩

Zd.

Introduce the graph GN

=

(VN , ∼), where throughout

the paper, the edge relation ”∼” means that two vertices x, x′ are connected by

an edge if and only if their Euclidean distance x − x′ = 1. Put G∞ = ⋃N∈N GN , which we call the outer Sierpinski carpet graph. Observe that G∞ is a subgraph

of (Z+)d. In this paper we will study the Gaussian free field on G∞.

Next, let IN =

N F

FN

∩

Zd +

1 2

,

1 2

,

⋯,

1 2

. Introduce the graph IN = (IN , ∼).

Put I∞ = ⋃N∈N IN , which we call the inner Sierpinski carpet graph. It is easy to

see that IN = mNF , and that there exist constants C1.1 and C1.2, independent of

61

N , such that C1.1mNF ≤ VN ≤ C1.2mNF .
For easy reference, we provide in Table 1.1 a side-by-side comparison of the relevant parameters on Zd and on the Sierpinski carpet graph (G∞ or I∞). It is known that simple random walk on the latter is transient if and only if ρF < 1 [BBSCGraph, McGillivray].

4.1.1 Main results

In what follows, F is a transient generalized Sierpinski carpet, with ρF < 1 (equivalently, ds(F ) > 2). This includes any generalized Sierpinski carpet whose cross-section contains a full copy of the 2-plane [0, 1]2 (cf. [BB99]*§9), as well as other, but not all, d-dimensional (d ≥ 3) carpets, such as the Menger sponge. Our analysis does not apply to any generalized Sierpinski carpet in R2, whereby ρF > 1 and is hence recurrent.
Let GG∞ ∶ V∞ × V∞ → R be the Green’s function for simple random walk on the outer Sierpinski carpet graph G∞ without killing. We denote G ∶= supx∈V∞ GG∞(x, x) and G ∶= infx∈V∞ GG∞(x, x). Since simple random walk on G∞ is transient, both G and G are positive and finite.

We also need the notion of the (0-order) capacity of the compact carpet F with respect to a Dirichlet form (E, F ) ∈ E on L2(F∞, ν∞), given by

Cap E

(F

)

∶=

inf{E(f, f )

∶

f

∈

F

∩

Cc(F∞), f

≥

1

a.e.

on

F },

See Proposition 4.3.2 for a more general definition of the capacity, as well as some important properties.

62

Let P be the law of the Gaussian free field on G∞ with covariance GG∞, and let Ω+VN denote the entropic repulsion event {ϕx ≥ 0 for all x ∈ VN }. Our first main result identifies the rate of exponential decay for P(Ω+VN ).

Theorem 4.1.1. There exists a point x0 ∈ V∞ such that for any Dirichlet form (E, F) ∈

E, there are positive constants C1.3(E) and C1.4(E) such that

−C1.3

⋅

G

⋅

Cap E

(F

)

≤

lim
N →∞

log ρ−FN

P(Ω+VN ) log(tNF )

(4.1)

≤

lim
N →∞

log P(Ω+VN ) ρ−FN log(tNF )

≤

−C1.4

⋅

GG∞(x0, x0)

⋅

Cap E

(F

).

The constants C1.3 and C1.4 are attributed to two sources: one coming from comparing the Dirichlet forms on G∞ and on I∞ (Lemma 4.2.4), and the other coming from comparing the (maximal or minimal) cluster point of the sequence of renormalized Dirichlet forms on I∞ with an element of E (Theorem 4.2.3). Due to the lack of precise control of the constants involved in these comparisons,
the authors deem it not possible to determine whether C1.3 equals C1.4.

Notwithstanding the small discrepancy between the lower and upper

bounds in Theorem 4.1.1, we are still able to give a precise description of en-

tropic

repulsion

on

G∞.

We

shall

prove

that

conditional

on

Ω+VN

,

the

local

sample √

mean of the free field on VN is pushed to a height which is proportional to N ,

and as N → ∞, the rescaled height converges in probability to a constant.

Theorem 4.1.2. For any > 0 and η > 0,

⎛

lim
N →∞

sup
x∈VN

P⎝

VN, (x)⊂VN

ϕ¯N, (x) log(tNF )

−

2G ≥ η

Ω+VN

⎞ ⎠

=

0,

(4.2)

where ϕ¯N, (x) ∶=

1 VN, (x)

ϕz and VN, (x) ∶=
z∈VN, (x)

z ∈ VN

∶ max 1≤i≤d

zi − [

N F

xi]

≤

⋅

N F

.

63

We comment that in the case of Zd, which has full translational invariance, one can replace GZd(0, 0) by GZd(x, x) for any x ∈ Zd. On the other hand, in the

case of the Sierpinski carpet graph, we have no explicit information about how

the our

on-diagonal Green’s function √
result says that 2G log tF N

GG∞(x, x) varies with , where G = infx∈V∞ GG∞

x ∈ V∞. Nevertheless, (x, x), sets the leading-

order asymptotic height for the free field above the hard wall on VN . A sketch

of the arguments leading to this result will appear at the beginning of §4.5.2.

The rest of the Chapter is organized as follows. In Section 4.2 we recapitulate the construction of local regular Dirichlet forms on Sierpinski carpets via graphical approximations (the Kusuoka-Zhou construction), and prove the convergence of the discrete Green forms, both on I∞ and on G∞, to a continuum Green form on F∞. We then proceed to prove Theorem 4.1.1 in Section 4.4, and Theorem 4.1.2 in Section 4.5.

4.2 Dirichlet Forms
In this section we provide the necessary potential theoretic lemmata to prove Theorems 4.1.1 and 4.1.2. Our main results are Theorem 4.2.3 and Lemma 4.2.5; only Lemma 4.2.5 will be used in subsequent sections.
Notations. If (X, m) denotes a measure space, then ⟨f, µ⟩X stands for ∫X f dµ, pairing a function f on X with a Borel measure µ.

64

4.2.1 Kusuoka-Zhou construction of Dirichlet forms

Let F be a generalized Sierpinski carpet, and I∞ = (I∞, ∼) be the inner Sierpinski

carpet graph introduced in §4.1. For each N ∈ N and each w ∈ I∞, let Ψw(N) be

the closed cube of side

−N F

centered

at

−N F

w.

We

define

the

mean-value

operator

P˜N ∶ L1(F∞, ν∞) → C(I∞; R) by

(P˜N f )(w) = ν∞

1 Ψw(N) ∩ F∞

Ψ(wN)∩F∞ f (y)ν∞(dy),

Similarly, if µ∞ is a Radon measure on F∞ such that µ∞ ≪ ν∞, then define

P˜N µ∞ =

P˜N

dµ∞ dν∞

νN ,

where

νN

=

1 mNF

1I∞

is

a

self-similar

measure

on

I∞.

As is customary, we define the discrete Dirichlet form on the graph I∞ by

EI∞(f1, f2)

=

1 2

(f1(w)
w,w′∈I∞

−

f1(w′))(f2(w)

−

f2(w′))

w∼w′

for all f1, f2 in the natural domain D(EI∞) = {f ∈ 2(I∞) ∶ EI∞(f, f ) < ∞}. Furthermore, let ENI = ρNF EI∞ be the renormalized Dirichlet form, where ρF ∈ (0, ∞)

is the resistance scale factor identified in §4.1.

Let F0 ∶= f ∈ L2(F∞, ν∞) ∶ sup ENI (P˜N f, P˜N f ) < ∞ . The following N
convergence result for (ENI )N is originally due to Kusuoka and Zhou
[KusuokaZhou]*Proposition 5.2 & Theorem 5.4, and later generalized in
[HKKZ]*Lemma 4.1 & Theorem 4.3.

Proposition 4.2.1. all f ∈ F0,

1. There exists a constant C2.1 such that for all N, M ≥ 1 and ENI (P˜N f, P˜N f ) ≤ C2.1ENI +M (P˜N+M f, P˜N+M f ).

2. There exists (E, F0) ∈ E and positive constants C2.2 and C2.3 such that for all

65

f ∈ F0, C2.2 sup ENI (P˜N f, P˜N f ) ≤ E (f, f ) ≤ C2.3 lim ENI (P˜N f, P˜N f ).
N N →∞

(4.3)

Remark 4.2.2. In their original work [KusuokaZhou], Kusuoka and Zhou identified a family of Dirichlet forms, denoted Dch, which are associated with cluster points of the sequence of suitably rescaled Markov processes on IN . Then they proved (4.3) for any E ∈ Dch, and showed that (E, F0) is a local regular Dirichlet form. Note that Dch ⊂ E
by virtue of [BBKT]*Theorem 3.2.

