พฤติกรรมแบบเกาส์ของเคอร์เนลความร้อน Gaussian

Transcription

พฤติกรรมแบบเกาส์ของเคอร์เนลความร้อน
Gaussian Behavior of Heat Kernels
สันติ ทาเสนา*
ภาควิชาคณิตศาสตร์ คณะวิทยาศาสตร์ มหาวิทยาลัยเชียงใหม่
Santi Tasena*
Department of Mathematics, Faculty of Science Chiang Mai University
บทคัดย่อ
สมการความร้อนคือสมการที่ใช้อธิบายการกระจายความร้อนบนปริภูมิหนึ่งๆ ณ เวลาใดๆ โดยทั่วไปค�ำตอบของสมการความร้อน
จะอยู่ในรูปปริพันธ์ของผลคูณระหว่างฟังก์ชันความร้อนเริ่มต้นและเคอร์เนลความร้อน ดังนั้นพฤติกรรมของเคอร์เนลความร้อนจึง
มีผลกระทบโดยตรงต่อพฤติกรรมของค�ำตอบของสมการความร้อน บทความนี้จะยกตัวอย่างความสัมพันธ์ข้างต้น ซึ่งก็คือความสัมพันธ์
ระหว่างพฤติกรรมแบบเกาส์ของเคอร์เนลความร้อนและอสมการฮาร์แนค รวมถึงผลกระทบที่พฤติกรรมทั้งสองมีต่อสมบัติทางเรขาคณิต
ได้แก่สมบัติทวีคูณและอสมการปวงกาเรของปริภูมินั้นด้วย โดยผู้เขียนหวังว่าบทความนี้จะเป็นความรู้เบี้องต้นให้กับผู้ที่สนใจท�ำวิจัย
ในสาขานี้ต่อไป
ค�ำส�ำคัญ :อสมการฮาร์แนค การประมาณค่าเคอร์เนลความร้อน สมบัติทวีคูณ อสมการปวงกาเร
Abstract
Heat equations are used to explain the distribution of heat over spaces and time. Typically, solutions of heat
equations can be written as integrations of their initial heat profiles and heat kernels. Thus, heat kernel behavior
directly influences the behavior of solutions of heat equations and vice versa. This article will review one such
relation: the Gaussian behavior of heat kernel and the (parabolic) Harnack inequality. This article will also discuss
how these two properties relate to the geometric properties such as doubling property and Poincare inequality of
the underlying spaces. Hopefully, this will serve as an introduction to those interested in working in this field.
Keywords : Harnack inequality, heat kernel estimates, doubling property, Poincare inequality
--------------------------------------------------
*E-mail: [email protected]
สันติ ทาเสนา / วารสารวิทยาศาสตร์บูรพา. 19 (2557) 1 : 169-177
169
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, where
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exhibits19
the same
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asto
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related
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the
perturbation
of
the
heat
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using
Dirich
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the
perturbation
of theassociated
heat
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Dirichlet
forms
approach.
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heat
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behavior
as
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the
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e1 heat
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the
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satisfying
the
above
equation
is
called
a
equation
is
given
by
,
where
the
function
14
(Aronson,
1967,
1968,
197
13
Less
than
sixty
years
ago,
Nash
(Nash,
1958),
Moser
(Moser,
1961,
1964,
1967),
and
Aronson
13
Less
than
sixty
years
ago,
Nash
(Nash,
1958),
Moser
(Moser,
1961,
1964,
1967),
and
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Introduction
1 19
Introduction
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measure
ν space
onthe
X
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the
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of temperature
over1964,
space1967
and
results
related
to
perturbation
oftime
the
equation
using
Dirichlet
forms
approach.
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simplify
thethe
analysis,
the
author
assumes
that
the
topological
is
locally
the14
temperature
at
point
and
at
Typically,
is
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as
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and
at
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(Aronson,
1967,
1968,
1971)
independently
worked
on
this
problem
which
can
be
deduced
that
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(Aronson,
1967,
1968,
1971)
independently
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on
this
problem
which
can
be
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that
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the
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for
unifo
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of
the
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heat
equation.
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Dirichlet
Forms
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Dirichlet
Forms
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(Aronson,
1967,
1968,
1971)
independently
worked
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this
problem
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the
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is called
awell-known
solution
of solution
the classical
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6 and
that
of
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form
(E,D(E))
onequation
L always
(X,
)satisfies
is regular, strongly
3classical
compact
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second
Hausdorff,
the
Borel
measure
on
Introduction
heat
equation
isand
given
by
, Dirichlet
where
the
function
2temperature
The
classical
heat
equation
is
given
by
,heat
where
theνform
function
1countable
Introduction
nsatisfying
ofi.e.
over
space
time.
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15
the
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for
uniformly
operators
on
i.e.
for
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operators
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the
result
holds
for
uniformly
elliptic
operators
on
i.e.
for
heat
operators
of
the
form
the
represents
the
evolution
of
temperature
over
space
and
time.
It
is
well-known
22
To
simplify
the
analysis,
the
author
assumes
that
the
16theontopological
22
To
analysis,
the
author
assumes
that
isthatlo
15 simplify
the result
holds
for
uniformly
elliptic
operators
i.e. for space
heatsuch
operator
and
issupport,
interpreted
asthe
the
temperature
point
interpreted
as
temperature
atatpoint
and
atlocal,
time
21 isfull
Dirichlet
Forms
4 satisfies
has
andthe
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onthe
isand
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strongly
local,
and
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the
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operator.
on
that
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of
the
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satisfies
al
satisfying
above
is
called
a
solution
of
the
classical
heat
The
classical
heat
equation
given
by
,
where
the
function
3
satisfying
the
above
equation
is
called
a
solution
of
classical
heat
23
compact
but
non-compact,
second
countable
and
Hausdorff,
16
such
that
is
smooth
for
all
and
there
exists
16
such
that
is
smooth
for
all
and
there
exists
23
compact
but
non-compact,
second
countable
and
Hausdorff,
the
Borel
measure
on
2
The
classical
heat
equation
is
given
by
,
where
22
To
simplify
the
analysis,
the
author
assumes
that
the
topological
space
is
locally
presents
the evolution
temperature
over
space
time. It is well-known
16 and
such that
is17smooth for all for which the
a
5
admits
the
carret >duof
champ
operator.
at
time
0
i.e.
the
solution
u represents
the
evolution
Definition
1.