4.2.2 Convergence of discrete Green forms

In this subsection we shall consider Dirichlet forms on a class of smooth measures (instead of functions), and derive a convergence result similar to Proposition 4.2.1. From now on let M+(F ) be the family of all nonnegative finite Borel measures on F , and let

M(00,a)c(F ) =

µ ∈ M+(F ) ∶ µ ≪ ν,

dµ dν

∈ F0

.

Let GIN ∶ V∞ × V∞ → R be the Green’s function for simple random walk on I∞ killed upon exiting IN . By the reproducing property of Green’s function, EI∞(GIN (w, ⋅), h) = h(w) for all h ∈ D(EI∞) with supp(h) ⊂ IN . Therefore, denoting by UNI the 0-order potential operator associated with ENI , we have

ENI (UNI µ, h) = ⟨h, µ⟩IN

=

1 mNF

w∈IN

h(w) dµ dνN

(w)

=

ENI

ρ−FN

1 mNF

w∈IN

GIN

(⋅,

w)

dµ dνN

(w), h

66

for all h ∈ D(EI∞) with supp(h) ⊂ IN , and all nonnegative measures µ with support in IN . It follows that

ENI (UNI µ, UNI µ) =

= ENI

ρ−FN

1 mNF

∑w∈IN

GIN

(⋅,

w)

dµ dνN

(w), ρ−FN

1 mNF

∑w′∈IN

GIN

(⋅,

w′

)

dµ dνN

(w′)

=

ρ−FN

1 m2FN

∑w,w′∈IN

GIN

(w

,

w′

)

dµ dνN

(w)

dµ dνN

(w′

)

(4.4)

for all such measures µ. The expression in (4.4) is what we shall call the Green

form corresponding to the Dirichlet form ENI . It has a kernel given by the (renor-

malized) Green’s function ρ−FN GIN , whence the name.

The following result proved in [CU] describes the convergence of the discrete

Green forms.

Theorem 4.2.3. [CU] There exist (E, F ) ∈ E and constants C2.4(E), C2.5(E) such that

C2.4E (U µ, U µ) ≤ lim ENI UNI P˜N µ, UNI P˜N µ N →∞

≤

lim
N →∞

ENI

UNI P˜N µ, UNI P˜N µ

≤ C2.5E(U µ, U µ)

(4.5)

for all µ ∈ M(00,a)c(F ), where U is the 0-order potential operator associated with E.

4.2.3 Comparison of discrete Dirichlet & Green forms
Recall that we are considering the free field on the outer Sierpinski carpet graph G∞, while the convergence from discrete Dirichlet forms to the continuum one is based on the inner Sierpinski carpet graph I∞. To bridge this gap, we shall compare the discrete Dirichlet (and Green) forms on G∞ and on I∞ in this subsection.
Observe (from Figure 1.2) that for each ”center vertex” w ∈ I∞, there is a unique set C(w) of 2d ”corner vertices” in V∞ which are nearest neighbors of
67

w, i.e., C(w) = {x ∈ V∞ ∶

x−w

=

√ d

2}.

Let Q˜

∶

C(V∞; R)

→

C(I∞; R) be the

projection operator given by

(Q˜f )(w)

=

1 2d

x∈C(w)

f (x).

As is customary, we introduce the discrete Dirichlet form on graph G∞ by

EG∞(f1, f2)

=

1 2

(f1(x) −
x,x′∈V∞

f1(x′))(f2(x)

−

f2(x′))

x∼x′

for all f1, f2 in the natural domain D(EG∞). Let GGN ∶ V∞ × V∞ → R denote the

Green’s function killed upon exiting GN .

The following Lemma from [CU] gives the required comparison of the Discrete Dirichlet forms.

Lemma 4.2.4. [CU] For all f ∈ D(EG∞), EG∞(f, f ) ≥ EI∞ Q˜f, Q˜f .

(4.6)

It follows that for all nonnegative functions f on VN ,

GGN (x, x′)f (x)f (x′) ≤ 22d

GIN (w, w′)(Q˜f )(w)(Q˜f )(w′).

x,x′∈VN

w,w′∈IN

(4.7)

4.2.4 The main lemma

In this subsection, we establish the limsup convergence of discrete Green forms on G∞, which will play a crucial role in the main proofs.

Let

G GN

∶

VN

× VN

→

R

be

the

restriction

of

GG∞

on

VN

× VN ;

we

have

added

a superscript to distinguish it from the Green’s function on G∞ killed upon

68

exiting GN . Also introduce the probability measure ηN

∶=

1 VN

1VN

on VN . Define,

for any h ∈ 1(VN ; R) ∩ ∞(VN ; R),

UNG (hηN ) ∶= ρ−FN

1 VN

GGN (⋅, x)h(x),
x∈VN

UNG

(hηN ) ∶= ρ−FN

1 VN

G GN

(⋅,

x)h(x).

x∈VN

Writing ENG = ρNF EG∞ for the renormalized discrete Dirichlet form on G∞, we have

ENG

UNG (hηN ) , UNG (hηN )

= ρ−FN

1 VN

2

GGN (x, x′)h(x)h(x′)
x,x′∈VN

by the reproducing property of GGN . Meanwhile, let us abuse notations slightly

and introduce the quadratic form

ENG (UNG

(hηN ), UNG

(hηN )) ∶= ρ−FN

1 VN

2

x,x′∈VN

G GN

(x,

x′)h(x)h(x′

),

as it is suggestive of another Green form.

Lemma 4.2.5 (The main lemma). [CU] For every h ∈ L1(F, ν) ∩ L∞(F, ν), define

hN

∶

VN

→

R

by

hN (⋅)

=

h(

−N F

⋅).

Then

the

following

hold:

1.

lim
N →∞

ENG

(UNG

(hN ηN ), UNG

(hN

ηN

))

=

lim
N →∞

ENG

(UNG

(hN

ηN

),

UNG

(hN

ηN

)).

For some (E, F) ∈ E:

2. There exists a constant C2.6(E) such that

lim
N →∞

ENG

UNG

for all µ ∈ M(00,a)c(F ).

dµ

dν

ηN
N

, UNG

dµ

dν

ηN
N

3. There exists a constant C2.7(E) such that

≤ C2.6E(U µ, U µ)

lim
N →∞

ρNF

⟨1VN , (GGN )−1(⋅, x)1VN (x)⟩

x∈VN

VN

≤

C2.7CapE (F ),

where

G GN

−1

denotes

the

matrix

inverse

of

G, GN

and

Cap E

(F

)

denotes

the

0-

capacity of F with respect to E.

69

4.3 Different Characterizations of Capacity

In this subsection we introduce the concepts of smooth measures and (0-order) capacity with respect to a (transient) regular Dirichlet form. Much of this can be found in [FOT]*Chapter 2 and [ChenFukushima]*Chapter 2.

Suppose (E, F) is a regular Dirichlet form on L2(X, m). Let O denote the

family of all open subsets of X, and for each A ∈ O, define LA = {u ∈ F ∶ u ≥

1 m-a.e. on A}. The 1-capacity of the set A ∈ O with respect to E is given by

Cap E

,1(A)

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

inf f ∈LA ∞,

E(f, f ) +

f

2 L2

,

LA ≠ ∅ LA = ∅

.

(4.8)

If A ⊂ X

is

an

arbitrary

subset,

then

put

Cap E

,1(A)

=

inf B∈O,A⊂B

Cap E

,1

(B

).

A

statement is said to hold quasi-everywhere (q.e.) on A if and only if there exists a

set

U

⊂

A

with

Cap E

,1

(U

)

=

0

such

that

the

statement

holds

everywhere

on

A

U.

A function f ∶ X → R is said to be quasi-continuous if for every > 0, there exists

an

open

set

Ω

with

Cap E

,1(Ω)

<

such that f is continuous on X Ω. We say that

v is a quasi-continuous modification of f if v is quasi-continuous and v = f m-a.e,

and denote v by f .

A positive Radon measure µ on X is called a measure of finite energy integral (with respect to E) if there exists a constant Cµ > 0 such that for all f ∈ F ∩ Cc(X),

X

f dµ ≤ Cµ

E(f, f ) +

f

2 L2

1 2.

(4.9)

We denote by S0 the family of all measures of finite energy integral.

If furthermore (E, F) is transient, then one may complete F in the E-norm, and (Fe ∶= F E , E) is a Hilbert space called the extended Dirichlet space. Then we have the following 0-order counterparts of the above notions: the 0-capacity of

70

a

set

A

∈

O,

denoted

by

Cap E

(A),

is

given

by

(4.8)

with

F

and

E(f, f )

+

f

2 L2

replaced respectively by Fe and E(f, f ). The 0-capacity of an arbitrary set A

then follows similarly. Likewise, a positive Radon measure µ on X is called a

measure of finite 0-order energy integral if (4.9) holds with the same replacements. Denote by S0(0) the family of all measures of finite 0-order energy integral.

There is an important connection between S0(0) and Fe, which is based on the Riesz representation theorem. For every µ ∈ S0(0), there exists a unique U µ ∈ Fe such that E(f, U µ) = ⟨f , µ⟩X for all f ∈ Fe. We shall refer to U ∶ S0(0) → Fe as the 0-order potential operator associated with E. Any h ∈ Fe which can be written in the form h = U µ for some µ ∈ S0(0) is called a 0-order potential relative to E.

Let us remark that S0(0) ⊂ S0 ⊂ S, where S is the family of smooth measures consisting of all positive Borel measures µ on X such that:

• µ charges no set of zero 1-capacity.

• There exists an increasing sequence (Fn)n of closed sets such that µ(Fn) <

∞

for

all

n,

and

that

limn→∞

Cap E

,1(K

Fn) = 0 for any compact set K.

In general, elements of S0 need not be absolutely continuous with respect to m, but each of them can be approximated by a sequence of absolutely continuous measures, cf. [FOT]*Lemma 2.2.2. Here we give the 0-order version of this statement.
Proposition 4.3.1. Let (E, F) be a transient regular Dirichlet form on L2(X, m), and let Gβ and U denote respectively the β-resolvent and the 0-order potential operator associated with E. Given each µ ∈ S0(0), let hβ ∶= β(U µ − βGβ(U µ)) for each β ∈ N. Then as β → ∞, hβ ⋅ m converges vaguely to µ.

71

Proof. This is the Yosida approximation (cf. [FOT]*(1.3.18)): hβ ≥ 0 m-a.e., and for all f ∈ F,

(hβ ,

f )L2(m)

=

(β(U µ

−

βGβ(U µ)),

f

)L2(m)

→
β→∞

E(U µ,

f ).

Therefore limβ→∞⟨f, hβ ⋅ m⟩X = ⟨f, µ⟩X for all f ∈ F ∩ Cc(X).

Last but not least, let us record several equivalent characterizations of the 0-capacity.
Proposition 4.3.2. Let (E, F) be a transient regular Dirichlet form on L2(X, m). Fix an arbitrary set B ⊂ X and suppose LB ≠ ∅.

1. There exists a unique element eB in LB minimizing E(⋅, ⋅). In particular,

Cap E

(B)

=

E

(eB ,

eB ).

2. eB is the unique element of Fe satisfying eB = 1 q.e. on B and E(eB, f ) ≥ 0 for any f ∈ Fe with f ≥ 0 q.e. on B.

3. There exists a unique measure µB ∈ S0(0) supported in B such that eB = U µB. In particular,

Cap E

(B)

=

E

(U

µB ,

U

µB )

=

⟨U

µB ,

µB⟩X .

4. If B is a compact set, then

Cap E

(B)

=

⟨1B, µB⟩X

=

sup E(U µ, U µ) ∶ µ ∈ S0(0), supp(µ) ⊂ B, U µ ≤ 1 q.e.

=

sup

⟨1B, µ⟩2X
E(U µ, U µ)

∶

µ

∈

S0(0),

supp(µ)

⊂

B

.

Proof. The first two items are the 0-order version of [FOT]*Theorem 2.1.5, as explained on [FOT]*p. 74. Item (iii) is proved in conjunction with [FOT]*Lemma 2.2.10. The first two equalities in Item (iv) follow directly from (ii) and (iii), while the third equality can be obtained by a variational argument.
72

The function eB and the measure µB are known as, respectively, the 0-order equilibrium potential and equilibrium measure of the set B (with respect to E).

4.4 Proof of Theorem 4.1.1

Notations. In the next two sections, Φ ∶ R → [0, 1], defined by

Φ(a) = √1

a e−ξ2 2dξ,

2π −∞

stands for the cdf of a standard normal random variable. For any measurable subset S of V∞, we denote by FS ∶= σ{ϕx ∶ x ∈ S} the sigma-algebra generated by the free field on S, and by Ω+S ∶= {ϕx ≥ 0 for all x ∈ S} the event that the field is nonnegative everywhere on S. Finally, we fix an element (E, F) from the family E of local, regular, conservative, non-zero Dirichlet forms on L2(F∞, ν∞) which
are invariant under the local symmetries of the carpet.

4.4.1 Lower bound

Let α > 2G, on G∞ with

where G √

∶=

supx√∈V∞

mean α log tF N

GG∞(x, x). Denote by and covariance GG∞.

PN

the

law

of

the

free

field

First we wish to show that limN→∞ PN (Ω+VN ) = 1. Observe that for any x ∈ V∞,

PN (ϕx < 0) = P(ϕx < − αN log tF ) = Φ ⎛⎝−

αN log GG∞ (x,

tF x)

⎞ ⎠

,

Using

the

fact

that

GG∞(x, x)

≤

G

and

Φ(a)

≤

1 2

e−a2

2

for

a

≤

0,

we

deduce

that

PN (ϕx

<

0)

≤

1 2

t−F(N

α)

(2G).

73

It follows that

PN

(Ω+VN )c

= PN

⋃ {ϕx < 0}
x∈VN

≤

VN

PN

(ϕx

<

0

∶

x

∈

VN )

≤

c(tNF

)(1− α ) 2G

→
N →∞

0,

which is what we want.

Next we adopt the relative entropy argument as used in the proof of

[BDZ95]*Lemma 2.3. Let ΠN

=

dPN dP

. Introduce the relative entropy of PN
FVN

to

P restricted to VN by

EntVN (PN P) =

RV∞

ΠN

log(ΠN )dP

=

1 αN
2

log tF

⟨1VN , (GGN )−1(⋅, x)1VN (x)⟩

x∈VN

VN

,

where

(G )−1 GN

denotes

the

matrix

inverse

of

(G ) GN

∶=

GG∞

.VN ×VN

Applying

the

entropy inequality

log

P(Ω+VN ) PN (Ω+VN )

≥

−

PN

1 (Ω+VN

)

EntVN (PN P) + e−1

,

cf. the end of the proof of [BDZ95]*Lemma 2.3, we obtain

lim
N →∞

log ρ−FN

P(Ω+VN ) N log tF

(4.10)

≥

lim
N →∞

−

PN

1 (Ω+VN

)

EntVN (PN P) + e−1 ρ−FN N log tF

+

log PN (Ω+VN ) ρ−FN N log tF

≥

lim
N →∞

−

PN

1 (Ω+VN

)

EntVN (PN ρ−FN N log

P) tF

+ lim N →∞

−

PN

1 (Ω+VN

)

ρ−FN

e−1 N log

tF

+

lim
N →∞

log PN ρ−FN N

(Ω+VN ) log tF

≥

−

lim
N →∞

1 PN (Ω+VN

)

⋅

lim
N →∞

EntVN (PN ρ−FN N log

P) tF

+0+0 ≥

−

1 2

αC2.7CapE

(F

)

by Lemma 4.2.5(iii). By making α arbitrarily close to 2G, we obtain the desired

lower bound.

Remark 4.4.1. If we instead use a constant multiple of the original Dirichlet form

(γE, F ), γ > 0, inequality (4.10) will hold under the substitutions C2.7 → γ−1C2.7 and

Cap E

(F

)

→

Capγ

E

(F

).