A
Dirichlet
form
is
a
pair
(E,D(E))
where
ypically,
isTypically,
interpreted
as24the
temperature
at
and
at
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24
has
full
theclassical
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on
4 23 equation.
issecond
interpreted
as point
the
temperature
atsupport,
point
and
at time
has
fullcountable
support,
and
the
Dirichlet
form
onon
local,
local
6classical
compact
but
non-compact,
Hausdorff,
the
Borel
measure
always
heat
equation
satisfies
satisfying
the
above
equation
is
called
a
solution
the
heat
3time. It is17well-known
satisfying
the
equation
a solution
ofregul
theh
carre
17
for
which
for
all
17 of temperature
for over
which
for
allabove
18is called
. Since
thenfor
theall
result
space
and
that
which
provided
that
the
initial
heat
profile
7carre
satisfies
some
25
admits
du
champ
tion
over
space
andfor
time.
Itthe
isspace
well-known
25
admits
du
champ
operator.
5 Definition
i.e.has
the
solution
represents
the
of
temperature
over
and
time.
is2 well-known
isis
aoperator.
closed,
positive
7 equation.
1.the
Aevolution
Dirichlet
form
is
a evolution
pairthe
where
aIttime
closed,
24represents
full
support,
and of
thetemperature
Dirichlet
form
on
isinterpreted
regular,
strongly
local,
and symmetric,
Typically,
is
interpreted
as
the
temperature
at
point
and
at
4
equation.
Typically,
is
as
the
temperature
at
point
and
19many
results
relatedThis
to the
perturb
26
18
.. Since
then
the
result
has
been
progressed
many
directions.
This
article
will
on
the
18 classical
Since
then
the
result
hasheat
been
progressed
inSince
manythen
directions.
This
article
will focus
focus in
on
the
26
18 satisfies
.in
the
result
has
been
progressed
directions.
article
w
solution
usome
of
the
classical
heat
equation
2
form
n8 i.e.
the
heat
equation
satisfies
6ofpositive
that
solution
of
the
classical
equation
satisfies
symmetric,
densely
defined,
bilinear
on
with
the
following
property:
for
25
admits
the
carre
du
champ
operator.
satisfies
regularity
assumptions.
The
densely
defined,
bilinear
form
on
L
(X,
ν
)
with
the
the
solution
represents
the
evolution
of
temperature
over
space
and
time.
It
is
well-known
8heat
function
is
the
heat and
kernel
asso
satisfies
some
regularity
assumptions.
The
provided
that
therelated
initialto
profileDefinition
5 of
i.e.
the
solution
represents
the
evolution
ofheat
temperature
over
space
time.
It
20
19
the
the
equation
using
Dirichlet
approach.
19 results
results
related
toheat
the perturbation
perturbation
of
the
heat
equation
using
Dirichlet
forms
approach.
27related
Definition
1.forms
A Dirichlet
form
is
acalled
pair
where
27
1.
A
Dirichlet
form
is
a
pair
where
is
a
cl
26
19
results
to
the
perturbation
of
the
equation
using
Dirichlet
forms
appro
9 that
anysolution of
, the
function
and
. for2 any f ∈ D(E), the function
the
classical
heat
equation
satisfies
following
property:
20
20
21
6
that
solution
of
the
classical
heat
equation
satisfies
densely
defined,
bilinear
on property
form
2w
28
positive
densely
defined,
bilinear
on Laplacian
with theform
following
20regularity
27 profile
Definition
1. Aassociated
Dirichlet
form
is is
asymmetric,
pair
where
is a closed,
is heat
called
the
heat kernel
to some
the
Laplacian
ial
satisfies
assumptions.
The
Introduction
9with
. symmetric,
called
the28
heatpositive
kernel
associated
to the
0 function
equipped
the
inner
product
is
a
1 any
Introduction
22
To
simplify
the
an
21
Dirichlet
Forms
21 The set
Dirichlet
Forms
29
any
,
the
function
and
and
29
,
the
function
and
.
Dirichlet
Forms
28
positive symmetric, densely defined, 21
bilinear form on
with
the following
property:
for
equations
10
General
heat
are
given
by
replacing
the
Laplacian
wi
1 the
space,
called
a
Dirichlet
space,
and
its
norm
associated
to
its
inner
product
is
referred
to
22
To
simplify
the
analysis,
the
author
assumes
that
the
topological
space
is
locally
22provided
To
simplify
analysis,
the
author
assumes
that
the
topological
space
is
locally
23
compact
but
non-compact,
The
classical
heat
equation
is
given
by
,
where
the
function
is
called
the
heat
kernel
associated
to
the
Laplacian
provided
that
the
initial
heat
profile
at
initial
heat
profile
satisfies
some
regularity
assumptions.
The
.
that
the
initial
heat
profile
7 Hilbert
satisfies
some
regularity
assumptions.
The
22
the analysis,
the.the
author
assumes
thatwith
the 2the
topological
spa
29
any
, the function
and
The classical
heat
equation
iswith
given
by
, where
th
any
The
set measure
equipped
with
30To simplify
The
produ
30 2
The
inner
product
Introduction
1 non-compact,
Introduction
11 setcompact
One
researcher
might
ask
is
that
whether
orthe
notinner
the
heat
k
2by replacing
as23
the Dirichlet
norm.
compact
but
second
countable
and
Hausdorff,
the
Borel
on
always
23
compact
but
non-compact,
second
countable
andquestion
Hausdorff,
theequipped
Borelsecond
measure
on
always
24
has full the
support,
and
the D
2
2
23
but
non-compact,
countable
and
Hausdorff,
Borel
measure
the
Laplacian
with
other
heat
operators.
satisfies
some
regularity
assumptions.
The
function
satisfying
theis31
above
equation
iskernel
called
aDirichlet
solution
of
the
classical
heat
General
heat
equations
are
given
by
replacing
the
Laplacian
with
other
heat
operators.
that
the
initial
heat
profile
satisfies
some
regularity
assumptions.
The
31
Hilbert
space,
called
a
Dirichlet
space,
and
its
norm
associated
to
Hilbert
space,
called
a
space,
and
its
norm
associated
to
its
inner
product
is
referr
3 provided
inner
product
is
a
Hilbert
space,
The
set
equipped
with
the
inner
product
is
a
provided
that
the
initial
heat
profile
7
satisfies
some
regularity
assu
called
the
heat
associated
to
the
Laplacian
2
8 3024
function
is
called
the
heat
kernel
associated
to
the
Laplacian
3
satisfying
the
above
equation
is
called
a
solution
of
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nctions
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21
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30
Definition
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the
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37
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37
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ififits
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ry,
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16
subject
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13
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13
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rmally,
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uous.
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information
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One
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15
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Informally,
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2012).
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further
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42
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further
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43
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Inintrinsic
the previous
section,
classical
heat
kernel
depends
on
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objects,
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is1986)
the
34
Santi
Tasena
/
Burapha
Sci.
J.