74

4.4.2 Upper bound

Just as in the Zd setting [BDZ95], the proof of the upper bound involves a series of coarse graining and conditioning arguments on the free field {ϕx}x∈V∞, though some modifications are needed to account for the fractal geometry.

Notations. If G = (V (G), ∼) is a finite subgraph of a larger graph G0 = (V (G0), ∼ ), then we denote the set of peripheral vertices of G by

∂G ∶= {x ∈ V (G) ∶ x ∼ y for some y ∈ V (G0) V (G)}.

The interior of the graph G will thusly be defined by G˚∶= (V (G) ∂G, ∼).

Following §4.1, we denote by Qj(Fj) (j ∈ N) the collection of closed cubes

of side

−j F

whose

vertices

are

in

−j F

Zd

,

and

which

are

contained in

Fj .

Then

to

each Q¯ ∈ Qj(Fj) corresponds a unique vector p = (p1, ⋯, pd) ∈ (N0)d such that

Q¯ =

p1

−j F

,

(p1

+

1)

−j F

×⋯×

pd

−j F

,

(pd

+

1)

−j F

.

Keeping

with this notation,

we

define two related cubes derived from Q¯:

Q⌞ =

p1

−j F

,

(p1

+

1)

−j F

×⋯×

pd

−j F

,

(pd

+

1)

−j F

,

Q

=

Q⌞ ∪ ⎛⎝Q¯

⋃
Q¯′∈Qj (Fj

)

⎞ Q′⌞⎠

.

Let Q○j (Fj) be the totality of all Q. Observe that Fj = ⋃Q∈Q○j(Fj) Q, and that Q1 ∩ Q2 = ∅ for any Q1, Q2 ∈ Q○j (Fj) with Q1 ≠ Q2.

Next we introduce, for each k ≤ N , the following collections of kth-level subgraphs of GN :

Sk(GN ) ∶= Sk○(GN ) ∶=

N F

Q¯

∩

GN

∶

Q¯

∈

QN −k (FN −k )

,

N F

Q

∩

GN

∶

Q

∈

Q○N −k (FN −k )

.

75

By construction, there is a bijection ιk ∶ Sk○(GN ) → Q○N−k(FN−k) which maps each g ∈ Sk○(GN ) to a Q ∈ Q○N−k(FN−k).

Now let us fix a sufficiently large k ∈ N, and designate a vertex x0 ∈ Vk ∂Gk as the ”representative interior point” of Gk. We don’t insist on where x0 is located within Vk, so long as it stays away from the periphery ∂Gk. (Contrast this setup with previous works on Zd [BDZ95, Kurt], where it is natural to designate the
center vertex of each block cell as the representative interior point.). Then for any N > k, let

CN

=

{x ∈ VN ∶ x = x0 +

k F

p

for

some

p

∈

(N0)d}

and

DN

=

{x

∈

VN

∶

∃i

∈

{1, ⋯, d}

such

that

xi

=

p

k F

for

some

p

∈

N0}

be, respectively, the set of all representative interior points and kth-level boundary points in VN ; see Figure 4.1. Note that CN = mNF −k.

With the setup complete, we can proceed with the main arguments. Coarse
graining means that we are sampling the free field φ at only one vertex from each subgraph g ∈ Sk(GN ). On top of that, we will analyze these Gaussian random variables conditional upon the sigma-algebra FDN generated by the free field on the ”conditioning grid” DN . The key observation is that under P(⋅ FDN ), {ϕx ∶ x ∈ CN } are independent Gaussian random variables with mean E(ϕx FDN ) =∶ µx and identical variance GG˚k(x0, x0). It is standard to check for every x ∈ CN , µx is nonnegative on Ω+ via a random walk representation.
DN

Let us now carry out the estimate of P(Ω+VN ). First of all,

P(Ω+VN ) ≤ P

Ω+ CN

∩

Ω+ DN

=E

x∈CN P[ϕx ≥ 0 FDN ] ⋅ 1Ω+DN

,

(4.11)

where the equality comes from a basic identity for conditioned random vari-

ables and the independence of {ϕx ∶ x ∈ CN } under P(⋅ FDN ).

76

Figure 4.1: The coarse-graining and conditioning scheme on the outer Sierpinski carpet graph G∞. Vertices indicated by filled dots are the representative interior points (CN ), while vertices covered by the solid lines (the conditioning grid) are where the free field ϕ is conditioned upon (DN ).

Now take a j ∈ N and consider all N > j + k. For each B ∈ SN○ −j(GN ), let BC ∶= B∩CN ; observe that BC = mNF −j−k. Let κ > 0, and αx0,κ ∶= 2(GG∞(x0, x0)−κ) log tF . Finally, for δ ∈ (0, 1), define the event

Γx0,B ∶= ϕ ∶ {x ∈ BC ∶ µx ≤ αx0,κN } ≥ δ BC .

Set Γx0 = ⋃ Γx0,B. B∈SN○ −j (GN )

By

writing

Ω+ DN

as

the

disjoint

union

of

(Ω+ DN

∩ Γx0)

and

(Ω+ DN

∩ Γcx0),

we

develop (4.11) further as

1 1P(Ω+VN ) ≤ E

P[ϕx ≥ 0 FDN ] ⋅
x∈CN

Ω+DN ∩Γx0

+E

P[ϕx ≥ 0 FDN ] ⋅
x∈CN

Ω+DN ∩Γcx0

.

(4.12)

The claim is that the first term on the RHS of (4.12) becomes negligible as

77

N → ∞. Since this result plays an essential role in Section 4.5, we record it as a separate lemma.

Lemma 4.4.2. Let γ ∈ (0, 1). Then for k large enough, there exists a constant C3.1(δ, k), independent of N , such that

1E

P[ϕx ≥ 0 FDN ] ⋅
x∈CN

Ω+DN ∩Γx0

≤ exp −C3.1mNF tF−N(1−γ) .

(4.13)

Proof 1−γ

of Lemma.

Since

↑

lim
k→∞

GG˚k

(x0,

x0)

=

GG∞

(x0,

for k large enough. So on Γx0, there exists

x0), we at least

have a B∗

G∈ GGS∞N○G˚(k−x(j0x(,G0x,N0x))0−)suκch≤

that Γx0,B∗ holds, and therefore

δ B∗C
P[ϕx ≥ 0 FDN ] ≤ P ϕx0 − µx0 ≥ − αx0,κN FDN

x∈CN

= ⎡⎢⎢⎢⎢⎢⎣1 − Φ ⎛⎜⎝−

αx0,κN GG˚k (x0, x0)

⎞⎟⎠⎤⎥⎥⎥⎥⎥⎦δ

B∗C

≤ ⎡⎢⎢⎢⎢⎢⎣1 − ≤ ⎡⎢⎢⎢⎢⎣1 −

GG˚k (x0, x0) exp αx0,κN

−

αx0,κN 2GG˚k (x0, x0)

⎤⎥⎥⎥⎥⎥⎦δ B∗C

2(1

−

1 γ)N

log

tF

tF−N (1−γ ) ⎤⎥⎥⎥⎥⎦δ

B∗C

≤ exp −cmNF t−FN(1−γ) .

Once again, we used the fact that ϕx0 − µx0 is a centered Gaussian random vari-

able under P(⋅ FDN ), and applied a standard Gaussian estimate. The last in-

equality derives from the inequality 1 − x ≤ e−x.

We turn to estimate the second term on the RHS of (4.12). The key is to obtain a lower bound for ∑x∈BC µx (B ∈ SN○ −j(GN )) on Γcx0. We have

µx =

µx +

x∈BC

x∈BC

µx> αx0,κN

µx.
x∈BC αx0,κN ≥µx≥0

78

The second summand can be bounded below by 0. As for the first summand, observe that on Γcx0, there are at least (1 − δ) BC many representative interior points x whose µx exceeds αx0,κN . Therefore

µx ≥ (1 − δ) BC αx0,κN on Γcx0.
x∈BC

Introduce arbitrary nonnegative numbers fB ≥ 0, B ∈ SN○ −j(GN ). We then have

P(Γcx0) ≤ P

1 BC

µx ≥ (1 − δ)
x∈BC

αx0,κN

=

⎛

P

⎜ ⎝B∈SN○

−j

(GN

)

fB

1 BC

µx ≥ (1 − δ)
x∈BC

⎞ αx0,κN B∈SN○ −j (GN ) fB⎟⎠

≤

exp

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−

⎛ ⎞2

(1

−

δ)2αx0,κN

⎜ ⎝B∈SN○ −j (GN

)

fB⎟⎠

2

⎛

Var

⎜ ⎝B∈SN○

−j

(GN

)

fB

1 BC

⎞ x∈BC µx⎟⎠

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

From the elementary random variable identity Var(X) = Var(E(X F)) +

E(Var(X F)) one deduces that Var(X) ≥ Var(E(X F)). Applied to our setting

we find

⎛ ⎞⎛ ⎞

Var

⎜ ⎝B∈SN○ −j (GN

)

fB

x∈BC

µx⎟⎠

≤

Var

⎜ ⎝B∈SN○ −j (GN )

fB

x∈BC

ϕx⎟⎠

.

Let the function Ξj ∶ F → R+ be given by Ξj =

1fB ιj(B). One verifies

B∈SN○ −j (GN )

that

fB =
B∈SN○ −j (GN )

1 BC

Ξj
x∈CN

x
N F

and

fB ϕx = Ξj

B∈SN○ −j (GN )

x∈BC

x∈CN

x N ϕx.
F

79

Hence

⎛

Var

⎜ ⎝B∈SN○

−j

(GN

)

fB

1 BC

⎞ x∈BC ϕx⎟⎠ =

1 Var
BC 2

Ξj
x∈CN

=

1 BC

2

x,x′∈CN

GG∞ (x,

x′)Ξj

x N Ξj
F

x′
N F

1=

1 BC

2

x,x′∈VN

GG∞ (x,

x′)

Ξj

−N F

CN

1x

ΞN

j

−N F

CN

F

x N ϕx
F
x′ N.
F

Putting things together,

P(Γcx0 )

≤

exp

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−

2

x,x′∈VN

(1 − δ)2αx0,κN

GG∞(x, x′)

1Ξj −N F

x∈CN CN

Ξj
x
N F

x2
N F

1Ξj

−N F

CN

x′
N F

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ .

It follows that

lim
N →∞

log ρ−FN

P(Ω+VN ) N log tF

≤

lim
N →∞

log P(Γcx0) ρ−FN N log tF

≤

lim
N →∞

−(1 − δ)2(GG∞(x0, x0) − κ)

1 mNF −k x∈CN Ξj

x
N F

2

1ρ−FN
m2F(N −k)

x,x′∈VN

GG∞ (x,

x′)

Ξj

−N F

CN

≤

−

(1

−

δ

)2

(GG∞ (x0 , C12.2m2Fk

x0

)

−

κ)

x
N F

1Ξj

−N F

CN

x′
N F

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1

lim
N →∞

mNF −k x∈CN Ξj

x
N F

2

1lim
N →∞

ρ−FN VN 2

x,x′∈VN

GG∞ (x,

x′)

Ξj

−N F

CN

1x

ΞN

j

−N F

CN

F

x′
N F

≤

−

(1

−

δ

)2

(GG∞ (x0 , C12.2C2.6

x0

)

−

κ)

⟨1F , Ξjν⟩2F
E(U (Ξjν), U (Ξjν))

(4.14)

for all Ξj ∈ F. In obtaining the convergence for the denominator, we applied

Lemma 4.2.5(ii) and identified the limit measure as m−FkΞjν.

80

By varying over the coefficients fB in Ξj and taking the limit j → ∞, we can

obtain any Ξν ∈ M(00,a)c(F ). Then we can recover any µ ∈ S0(0), supp(µ) ⊂ F , by an approximating sequence of measures in M(00,a)c(F ) a` la Yosida (Proposition 4.3.1). We supremize the bracketed expression on the RHS of (4.14) over all µ ∈ S0(0) and apply Proposition 4.3.2(iv), then take δ, κ → 0 to get

lim
N →∞

log ρ−FN

P(Ω+VN ) N log tF

≤

−

1 C12.2C2.6

⋅

GG∞

(x0

,

x0)

⋅

Cap E

(F

).

(4.15)

This essentially proves the upper bound in Theorem 4.1.1, though a priori not

the sharpest possible bound. In principle, one can choose the interior point x∗0 ∈ Vk ∂Gk with the biggest on-diagonal Green’s function value GG∞(x∗0, x∗0), and run through the preceding argument to get (4.15) with GG∞(x∗0, x∗0) in place of GG∞(x0, x0).

4.5 Proof of Theorem 4.1.2

The purpose of this section is to prove that for any > 0 and any η > 0,

lim sup P ϕ¯N, (x) ≤
N →∞ x∈VN VN, (x)⊂VN
lim sup P ϕ¯N, (x) ≥
N →∞ x∈VN VN, (x)⊂VN

2G log tF − η

√ N

Ω+VN

2G log tF + η

√ N

Ω+VN

= 0. = 0.

(4.16) (4.17)

4.5.1 Lower bound

In this subsection, LS ∶=

1 S

δϕx denotes the empirical measure of the free field
x∈S

ϕ on a measurable subset S of V∞.

Equation (4.16) is a direct consequence of the following lemma.

81

Lemma 4.5.1. For any α < 2G log tF and δ > 0,

lim P
N →∞

LVN

√ 0, αN

≥δ

Ω+VN

= 0.

(4.18)

Proof. For the sake of clarity, we present the proof in two steps.

Step 1. Fix a representative interior point x0 ∈ Vk ∂Gk as in Section 4.4.2.

Also recall the definition of CN . Our interim goal is to show that for any α <

2GG∞(x0, x0) log tF and δ > 0,

lim P
N →∞

LCN

√ 0, αN

≥δ

Ω+VN

= 0.

(4.19)

Following the proof of [BDZ95]*Lemma 4.4, we define, for each α > 0, the

events

ΘN (α) =

x

∈

CN

∶

ϕx

≤

√ αN

and

Θ¯ N (α) =

x

∈

CN

∶

µx

≤

√ αN

.

Then for each δ > δ′ > 0 and α < α′ < 2GG∞(x0, x0) log tF ,

LCN

√ 0, αN

≥δ

= { ΘN (α) ≥ δ CN }

= ΘN (α) ≥ δ CN , Θ¯ N (α′) ≥ δ′ CN ∪

ΘN (α) ≥ δ CN , Θ¯ N (α′) < δ′ CN

⊂ Θ¯ N (α′) ≥ δ′ CN ∪ ΘN (α) ∩ Θ¯ N (α′)c ≥ (δ − δ′) CN

= ∶ J0 ∪ J1.

By Lemma 4.4.2, for each γ ∈ (0, 1) there exists a positive constant C3.1 such that

P J0 ∩ Ω+VN ≤ exp −C3.1mNF tF−N(1−γ) .

82

On the other hand, the lower bound of Theorem 4.1.1 implies that for all suffi-

ciently large N , P(Ω+VN ) ≥ exp (−cρ−FN N log tF ). Therefore

P J0 ∩ Ω+VN P(Ω+VN )

≤ exp −cρ−FN tFN(1−γ) − c′N

→ 0 as N → ∞.

Thus it remains to show that

P J1 ∩ Ω+VN P(Ω+VN )

→ 0 as N → ∞.

Note first that µx − ϕx ≥ (√α′ − √α)√N whenever x ∈ ΘN (α) ∩ Θ¯ N (α′)c. So on

J1,

1 CN

x∈CN

ϕx − µx

≥ (δ − δ′)(√α′ − √α)√N .

Using the fact that under P(⋅ FDN ), {ϕx − µx ∶ x ∈ CN } are independent centered

Gaussian random variables with variance GG˚k(x0, x0), we then find

P J1 ∩ Ω+VN

≤

P

J1

∩

Ω+ DN

≤ ≤

EP exp

−

1 CN (δ −

ϕx − µx ≥ (δ −

x∈CN
δ′)2

(√α′

−

√α)2N

2 CN GG˚k (x0, x0)

δ′)(√α′ CN 2

−

√α)√N

FDN

1⋅ J1∩Ω+DN

≤ exp −CN ρ−FN tNF

for some positive constant C which depends on anything but N . This shows that P(J1 ∩ Ω+VN ) decays faster than P(Ω+VN ) as N → ∞, and hence proves (4.19).