19
(2014)
1
:
169-177
25
is
a
(weak)
solution
of
the
heat
equation
associated
to
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172
24
kernels.
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heat
operator
where
is
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24
kernels.
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be
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In
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24
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ussian
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1967),
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27
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ical
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1971).
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28
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Nash
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1958),
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(Moser,
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1967),
1967),
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and
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Aronson
estimates
found
in
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Nash
(Nash,
1958),
Moser
20
the
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measure
called
doubling
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1996).
ean distance
on
, and another
is the dimension
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ance
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,
and
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define
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29
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(Aronson,
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1968,
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1971)
1971)
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isone
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th
21 other
1964,
1967),
and
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(Aronson,
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to
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for
other
heat
of
this
property,
one
might
instead
ask
whether
a
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kernel
has
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is
again
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of
this
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oation
know
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behavior
happens
for
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kesting
whether
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,
and
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is
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30
30
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Li
and
and
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1986)
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isinstead
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er
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(Nash,
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(Moser,
1961,
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1967),
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1968,and
1971).
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to
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oser,
1964,
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one
ask
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ng
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know
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31
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Li
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17elliptic
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7,
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isonly
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Nash
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(Moser,
1961,
1964,
16
regardless
regardless
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Even
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24measure.
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200
Gaussian
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lower
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Nash
(Nash,
1958),
32
32
should
should
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with
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volume
measure.
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definition
gives
gives
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current
form
form
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14
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de
& Yao,
1986)
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18
as
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7),
andon
Aronson
(Aronson,
1967,
1971).
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astarting
generalization
of
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17
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orThere
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to
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zero
acians
manifolds
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nonnegative
25
good
point.or
areitit
two
closely
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property,
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33
33 Gaussian
Gaussian
behavior
behavior
of
of
heat
heat
kernel.
kernel.
quation
associated
to
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and
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.
Moser
(Moser,
1961,
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1967),
and
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(Aronson,
general.
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26 the
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, which
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34to is 1967, 1968, 1971) concluded
that
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20 that
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1996).
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The
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1919
general.
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estimates
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imply
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27 nature.
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ng
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rty,
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might
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21
vior
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kernel.
28
measure
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1998).
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only
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pr
20
20
the
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volume
measure
measure
called
called
doubling
doubling
property(Sturm,
property(Sturm,
1996).
1996).
ellipticthe
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LiGaussian
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17 bylimsup or liminf, it still possible
rdlessMoser
of whether
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estimates.
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limit
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Nash,
1958),
Moser (Moser,
1961,
1964,
1967),
and
Aronson
29
of volume
measures usually
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does
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958),
(Moser,
1961,
1964,
1967),
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Aronson
22
Property
21
21
one
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whether
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18limitasofDoubling
converges
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extended
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still
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30
cluded
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uniformly
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23
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Doubling
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kernel
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Even
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h,
1958),
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1961,
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1967),
and
Aronson
19
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31there
Definition
6.also
A non-zero
Borel measure
on a metric
is doubling if there
converges
to infinity.
In
other
words,
noYao
universal
definition
of
dimension
in space
nonnegative
Ricci
curvature.
However,
Li is
and
ended
this
result
to Laplacians
on manifolds
with
nonnegative
is
result
to
Laplacians
on
manifolds
with
nonnegative
24
forms.
It
has
been
studied
as
a
subject
by
itself.
The
book
of
Hein
23
23 on,
Actually,
Actually,
doubling
property
property
isitisthe
astill
aproperty
property
of
the
thethe
underlying
underlying
spa
sp
ed that this Nevertheless,
is true for uniformly
elliptic
operator.
Later
20 of
volume
called
bydoubling
limsup
or
liminf,
possibleofmeasure
that
32 longer
constant
suchreplaced
thatrelated
for any
and
. doub
eral.
Gaussian
estimates
still
imply
aand
property
to the
growth
that
the
is
no
correct
nd
Yao
also showed
showed
that
the term
term
is
no
longer
correct
and
25
good
starting
point.
There
are
two
closely
related
versions
of
dou
lso
showed
that
the
term
is
no
longer
correct
and
24
24
forms.
forms.
It
It
has
has
been
been
studied
studied
as
as
a
a
subject
subject
by
by
itself.
itself.
The
The
book
book
of
of
Heinom
Heino
33
ed this result
to Laplacians
on manifolds
with nonnegative
21is not the same as the limit
volume
doubling
property(Sturm,
1996).
limit
t convergesand
to zero
me
measure.
Thecalled
following
definition
gives
current
form
ofstarting
26
space
to Saloff-Coste(Grigor’yan
beaspoint.
doubling,
another
is for related
the
volume
measure
of t
ure.
Themeasure
following
definition
should
bethe
replaced
with
the
volume
measure.
The
34 theform
Grigor’yan
and
& Saloff-Coste,
2005)
extended
the
conc
2525of
good
good
starting
point. There
Thereare
are
two
two
closely
closely
related
versions
versions
ofof
doubli
doub
Yao
also
showed
that
termgives
isthe
nocurrent
longer
correct
and
22
t
converges
to
infinity.
In
other
words,
there
is
no
as
35 current
doubling
property
that
of remote
and
-anchored
balls.
They
showedthe
thatformer
doubling
pr
turnstoto
that
the latter
implies
the
former
while
2626
space
space
toout
be
bedoubling,
doubling,
and
andanother
another
is
for
for
the
the
volume
volume
measure
measure
of
ofimp
tha
th
definition
givesgives
the current
form
of27Gaussian
measure. Thefollowing
following
definition
the
form
of
23 is
Actually,
doubling
prope
Doubling
36 Property
of 27
remote
and
S-anchored
balls
together
withthe
another
property
called
volumeimplie
comp
measure
which
is
doubling(Luukkainen
&
Saksman,
1998).
There
universal
definition
of
dimension
in
general.
Nevertheless,
2728 turns
turns
out
out
that
that
the
the
latter
latter
implies
implies
the
former
former
while
while
the
the
former
former
impli
24
forms. It has been studied as a s
of heat
kernel.is a property
Actually,behavior
doubling
property
the underlying
spacesmeasures
and not for
of
the
Dirichlet
37 ofcondition
imply
doubling
property
allbe
balls
providedhere.
that
the set S1998).
is fully
acce
of volume
will
discussed
282829 measure
measure
which
which
is
is
doubling(Luukkainen
doubling(Luukkainen
&&
Saksman,
Saksman,
1998).
Therefo
Theref
Gaussian(Heinomen,
estimates
imply
related
to the
25 isaaproperty
good
starting
point.