Step 2. Observe that the proof in Step 1 continues to hold for any other

interior point x0 ∈ Vk ∂Gk with the obvious replacements. Thus we can deduce

that for any α < 2 minx0∈Vk ∂Gk GG∞(x0, x0) log tF and δ > 0,

lim P
N →∞

LVN DN

√ 0, αN

≥δ

Ω+VN

= 0.

This falls short of (4.18) because DN has been excluded from the empirical mea-

sure. To redress this shortcoming, we need to translate the conditioning grid

83

Figure 4.2: The coarse graining and conditioning scheme upon translation. As in Figure 4.1, the filled dots indicate the original representative interior points (CN ). Applying a translation by z − x0 for some z ∈ Vk (one of the hollow dots), one obtains the new representative interior points (C˜Nz , hollow dots) and conditioning grid (D˜Nz , solid lines).

relative to the underlying graph V∞, so that points on DN lie within the grid, and then carry out the conditioning scheme. Let us take a moment to describe the translation procedure, as it will be used again in §4.5.2.

As before we fix a representative interior point x0 ∈ Vk ∂Gk. For each z ∈

[0,

k F

)d

∩

Vk

=∶

Vk⌞,

define

C˜Nz

=

{x ∈ VN ∶ x = z +

k F

p

for

some

p

∈

(N0)d},

D˜Nz

=

{x ∈ VN

∶ ∃i ∈ {1, ⋯, d} such that

xi

=

p

k F

+ (z − x0)i

for some

p

∈

N0}.

In effect, we are translating the set of coarse-graining points and the conditioning grid by a vector z − x0; see Figure 4.2. Since D˜Nz separates points in C˜Nz , we can associate to each x ∈ C˜Nz a unique subgraph gx = (V (gx), ∼) of GN such that:

84

• ∂gx ⊂ D˜Nz . • V (gx) contains all vertices in VN which are inscribed by ∂gx. • x is the only element of C˜Nz which lies in V (gx).

The conditioning argument now reads as follows: Under P(⋅ FD˜Nz ), {ϕx ∶ x ∈ C˜Nz } are independent Gaussian random variables, each having mean E(ϕx FD˜Nz ) =∶ µ˜zx and variance G˚gx(x, x). Keep in mind that the variances are not all identical because the subgraphs (gx)x∈C˜Nz no longer retain the symmetries
of the original carpet. Nevertheless, we still have the resistance shorting rule

G˚gx(x, x) = Reff (x, V ((˚gx)c)) ≤ Reff (x, {∞}) = GG∞(x, x),
where Reff(A, B) is the effective resistance between two (finite) subsets A, B of V∞ on the graph G∞.

Now

define

B˜ z C

=

V (B) ∩ C˜Nz

for

each

B

∈

SN○ −j(GN )

and

each

z

∈

Vk⌞.

Note

that

V

(B)

equals

the

disjoint

union

⋃z∈Vk⌞

B˜ z C

,

and

that

the

B˜ z C

are not the same

for all z and B due to inclusion/exclusion of kth-level boundary points. Never-

theless we still have

B˜ z C

= O(mNF −k).

Let κ > 0 and ακ ∶= 2(G − κ) log tF . Define, for δ ∈ (0, 1), the event

Γ˜zB ∶=

ϕ∶

{x

∈

B˜ z C

∶

µ˜zx

≤

ακN

≥

δ

B˜ z C

,

(4.20)

and put Γ˜z = ⋃B∈SN○ −j(GN ) Γ˜zB. We have the following analog of Lemma 4.4.2:

Lemma 4.5.2. Let γ ∈ (0, 1). Then for k large enough, there exists a constant C4.1(δ, k),

independent of N , such that

⎛⎞

E

⎜ ⎝x∈C˜Nz

P[ϕx

≥

0

FD˜Nz

]

⋅

1Ω+D˜Nz

∩Γ˜z ⎟⎠

≤

exp

−C4.1mNF t−FN(1−γ)

.

(4.21)

85

The proof is essentially identical to that of Lemma 4.4.2, except that we can-

not

peg

the

height

to

be

anything

higher

than

√ ακN

in

the

event

Γ˜ zB ,

due

to

the

unequal variances amongst the conditioned variables.

At last we can describe how to adapt the proof in Step 1 to the translated
conditioning grid. The events ΘN (α), Θ¯ N (α), J0 and J1 are as before, except that one replaces CN , DN , and µx with, respectively, C˜Nz , D˜Nz , and µ˜zx, and puts α < α′ < 2G log tF . Then by the aforementioned conditioning argument and Lemma 4.5.2, one shows that limN→∞ P(J0 Ω+VN ) = 0. Similarly, using conditioning and a standard Gaussian estimate, one finds limN→∞ P(J1 Ω+VN ) = 0. Upon varying over all z ∈ Vk⌞ one proves Lemma 4.5.1.

4.5.2 Upper bound

In this subsection we prove the upper bound (4.17). The overall strategy is to show that on Ω+VN , the coarse-grained averages of ϕx and of µx differ by O(1) as N → ∞. Since the µx are independent under the conditioning, we can use standard Gaussian estimates to bound them below uniformly by a threshold
2G log tF N . It follows that the local sample mean of the actual field ϕx is bounded below by the same threshold plus an O(1) error. Finally, we invoke the convergence of discrete Green forms (Lemma 4.2.5(ii)) and a capacity argument
(like the one used at the end of the proof in §4.4.2) to establish the asymptotic
sharpness of the threshold. Our approach is inspired by [Kurt].

In what follows, we fix an y ∈ F and an > 0 such that the cubic neighbor-

hood of y,

B(y, ) =

y′ ∈ F

∶ max 1≤i≤d

yi′ − yi

≤

,

86

is contained in F . We then let

VN, (y) =

z

∈ VN

∶ max 1≤i≤d

zi − [

N F

yi]

≤

⋅

N F

,

and denote by GN, (y) = (VN, (y), ∼) the corresponding graph. In essence, GN, (y)

(relative to GN ) can be viewed as the graphical approximation of B(y, ) (relative

to F ).

The quantity of interest is the average of ϕ over VN, (y), i.e., the local sample

mean of the free field,

ϕ¯N, (y) =

1 VN, (y)

ϕz .
z∈VN, (y)

For each η > 0, denote

MN,η ∶= ϕ¯N, (y) ≥ ( 2G log tF + η)√N

.

Our goal is to prove that for any η > 0,

lim P MN,η
N →∞

Ω+VN

= 0.

(4.22)

To begin the proof, we fix a sufficiently large k ∈ N, take a j ∈ N, and consider

all N > k + j. Fix a representative interior point x0 ∈ Vk ∂Gk as usual. Let κ > 0,

and denote ακ = 2(G − κ) log tF . Two events are introduced as follows. The first

event is Γ˜z = ⋃B∈SN○ −j(GN ) Γ˜zB, where Γ˜zB is given in (4.20) and is defined for each

B ∈ SN○ −j(GN ), z ∈ Vk⌞ and δ ∈ (0, 1). The second event, defined for each s > 0 and

z ∈ Vk⌞, is

D˜sz ∶= ⎧⎪⎪⎪⎨⎪⎪⎪⎩ϕ ∶ there exists an B ∈ SN○ −j(GN ) such that

1 B˜ z
C

(ϕx
x∈B˜ zC

−

µ˜zx)

<

−s⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

Observe that MN,η equals the disjoint union (MN,η∩J2)∪(MN,η∩J3)∪(MN,η∩

J4), where

J2

∶=

⎛⎝z⋃∈Vk⌞

Γ˜z

⎞ ⎠

,

J3 ∶= ⎛⎝z⋂∈Vk⌞

Γ˜z

c⎞⎠∩⎛⎝z⋃∈Vk⌞

D˜ sz

⎞ ⎠

,

J4 ∶= ⎛⎝z⋂∈Vk⌞

Γ˜z

c⎞⎠∩⎛⎝z⋂∈Vk⌞

D˜ sz

c⎞ ⎠

.