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are t
6 still
ms. It has been
studied5.asA aheat
subject
The
book
of Heinomen
2001)
Definition
kernelbyp itself.
is said to
have
Gaussian
30
2929 ofofvolume
volume
measures
measures
will
willbebediscussed
discussed
here.
here.
26 the
space
to on
be adoubling,
and anoth
growth
ofAthenon-zero
volume
called
doubling
property
d starting point.
twoexist
closely
related
of
doubling
property,
one measure
is
for
31
Definition
6.
Borel
measure
metric space
30
30
c1, cversions
>
0
such
that
upper There
bound are
if there
constants
2
27 Itturns out that the latter implies
ce
to 5.be Adoubling,
and another
forGaussian
the volume
measure
that
tosuch
be that
doubling.
(Sturm,
1996).
Definition
Definition
6.space
6.AA
non-zero
non-zero
Borel
Borel
measure and
onona ametric
metricspace
space
i
constant
for measure
any
tion
heat
kernel
said
to is
have
upper
bound
ifof
there
exist
constants
for
all
x, y inisthe
underlying
space
and
t >31031,32
28
measure which is doubling(Luu
s out
that
the
latter
implies
the
former
while
the
former
implies
the
existence
of
a
Borel
33
such that for all x,y in the underlying space and t>0, 3232 constant
constant
such
suchthat
thatfor
forany
any
and
and
29
of volume measures
will be disc
Grigor’yan
and Saloff-Coste(Grigor’yan
& Saloff-Coste,
2
sure which is doubling(Luukkainen & Saksman,33
1998).
Therefore,
only doubling
property
3334
30
Doubling
Property
doubling
property
to Saloff-Coste(Grigor’yan
that
of remote and -anchored
balls. They
sh
olume measures will be discussed here.
343435
Grigor’yan
Grigor’yan
and
and
Saloff-Coste(Grigor’yan
Saloff-Coste,
Saloff-Coste,
2005
200
31
Definition &
6.&A
non-zero Borel
Actually,
doubling
property
is
a
property
of
the
36
of
remote
and
S-anchored
balls
together
with
another
property
3535 doubling
doublingproperty
propertytotothat
thatofofremote
remoteand
and -anchored
-anchoredballs.
balls. They
Theyshow
sho
32 aconstant
such that for an
inition 6. A non-zero Borel measure on a metric
space
isand
doubling
if thereballs
exists
imply
doubling
for
all
balls
provided
that
the
363637
ofofcondition
remote
remote
and S-anchored
S-anchored
balls
together
together
with
with
another
another
property
property
c
underlying
spaces
andproperty
not
of
the
Dirichlet
forms.
It
33
stant
thattoforhave
anyGaussianand
. property
A heat kernel such
is said
lower bound 37
if37therecondition
exist constants
condition
imply
imply
doubling
doubling
property
for
for
all
all
balls
balls
provided
provided
that
that
the
the
set
se
34
has been studied as a subject
by itself.Grigor’yan
The book ofand Saloff-C
is said
the underlying
A heat kernel
hat for all x, y in
spacep and
t>0,to have Gaussian lower
35 of
property
Grigor’yan
andif Saloff-Coste(Grigor’yan
extended
the concept
Heinomen
(Heinomen,
2001)
is doubling
a good starting
point.to that of rem
bound
there exist constants c3, c4 &
> 0Saloff-Coste,
such that for 2005)
36
of
remote
and
S-anchored bal
bling property to that of remote and -anchored balls. They showed
doubling
There that
are two
closelyproperty
related
versions
of
doubling
all
x, y in the underlying space and t > 0,
37
condition imply doubling prope
emote and S-anchored balls together with another property called volume comparison
property, one is for the space to be doubling, and another
dition imply doubling property for all balls provided that the set S
is fully accessible.
is the volume measure and is the intrinsic distance.
A heat kernel is said to satisfy Gaussianสันestimates
has both
Gaussian
upper
bound1 : 169-177
ติ ทาเสนาif/itวารสารวิ
ทยาศาสตร์
บูรพา.
19 (2557)
aussian lower bound. A heat operator or a Dirichlet form is said to satisfy Gaussian
tes if its associated heat kernel exists and satisfies Gaussian estimates.
173
Gaussian
estimates.
Evenofwhen
the limit
is replaced by
71).
As aweighted
generalization
Euclidean
dimensions,
2asasasconverges
measures
with
finite
unbounded
weight
Moshimi
and
Tesei(Moshimi
s,1810
there
is
no
converges
universal
toclosed
definition
infinity.
of
In
dimension
other
inand
there
isfunctions.
no
definition
of
dimension
10
,
S
is
closed
and
null,
and
Assume
the
following
conditions
hold.
10
,
,
S
S
is
is
closed
and
and
null,
null,
and
and
..universal
. universal
Assume
Assume
the
the
following
following
conditions
conditions
hold.
2 limit
weighted
measures
with
finite
weight
Moshim
18
as
converges
to
to
infinity.
infinity.
In
In
other
other
words,
there
there
is
is
no
no
universal
definition
definition
ofunbounded
of
dimension
dimension
inhold.
inin&functions.
10
,words,
Sas
is
closed
and
null,
and
. of
Assume
the following
condi
imit
converges
zero
is
not
the
same
aswords,
the
Using
this
fact
when
is
singleton,
Grigor’yan
Saloff-Coste
proved
doubling
property
al) 3dimension
of
the
underlying
space
Tesei,
2007)
also
used
this
fact
to
show
doubling
property
of
weighted
Lebesgue
measures
with
mates
19
still
general.
imply
a
property
Nevertheless,
related
to
Gaussian
the
growth
estimates
of
still
imply
a
property
related
to
the
growth
of
3
Tesei,
2007)
also
used
this
fact
to
show
doubling
property
of
weighted
19
general.
general.
Nevertheless,
Nevertheless,
Gaussian
Gaussian
estimates
estimates
still
still
imply
imply
a
property
a
property
related
related
to
the
the
growth
growth
of
of
11
a)
The
set
S
satisfies
-skew
condition
i.e.
for
any
and
there
exists
such
that
11
11
a)
a)
The
The
set
set
S
S
satisfies
satisfies
-skew
-skew
condition
condition
i.e.
i.e.
for
for
any
any
and
and
there
there
exists
exists
such
such
that
that
rds,
there ismeasures
no universal
ofa)dimension
infunctions.
weighted
with definition
finite11
unbounded
Moshimi
and
&
The weight
set S satisfies
-skew
condition
i.e.Tesei(Moshimi
for any
and
there exists
nature.
However,
this
limit
usually
doesproperty(Sturm,
not
exist
4 the
unique-singularity
weight
To4 be property
precise,
they
provedweight
doubling
property
measures
y(Sturm,
20Tesei,
the
1996).
volume
measure
called
doubling
property(Sturm,
1996).
unique-singularity
functions.