87

So the task boils down to proving that each of P(MN,η ∩ J2 ∩ Ω+VN ), P(MN,η ∩ J3 ∩ Ω+VN ), and P(MN,η ∩ J4 ∩ Ω+VN ) decays faster than P(Ω+VN ) as N → ∞.

For J2, we combine Lemma 4.5.2 with a union bound to find that for any γ ∈ (0, 1),
P(J2 ∩ Ω+VN ) ≤ Vk⌞ exp −CmNF t−FN(1−γ) ,
which decays faster than P(Ω+VN ).

For

J3,

we

use

the

fact

that

under

P(⋅

FD˜Nz

),

{ϕx

−µ˜zx

∶

x

∈

B˜ z } C

are

independent

(though not identically distributed) Gaussian random variables to find

Var

⎛⎜ ⎝

1 B˜ z
C

⎞

(ϕx − µ˜zx)
x∈B˜ zC

FD˜Nz

⎟ ⎠

=

1

B˜ z C

2

x∈B˜ zC

Var

ϕx − µ˜zx FD˜Nz

≤

1 B˜ z

⋅
2

B˜ z C

G

=

G B˜ z

.

CC

By applying a union bound followed by a Gaussian estimate, we see that there exists z′ ∈ Vk⌞ such that

P(J3 ∩ Ω+VN ) ≤ Vk⌞ ⋅ P Γ˜z′ c ∩ D˜ sz′ ∩ Ω+VN

≤

Vk⌞

⎛⎛ ⋅ E ⎜⎝P ⎜⎝∃B ∈ SN○ −j(GN ) ∶

1 B˜ z′
C

x∈B˜ zC′

ϕx − µ˜zx′

< −s

⎞⎞
1F ⎟⎠ ⋅ ⎟⎠D˜Nz′ Ω+D˜Nz′ ∩(Γ˜z′ )c

≤

Vk⌞ ⋅ exp

−C s2 mNF 2G

,

where C depends on anything but N . This decays faster than P(Ω+VN ) as N → ∞.

It remains to estimate P(MN,η ∩ J4 ∩ Ω+VN ). Observe that for every z ∈ Vk⌞, B ∈ SN○ −j(GN ), and s > 0,

1 B˜ z
C

(ϕx − µ˜zx) ≥ −s
x∈B˜ zC

on

(D˜ sz)c ∩ Ω+VN ,

1 B˜ z
C

µ˜zx ≥ (1 − δ)
x∈B˜ zC

ακN

on

(Γ˜z)c ∩ Ω+VN .

88

Therefore

1 B˜ z
C

ϕx =
x∈B˜ zC

1 B˜ z
C

µ˜zx +
x∈B˜ zC

1 B˜ z
C

(ϕx − µ˜zx) ≥ (1 − δ)
x∈B˜ zC

on (Γ˜z)c ∩ (D˜ sz)c ∩ Ω+VN .

ακN − s

We then take the intersection over all z ∈ Vk⌞ to conclude that for every B ∈ SN○ −j(GN ),

ϕ¯B ∶=

1 V (B)

ϕx ≥ (1 − δ)
x∈V (B)

ακN − s

on

J4 ∩ Ω+VN .

From now on put s = O(1).

(4.23)

Motivated by [Kurt]*§3, we define, for each θ ∈ [0, 1) and κ′ > 0, the event

Cθ,κ′ ∶= ϕ ∶ there exist ⌊mF(1−θ)j⌋ many B0 ∈ SN○ −j(GN, (y))

such that ϕ¯B0 ≥

√ακ + mθFjκ′

√ N

,

where

SN○ −j(GN, (y)) = B ∈ SN○ −j(GN ) ∶ B ⊂ GN, (y) .

Denote by SNθ −j the collection of B0 in the event Cθ,κ′. By (4.23), for every η > 0 and every θ ∈ [0, 1), there exists κ′ > 0, independent of j, such that for all

sufficiently large N ,

P(MN,η )

≤ P ϕ¯B ≥ (1 − δ) ακN − O(1), ∀B ∈ SN○ −j(GN, (y)) ∩ Cθ,κ′

= P ϕ¯B ≥ (1 − δ) ακN − O(1), ∀B ∈ SN○ −j(GN, (y)) , ϕ¯ ≥ (√ακ + mθFjκ′)√N , ∀B0 ∈ SNθ −j

⎛

≤

P ⎜⎝B∈SN○ −j (GN,

fBϕ¯B
(y))

≥

(1 − δ)

ακN − O(1)

B∈SN○ −j (GN,

fB
(y))

+

⌊m(F1−θ)j

⌋κ′

mθFj

√ N

⎞⎟⎠

,

89

where in the last line we inserted arbitrary fB ≥ 0, B ∈ SN○ −j(GN, (y)) SNθ −j, and fixed fB0 = 1 for all B0 ∈ SNθ −j. Since the ϕ¯B are centered Gaussian variables, we
employ a standard estimate to find

P (MN,η) ≤ exp ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜− ⎛⎜⎝ (1 − δ) ⎝

ακ

N

− O(1)

fB +

B∈SN○ −j (GN, (y))

⎛⎞

2Var ⎜ ⎝B∈SN○ −j (GN,

(y)) fBϕ¯B⎟⎠

C4.2

mjF

√ κ′ N

⎞2 ⎟ ⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟ ⎠

,

(4.24)

for some constant C4.2 independent of N and j.

Now let Ξj ∶ F → R+ be defined by Ξj =

1fB .ιN−j(B) (Note that

B∈SN○ −j (GN, (y))

supp(Ξj) ⊂ B(y, ).) Then

Ξj
x∈VN, (y)

x
N

=

1fB ιN−j (B)

F x∈VN, (y) B∈SN○ −j (GN, (y))

x
N F

⎛⎞

≤

VN −j

⎜ ⎝B∈SN○ −j (GN,

(y)) fB⎟⎠ ,

and

Ξj
x∈VN, (y)

x
N

ϕx =

1fBϕx ιN−j(B)

F x∈VN, (y) B∈SN○ −j (GN, (y))

x
N

=

fB ϕx.

F

B∈SN○ −j (GN, (y))

x∈B

Consequently,

⎛⎞

Var

⎜ ⎝B∈SN○ −j (GN,

(y))

fBϕ¯B

⎟ ⎠

≤

≤

1 VN⌞−j

2

x,x′∈VN,

GG∞(x, x′)Ξj
(y)

x N Ξj
F

x′
N F

C4.3 VN −j

2

x,x′∈VN

GG∞ (x,

x′)Ξj

x N Ξj
F

x′
N F

for some constant C4.3 independent of j and N . Plugging these into (4.24) and

90

then applying Lemma 4.2.5(ii), we find

≤

lim
N →∞
lim
N →∞

log P(MN,η)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ρ−−FN

N log tF (1 − δ)

G C4.3

−κ−
ρ−FN VN −j

O
2

√1 N
x,x′∈VN

⋅ GG∞

1 VN −j x∈VN
(x, x′)Ξj

Ξj
x
N F

x
N F
Ξj

+ C√4.2mjF κ′ 2 log tF
x′
N F

2 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lim ≤ − N→∞

(1 − δ) G − κ − O √1 N

⋅

1 VN

Ξj
x∈VN

x
N F

+ √C4.4κ′ 2 2 log tF

C4.3 lim N →∞

ρ−FN VN 2

GG∞(x, x′)Ξj
x,x′∈VN

x N Ξj
F

x′
N F

≤

−

1 C4.5

⋅

(1 − δ)

G

−

κ⟨1F

, Ξjν⟩F

+

√C4.4κ′ 2 log tF

E(U (Ξjν), U (Ξjν))

2
,

(4.25)

where C4.5 = C4.3C2.6 and C4.4 are independent of j (and N ).

Following the end of §4.4.2, one would expect to optimize the coefficients fB and take the j → ∞ limit to retrieve the capacity in the RHS of (4.25). But we

have decided prior to (4.24) to fix some of the coefficients fB0, in order to leave the κ′ term intact, viz.