Toforbe
precise, they proved doubli
20
the
volume
volume
measure
measure
called
called
doubling
doubling
property(Sturm,
1996).
1996).
imates
still
imply
a used
property
related
to the
growth
of
2007)
also
this
fact
tofunctions.
show
doubling
of weighted
Lebesgue
measures
with
5estimates.
where
is
aremotely
positive
number
less
than
the
dimension
ofathe
underlying
Euclidean
21
sian
when
the
limit
replaced
by
21
5 there
where
is
positive
number
less
thanspace.
the dimension
the for
und
b)
The
function
is
constant
i.e.
there
exists
constant
12
such
that
for
any
12
12
b)
b)The
TheEven
function
function
is
isfunctions.
remotely
remotely
constant
i.e.
i.e.
there
exists
exists
aaaconstant
constant
such
that
that
for
for
any
rty(Sturm,
1996).
unique-singularity
weight
ToThe
be precise,
they
doubling
property
for
measures
b)
function
is proved
remotely
constant
i.e.such
there
exists
aany
constant
12 is constant
suchofthat
ling
22
Property
Doubling
Property
6
Later,
the
fully
accessible
assumption
was
weaken
to
that
of
-skew
condition
and
the
uniqueas
converges
to
zero
is
not
the
same
as
the
limit
22 13
Doubling
Doubling
Property
Then
is for theisvolume
measure
space
to6 the
beProperty
doubling.
isunderlying
doubling
if Euclidean
and only if was
there
exists a constant
Later,
the fully
assumption
weaken
to that of -skew c
13
13
where
a positive
number
than
dimension
of theµaccessible
space.
13of thatless
3
3
erty
23
of
the
underlying
Actually,
spaces
doubling
and
property
not
of
the
is
Dirichlet
a
property
of
the
underlying
spaces
and
not
of
the
Dirichlet
7
singularity
assumption
was
removed
yielding
the
following
result.
there
is
no
universal
definition
of
dimension
in
23
Actually,
Actually,
doubling
doubling
property
property
is
a
is
property
a
property
of
of
the
underlying
underlying
spaces
spaces
and
and
not
not
of
of
the
the
Dirichlet
Dirichlet
7
singularity
assumption
was
removed
yielding
the
following
ubling
Property
Itfully
turns
out
that the
latter
implies
the weaken
former
the
such
that
for
and
14
Then
doubling
and
only
there
exists
aaconstant
constant
such
that
for
any
and
14
14
Then
Then
isisisdoubling
doubling
ifififassumption
and
and14
only
only
ififThen
ifthere
there
exists
exists
awhile
constant
such
such
that
that
for
forany
any
any
and
and such that forresult.
Later,
the
accessible
was
to
that
of only
-skew
condition
and
the uniqueis
doubling
if
and
if
there
exists
a
constant
any
8
8is
self.
24still
The
forms.
book
of
has
Heinomen
been
studied
(Heinomen,
asubject
subject
2001)
byof
itself.
a The
The
book
of
Heinomen
(Heinomen,
2001)
3Heinomen
es
imply
aIthas
property
related
to
growth
24
forms.
forms.
It
has
Itunderlying
been
been
studied
studied
asand
as
aassubject
athe
by
by
itself.
itself.
The
book
book
of of
Heinomen
(Heinomen,
(Heinomen,
2001)
2001)
is is
a isa a
operty
of
the
spaces
not
of
the
Dirichlet
singularity
assumption
was
removed
yielding
the
following
result.
former
implies
the
existence
of
a
Borel
measure
which
is
9
Theorem
1.(Tasena,
2011)
Let
be
a
doubling
measure
on
a
metric
space
,
9
Theorem
1.(Tasena,
2011)
Let
be
a
doubling
related
25
good
versions
starting
of
doubling
point.
There
property,
are
two
one
closely
is
for
the
related
versions
of
doubling
property,
one
is
for
themeasure on a metric s
turm,
1996).
25
good
good
starting
starting
There
There
are
are
two
two
closely
closely
related
related
versions
of of
doubling
doubling
property,
property,
oneone
is is
forfor
thethe
15
15
15
itself.
The
bookpoint.
ofpoint.
Heinomen
(Heinomen,
2001)
is a versions
15
k
-skew
subsets
of
Euclidean
doubling
(Luukkainen
&
Saksman,
1998).
Therefore,
only
Examples
of
10
,
S
is
closed
and
null,
and
.
Assume
the
following
conditions
hold.
volume
space
measure
tobe
be
doubling,
ofdoubling
that
space
and
to
another
isfor
for
the
It
volume
measure
ofthat
that
space
to
bedoubling.
doubling.
10
Sspaces
isinclude
closed
and
null,
and
. Assume
the fo
Theorem
1.(Tasena,
2011)
Let
besubsets
adoubling.
doubling
measure
on
a metric
space
,subspaces,
26
space
space
to
to
be
doubling,
doubling,
and
another
another
is
is
for
the
the
volume
volume
measure
of
of
that
space
space
to
to
be
be
doubling.
It include
ItIt spaces
16
Examples
of
-skew
subsets
of
Euclidean
spaces
include
proper
subspaces,
spheres,
16
Examples
Examples
of
ofand
-skew
-skew
subsets
of
Euclidean
Euclidean
spaces
include
proper
proper
subspaces,
spheres,
spheres,
ye2616
related
versions
of
property,
one
isof
for
the
16
Examples
of,measure
-skew
subsets
of
Euclidean
spaces
proper
subspace
11
a)
The
set
S
satisfies
-skew
condition
i.e.
for
any
and
there
exists
such
that
27
while
turns
the
former
out
that
implies
the
latter
the
existence
implies
the
of
former
a
Borel
while
the
former
implies
the
existence
of
a
Borel
include
proper
subspaces,
spheres,
lattices
with
positive
doubling
property
of
volume
measures
will
be
discussed
11
a)
The
set
S satisfies
-skew
condition
i.e.
any
and
there
gthe
Property
27
turns
turns
out
that
that
the
the
latter
latter
implies
the
former
former
while
while
the
the
former
implies
implies
the
existence
existence
of of
ahold.
Borel
afor
Borel
,lattices
S out
is
closed
and1implies
null,
.former
Assume
the
following
conditions
17
with
positive
codimension,
cylinders,
etc.
Examples
of
remotely
constant
function
17
17
lattices
lattices
with
with
positive
positive
codimension,
codimension,
cylinders,
cylinders,
etc.
etc.
Examples
Examples
of
ofthe
remotely
remotely
constant
function
function
Using
this
fact
when
is
singleton,
Grigor’yan
and constant
Saloff-Coste
proved
doubling
proper
volume
measure
of
that
space
tothebe
doubling.
17and
lattices
with
positive
codimension,
cylinders,
etc.
Examples
of
constan
1 It
Using
this
fact
when
isonly
singleton,
Grigor’yan
andremotely
Saloff-Coste
pr
Saksman,
28
measure
1998).
which
Therefore,
is
doubling(Luukkainen
only
doubling
property
&
Saksman,
1998).
Therefore,
doubling
property
yer
of
the
underlying
spaces
and
not
of
the
Dirichlet
28
measure
measure
which
which
is
is
doubling(Luukkainen
doubling(Luukkainen
&
&
Saksman,
Saksman,
1998).
1998).
Therefore,
Therefore,
only
only
doubling
doubling
property
property
18
include
positive
rational
functions,
polynomial
growth
and
subpolynomial
growth
functions
such
18
18
include
positive
positive
rational
rational
functions,
functions,
polynomial
polynomial
growth
growth
and
and
subpolynomial
subpolynomial
growth
growth
functions
functions
such
such
codimension,
cylinders,
etc.
Examples
of
remotely
2 the
weighted
with
finite
unbounded
weight
functions.
Moshimi
and
Tesei(Moshim
while
thehere.
implies
existence
of
Borel
a)
Theinclude
set
Sformer
satisfies
-skew
condition
i.e.measures
forapositive
any
and
there
exists
such
that
18
include
rational
functions,
polynomial
growth
and
subpolynomial
growth
fun
2there
weighted
measures with finite
unbounded
weight functions. Mosh
12
b)
The
function
is
remotely
constant
i.e.
exists
a details,
constant
such
that
for
any
of
volume
measures
will
be
discussed
here.
f.29
The
book
of
Heinomen
(Heinomen,
2001)
is
a on
29
of of
volume
volume
measures
measures
will
will
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40
equivalent
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40
40
equivalent
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(Saloff-Coste,
2002).
2002).
24
Open
Problem.
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be
a
doubling
measure
on
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metric
space
Find
necessary
ghted
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singularity
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2477accessible
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1 35 Using
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isspace.
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associated
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. strongly
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isProblem.
said
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the property
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35
energy
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7 Recent
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asures
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bling
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Unlike
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bling
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ly
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e the
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2002).
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L
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)
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Under
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2011)
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14
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bling
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34 the
Definition
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be
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strongly
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34
Definition
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15
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.
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r any 2011)
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and
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36
inequality
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.
.
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Assume
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the
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following
conditions
conditions
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hold.
17
lattices
with
positive
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cylinders,
etc.
Examples
of
remotely
fully accessibility
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-skew
is closed
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null,Note
andthat
Assume
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and
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include
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Examples
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check
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ere exists
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any
37 whether
If such
,________________________________________________________
thenfor any
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and
17
lattices
with
positive
codimension,
cylinders,
etc.
Examples
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remotely
bling
if andinclude
only
exists
a constant
such
that
for
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and
ean
spaces
proper
subspaces,
spheres,
Examples
of if there
-skew
subsets
of Euclidean
spaces
proper
subspaces,
38
21
Oneinclude
interesting
question
is to spheres,
classify
functions
that
fully
accessibility
is equivalentweight
to 1‐skew
condition. for which th
38
15
18 etc.
include
positive
rational
functions,
polynomial
growth
subpolynomial
grow
etc.
Examples
ofsuch
remotely
constant
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ces
with
positive
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ofUnder
remotely
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function
39-skew
Under
doubling
property,
weak
Poincare
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andandPoincare
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22
measures
satisfy
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property.
39Examples
doubling
property,
weak
Poincare
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aabling
constant
constant
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for
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and
and
16
Examples
of
subsets
of weighted
Euclidean
spaces
include
proper
subspaces,
spheres,
if
and
only
if
there
exists
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constant
such
that
for
any
and
19
as
functions
with
logarithmic
growth
rates.
For
further
details,
examples,
as w
23
rowth
and
subpolynomial
growth
functions
such
ude
positive
rational
functions,
polynomial
growth
and
subpolynomial
growth
functions
such
40
equivalent
(Saloff-Coste,
2002).
40 etc.
equivalent
(Saloff-Coste,
2002).
17
with
positive
codimension,
cylinders,
Examples
of remotely
function doubling property, see
uclidean
spaces
include
proper
subspaces,
ples
oflattices
-skew
subsets
of
Euclidean
spaces
include
proper
20 spheres,
check
whether
weighted
ofconstant
this
type
24
Open
Problem.
Let
be
awell
doubling
measure
on a metric space
. F
further
details,
examples,
asgrowth
well
asrates.
methods
to
unctions
with
logarithmic
For
further
details,
examples,
as measure
as
methods
to satisfies
Santi
Tasena
/growth
Burapha
Sci.asubspaces,
J. 19 (2014)
1spheres,
: 169-177
174
18
include
positive
rational
functions,
polynomial
and
subpolynomial
growth
functions
such functions for which th
ers,
etc.
Examples
of remotely
constant
function
positive
codimension,
cylinders,
etc.
Examples
of
remotely
constant
function
21
One
interesting
question
is
to
classify
weight
25
sufficient
conditions
of
for
which
the
weighted
measure
tisfies
doubling
property,
see
(Tasena,
2011).
ck
a weighted
measure
of
this
type
satisfies
doubling
property,
see (Tasena,
2011).
clidean
clidean
spaces
include
include
proper
proper
subspaces,
subspaces,
spheres,
spheres,
19 whether
functions
with
logarithmic
growth
rates.
For
further
details,
examples,
as well
as methods to
ples
ofasspaces
-skew
subsets
of Euclidean
spaces
include
proper
subspaces,
spheres,
ial
growth
and
subpolynomial
growth
functions
such
ve
rational
functions,
polynomial
growth
and
subpolynomial
growth
functions
such
22 weight
weighted
measures
satisfy
doubling
property.
33
26
weight
functions
for
which
their
corresponding
One
interesting
question
is
to
classify
functions
for
which
their
corresponding
3property,
Note
Note
that
that
fully
fully
accessibility
accessibility
is
is
equivalent
equivalent
to
to
-skew
-skew
condition.
condition.
ers,
rs,
etc.
etc.
Examples
Examples
of
of
remotely
remotely
constant
constant
function
function
20
check
whether
a
weighted
measure
of
this
type
satisfies
doubling
see
(Tasena,
2011).
positive
codimension,
cylinders,
etc.
Examples
of
remotely
constant
function
For further details, examples, as well as methods
to
23
Note that fully accessibility
is equivalent to -skew condition.