⎛⎞

1 1 1Ξ = ⎜ f ⎟ +j B ιN−j(B)

⎝ ⎠B∈SN○ −j (GN, (y)) SNθ −j

B0∈SNθ −j

ιN−j(B0) =∶ Ξj,0 +

,ιN−j (SNθ −j )

where the fB ≥ 0 are arbitrary. This is done with intention to create a ”rescaling

imbalance” between the two terms in the numerator, as the end of the proof re-

veals. On the other hand, in order to have complete control on Ξj, we would like

to exclude the fixed coefficients. The next arguments show that this is indeed possible: as j → ∞, Ξj can be replaced by Ξj,0 in the RHS of (4.25).

91

Let’s write

E (U (Ξjν), U (Ξjν))
= E (U (Ξj,0ν), U (Ξj,0ν)) + 2 ⋅ E 1U (Ξj,0ν), U ( ν)ιN−j(SNθ −j) 1 1+ E U ( ιN−j(SNθ −j)ν), U ( ν)ιN−j(SNθ −j)
=∶ K1 + 2K2 + K3.

Suppose, without loss of generality, that K1 is bounded above by a constant

independent of j. The key estimate is on K3. Let Qj denote the closure of any

one of the ιN−j(B0), which is a subset of B(y, ) inscribed by a hypercube of

side

−j F

.

Also

let

G

∶

F∞

×

F∞

→

R+

be

the

integral

kernel

associated

with

U.

By

[BB99]*Corollary 6.13(a), there exists C8 such that G(x, x′) ≤ C8 x − x′ dw−dh for

all x, x′ ∈ F∞. Therefore

K3

≤

⌊m(F1−θ)j ⌋

G(x, x′)dν(x)dν(x′)
Qj ×Qj

≤

C m(F1−θ)j

dν(x)dν(x′) Qj ×Qj x − x′ dh−dw

≤

C m(F1−θ)j

dν(

−j F

x)dν

(

−j F

x′)

Q×Q

(

−j F

x − x′

)dh−dw

≤

C m(F1−θ)j

−j (dh +dw ) F

dν(x)dν(x′) Q×Q x − x′ dh−dw

= O(m−Fθjt−Fj),

where Q is a cubic region of side O(1), and the last integral is finite by the same

argument as in the proof of Lemma 4.2.5(i). This then allows us to estimate K2 via Cauchy-Schwarz, namely, (K2)2 ≤ K1K3 ≤ K1O(m−Fθjt−Fj). Hence as j → ∞,

E (U (Ξjν), U (Ξjν)) = E (U (Ξj,0ν), U (Ξj,0ν)) + O(m−Fθj 2t−Fj 2).

(4.26)

We can now bound the RHS of (4.25) from above. First use the trivial in-
equality ⟨1F , Ξjν⟩F ≥ ⟨1F , Ξj,0ν⟩F and apply it to the numerator. Then plugging
(4.26) into the denominator, and taking the limit j → ∞ on both sides of (4.25),

we obtain

lim
N →∞

log P(MN,η) ρ−FN N log tF

≤

−

1 C4.3

⋅

(1 − δ)

G

−

κ⟨1F ,

Ξ0 ν ⟩F

+

√C4.4κ′ 2 log tF

E(U (Ξ0ν), U (Ξ0ν))

2
,

(4.27)

92

where Ξ0 ∈ F is any nonnegative function supported on B(y, ) minus a set of capacity zero. By the Yosida approximation (Proposition 4.3.1), we may then replace Ξ0ν by any µ ∈ S0(0) with the same maximal support set.

Now comes the simple but crucial rescaling argument:

E(U µ, U µ) = γ ⋅ (γE)(U γµ, U γµ) for each γ > 0 and for all µ ∈ S0(0),

where U γ = γ−1U is the 0-order potential operator associated with γE. Thus the

RHS of (4.27) can be rewritten as

−

1 γC4.3

⋅

(1 − δ)

G

−

κ⟨1F ,

µ⟩F

+

√C4.4κ′ 2 log tF

(γE)(U γµ, U γµ)

2
.

Fix a compact set K within B(y, ) minus the aforementioned set of capacity zero, and choose µ to be µK, the 0-order equilibrium measure of K with respect to
γE. According to Proposition 4.3.2, ⟨1F , µK⟩F = (γE)(U γµK, U γµK) = CapγE (K).

(This achieves the aforementioned ”rescaling imbalance.”) Then upon taking

δ, κ → 0, and combining with the lower bound of Theorem 4.1.1 (see also Remark

4.4.1), we arrive at

lim
N →∞

log P(MN,η Ω+VN ρ−FN N log tF

)

≤ ≤

−Nli→γm∞C14lρ.o3−Fg⎡⎢⎢⎢⎢⎣NPGN(CMlaopgNγt,ηEF)(K−)Nl+i→m∞

log P(Ω+VN ) ρ−FN N log tF 2√G(C4.4κ′)
log tF

+

(C4.4κ′)2 2 log tF CapγE (K)

+ γ−1C1.3GCapγE (F ).

(4.28)

Solving a quadratic inequality shows that the RHS of (4.28) is negative if

κ′ > C4−.14 ⋅ CapγE (K) ⋅

⎛ 2 log tF ⋅ ⎜⎝

C1.3C4.3G

⋅

CapγE (F ) CapγE (K)

−

G⎞⎟⎠ .

93

Observe that CapγE (K) depends linearly on γ, while the rest of the expression on the RHS is manifestly independent of γ. So by tuning γ, we can make κ′ > ∆

for any ∆ > 0, and thus

lim
N →∞

log P(MN,η Ω+VN ) ρ−FN N log tF

<

0

for any η > 0. This proves (4.22) for any y ∈ F and > 0, whence (4.17).

94

APPENDIX A CHAPTER 1

We collect some of the results about Green’s function we used in Chapter 2 here. Most of this material is known to experts and drawn from the Appendix of [LS]. In this chapter, we freely use the notation from Sections 2.1 and 1.2.
Theorem A.0.3. For a continuous function f , consider the following Dirichlet problem on SG
−∆u = f on SG ∖ V0 u = 0 on V0.
The Dirichlet problem has a unique solution in dom(∆) given by u(x) = G(x, y)f (y)dy
SG
for G(x, y) = limM→∞ ∑Mk=1 ∑s,s′∈Vk∖Vk−1 g(s, s′)ψsk(x)ψsk′(y) where

g(s,

s′)

=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

3 10
1 10

3k 5
3k 5

for s = s′ ∈ Vk ∖ Vk−1, for s = s′ ∈ FwK, w = k − 1 and s ≠ s′.

As we also noted in Section 2.1 we drop the superscript “m” in ψxmm, ψymm and ψzmm and instead write ψxm, ψym and ψzm. Because, unless otherwise is noted, the superscript index well matched the subscript index. We can get from the definition that

ψxm dy =
SG

ψym dy =
SG

SG

ψzm

dy

=

2 3m+1 .

95

(A.1)

since ψxm 2 ≤ ψxm , we can further develop this as

ψxm 2dy =
SG

ψym 2dy =
SG

SG

ψzm

2dy

≤

2 3m+1 .

For simplicity, we define Ψm(a, b, c)(x) = aψxm(x) + bψym(x) + cψzm(x).

Observe that putting together (A.1) and (A.2) yields

(A.2) (A.3)

SG

Ψm(a, b, c)(x)

dx

≤

C1 3m

and

SG

Ψm(a, b, c)(x) 2dx

≤

C2 3m

(A.4)

for constants C1 = C1(a, b, c) and C2 = C2(a, b, c).

Lemma A.0.4. We obtain

G(xm, y)

=

2 15

3 5

m

m m=1

Ψk(1,

1,

1)(y)

+

1 6

3 5

m
Ψm(1, −1, −1)(y).

Lemma A.0.5. We have

G(zm, y)

=

1 10

3 5

m

m m=1

Ψk(1,

2,

2)(y)

+

1 10

3 5

mm
3kΨk(0, −1, 1)(y).
k=1

Lemma A.0.6. Suppose u = 0 on V0 and ∆u exists on SG then we have

(A.5) (A.6)

ηm

=

∂nu(xm)

=

3 5

1

mm
3k

3 k=1

Ψk(0, −1, 1)∆udy
SG

+

−1 2

Ψm(1, −1, 1)∆udy − φm
SG

where φm = ∫Zm ψzm∆udy.

96

(A.7)

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