17
is path-connected.
17
is19
path-connected.
Theorem 2.(Tasena, 2011) Let
be a strongly loca
18
18 with
strongly
local
regular
Dirichlet
form
on
7
is19path-connected.
20
with
associated
energy
be alocal
k-accessibl
Theorem 2.(Tasena, 2011) Let
be
a strongly
local regular
Dirichlet
form on
Theorem
2.(Tasena,
2011)
Let measure
be, a strongly
regular
n8
is said to satisfy weak (uniform) 19
Poincare
21
.
Define
20
with
associated
energy
measure
,
be
a
k-accessible
null
set,
,
and
20
with
associated
energy
measure
,
be
a
k-accessible
null se
9
Theorem
2.(Tasena, 2011) Let
be a strongly local regular Dirichlet form on
and
,
21
.
Define
21 a k-accessible
. Define
0
with associated energy measure ,
be
null set,
, and
for
any 22
for
any
. Then
1
. Define
23
regular
Dirichlet
form
,
called
the
weighted
Diri
Iffork =any1, then (E,D(E)) is said to satisfy
Then
extended toto aastrongly
local
regular
22
. Then
extended
strongly
22 (uniform)
ext
for any
. local
Then
sfy (uniform) Poincare inequality.
24
Moreover,
if
satisfies
Poincare
inequality,
23
regular
Dirichlet
form
,
called
the
weighted
Dirichlet
form
,
on
.
Dirichletextended
form
called
theweighted
weightedDirichlet for
23 . regular
form
, ,called
the
2
for anyPoincare inequality.
to a strongly
local
Then Dirichlet
25
is
a
nonincreasing
remotely
constant
function,
then satisfie
24
Moreover,
if
satisfies
Poincare
inequality,
satisfies
doubling
property,
and
24 weighted
Moreover,
if ,,on
satisfies
Poincare inequality,
Dirichlet
Underform
doubling
property,
weak
Poincare
Dirichlet form
form
3
regular
, called
the
on
.
weak
Poincare
inequality
and Poincare
inequality
are Dirichlet
26
and
only
if
the
measure
is
doubling.
25
is
a nonincreasing
constant
function,
satisfies
Poincare
inequality if
25
is a then
nonincreasing
remotely
constant
function,
sati
4
Moreover,
ifPoincareremotely
satisfies
Poincare
inequality,
doubling
property,
and
satisfies
Poincarethen
inequality,
inequality
and
inequality
are equivalent
(Saloff satisfies
Moreover,
if (E,D(E))
27
26
and
only
if
the
measure
is
doubling.
26 then
and only if the measure
is doubling.
5
is a nonincreasing
satisfies
Coste, 2002). remotely constant function,
ν satisfies
doublingPoincare
property,inequality
and a is a ifnonincreasing
28
Sturm’s Result
27
27
6
and only if the measure is doubling.
h 1996)
h
29
Sturm(Sturm,
1994,
1995b,
showed in a series
Grigor’yan and Saloff-Coste (Grigor’yan
& Saloff- Result
remotely constant function, then (ESturm’s
,D(E ))Result
satisfies
28
28 Sturm’s
y7is equivalent to -skew condition.
30 of Sturm(Sturm,
parabolic
Harnack
Poincare
inequality,
29
Sturm(Sturm,
1994, 1995b,
1996)
in
a seriesinequality
of articles
that
estimates,
Coste, 2005)
extended Poincare
inequality
that
Poincare
if1995b,
and Gaussian
only
if the
measure
is andofdoubli
29 to showed
1994,inequality,
1996)
showed
in aµ series
articl
8
Sturm’s
Result
31
30
parabolic
Harnack
inequality,
Poincare
inequality,
and
doubling
property
are
closely
related.
Harnack
inequality,
Poincare
inequality, be
anda doubling
balls, where
⊂30X is aparabolic
fully
remote
and S-anchored
doubling.
9
Sturm(Sturm,
1994, 1995b,
1996)S showed
in32
a series
of articles
that Gaussian
estimates,
Theorem
3.(Sturm,
1996)
Let
strongly prope
local
31
31
0
parabolic
Harnack
inequality,
Poincare
inequality,
and
doubling
property
are
closely
related.
33
Then
the
following
statements
are
equivalent.
accessible
set.
They
used
this
fact
to
prove
Poincare
32
Theorem 3.(Sturm, 1996) Let
a strongly
local regular
form on
. local regular
32 beTheorem
3.(Sturm,
1996)Dirichlet
Let
be a strongly
1
34
a)
The
Dirichlet
form
satisfies
parabolic
Harnack
8
8
33
Then
the
following
statements
are
equivalent.
inequality
on weighted
when
S is alocal
Then
theSturm’s
following
statementsform
are equivalent.
2
Theorem
3.(Sturm,
1996) LetDirichlet spaces
be 33
a strongly
regularResult
Dirichlet
on
.
35
b)
The
Dirichlet
form
satisfies
Gaussian
estimates
34 the
a)following
The Dirichlet
Harnackform
inequality.
34 parabolic
The Dirichlet
parabolic
Harnack
inequalit
singleton
andstatements
the form
geodesic
Xsatisfies
satisfies
aa)mild
3
Then
arespace
equivalent.
The Sturm
(Sturm,
1994, satisfies
1995b, 1996)
showed
in a inequality
36
c)
Dirichlet
form
satisfies
Poincare
35
b)
The
Dirichlet
form
satisfies
Gaussian
estimates.
35
b)Harnack
The
Dirichlet
form
satisfies Gaussian estimates.
4 and
a)
The
Dirichlet form
satisfies
parabolic
inequality.
nand
&&
Saloff-Coste,
Saloff-Coste,
2005)
2005)
extended
extended
Poincare
additional
assumption
called
relatively
connected
series
ofPoincare
articles
that
Gaussian
estimates,
parabolic
37
doubling
property.
36
c)
The
Dirichlet
form
satisfies
Poincare
inequality
and
the
volume
measure
satisfies
36
c)
The
Dirichlet
form
satisfies
Poincare
inequality and the
b) The
Dirichlet
form
satisfies
Gaussian
estimates.
f5of
remote
remote
andand
-anchored
-anchored
balls,
balls,
where
where
is
is
a fully
a fully
accessible
set.
set.
They
They
38accessible
annuli.
Their
weight
functions
include
strictly
positive
Harnack
inequality,
Poincare
inequality,
and
doubling
37
doubling
property.
37
doubling
6 Poincare
c)38
The Dirichlet
form
satisfies
Poincare
inequality
the volume
measure
ove
prove
Poincare
inequality
inequality
on on
weighted
weighted
Dirichlet
Dirichlet
spaces
spaces
when
when isproperty.
aisand
singleton
a singleton
39
The
fact 8that
a) and satisfies
b) are equivalent is perhaps
38singularity. property are closely related.
unbounded
real-valued
functions
with
no
7space
doubling
property.
40
equivalent
to
c).
Since
solutions
heatisare
equations
are inte
pace
satisfies
satisfies
a
mild
a
mild
additional
additional
assumption
assumption
called
called
relatively
relatively
connected
connected
39
The fact that a) and b) are 39
equivalent isThe
perhaps
less a)surprising
thanequivalent
thatofthey
fact that
and b) are
perhaps less
sur
8
Later,
Moshimi
and
Tesei(Moshimi
&
Tesei,
2007)
extended
Theorem
3.
(Sturm,
1996)
Let
(E,D(E))
be
a
strongly
41 functions
heat
kernels
the
initial
heat
profiles,
the are
behavior
of hea
ht
ight
functions
functions
include
include
strictly
strictly
positive
positive
unbounded
unbounded
real-valued
real-valued
functions
with
with
noand
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40
equivalent
to
c).
Since
solutions
of
heat
equations
are
integrals
the
multiplication
between
40
equivalent
to
c).
Since
of
heat
equations
integrals
of
yan
and
&
Saloff-Coste,
2005)
extended
Poincare
9
The fact and
thatSaloff-Costes’
a) and b) are
equivalent
is
perhaps less surprising than that they
are
2
Grigor’yan
work
to
include
weight
42
behavior
of
solutions
of
heat
equations.
On
the
contrary,
hea
Moshimi
,atMoshimi
and
and
Tesei(Moshimi
Tesei(Moshimi
&
&
Tesei,
Tesei,
2007)
2007)
extended
extended
Grigor’yan
Grigor’yan
and
and
SaloffSalofflocal
regular
Dirichlet
form
on
L
(X,
ν
)
.
Then
the
41
heat
kernels
and
the
initial
heat
profiles,
the
behavior
of
heat
kernels
will
directly
influence
the
41is a heat
andof
thethe
initial
heat
the behavior of heat kernels
remote andto c).
-anchored
balls, where
fully
accessible
set.
They
0 of equivalent
Since solutions
of heat equations
arekernels
integrals
multiplication
between
88 profiles,
functions
where
nclude
weight
weight
functions
functions
where
where 42 spaces
. when
Nevertheless,
.contrary,
Nevertheless,
the
the
42
behavior
of
solutions
of
heat
equations.
On
the
heat
kernels
can
be
viewed
solutions
following
statements
are
equivalent.
behavior
of
solutions
of
heat
equations.
On
the
contrary, heat kernels
olude
prove
Poincare
inequality
on
weighted
Dirichlet
is
a
singleton
1
heat kernels and the initial heat profiles, the behavior of heat kernels will directly influenceasthe
an
be
be
applied
applied
to
other
to
other
weight
weight
functions
functions
with
with
unique
unique
singularity.
singularity.
4
8
Nevertheless,
the
same
technique
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be
applied
to
(E,D(E))
satisfies
parabolic and relatively
a) The
Dirichlet
form
sic
a mild
additional
assumption
called relatively
connected
2 space
behaviorsatisfies
of solutions
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equations.
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heat
kernels
can In
be
viewed
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case
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, fully accessibility
Tasena(Tasena,
sena(Tasena,
2011)
extended
extended
these
these
results
toSaloff-Coste,
to
include
include
weight
weight
functions
functions
with
with
8results
gor’yan
and 2011)
& Saloff-Coste,
2005)
extended
Poincare
or’yanfunctions
and
&unbounded
2005)
extended
Poincare
4
accessibility.
The converse is not true, however.
weight
include
strictly
positive
real-valued
functions
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no
other weight functions with unique singularity.
Harnack
inequality.
In
case of singleton
fully
accessibility that
andthat
relatively
connected
annuli together
implies - and relatively connected
In
caseset.
of singleton
accessibility
es.
ities.
Heof
He
proved
Poincare
Poincare
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inequality
under
under
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assumptions
assumptions
thethe
weight
weight
that
ofproved
remote
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-anchored
balls,
where
isaaand
fullySaloff-Coste(Grigor’yan
accessible
They ,&fullySaloff-Coste,
oto
that
remote
and
-anchored
balls,
where
is
fully
accessible
set.
They
er,
Moshimi
and
Tesei(Moshimi
&
Tesei,
2007)
extended
Grigor’yan
and
Saloff1
Grigor’yan
2005) extended
accessibility.
The
converse
is
not
true,
however.
2011)
extended
these
42011, Tasena (Tasena,
The converse
is not
and Poincare
2005)
Poincare
b)extended
The accessibility.
Dirichlet
form
(E,D(E))
satisfies
Gaussian
eor’yan
the
form
form
,Inwhere
, In
where
is
the
the
singularity
singularity
setset
andand
is is
a when
nonincreasing
awhen
nonincreasing
case ofissingleton
,&
fullySaloff-Coste,
accessibility
and
relatively
connected
annuli
together
implies
- true, however.
act
to
prove
inequality
on
weighted
Dirichlet
spaces
is
a
singleton
ct
to
prove
Poincare
inequality
on
weighted
Dirichlet
spaces
is
a
singleton
off-Coste,
include
weight
functions
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. accessible
Nevertheless,
the
2balls,
inequality
to that
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remote
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2005)
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that
ofspace
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isadditional
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The
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issatisfy
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unction.
tcan
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The
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assumption
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relatively
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odesic
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be
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used
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Poincare
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ct
Poincare
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ility.
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This
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ais
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version
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mentioned
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previously.
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He
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Poincare
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under
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eir
weight
functions
include
strictly
positive
unbounded
real-valued
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with
no
c)
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form
(E,D(E))
satisfies
Poincare
weight
functions
include
strictly
positive
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real-valued
functions
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,ir
Tasena(Tasena,
2011)
extended
these
results
to
include
weight
functions
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4
and the
geodesic called
space relatively
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a mild additional assumption called relatively
richlet
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is aa mild
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assumption
.Later,
Later,space
Moshimi
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&
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2007)
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Grigor’yan
andthe
Saloffassumptions
that
the
weight
functions
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of
the
form
Moshimi
and
&
Tesei,
2007)
extended
Grigor’yan
and
Saloffν satisfies doubling
inequality
and
volume
measure
arities.
He
proved
Poincare
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under
the
assumptions
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the
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5
annuli.
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weight
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umption
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edweight
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said
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ir
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ork
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.
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.
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the
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where
S
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and
a
is
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property.
ounded
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nor
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some
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, the
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Later,
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and Saloff- & Tesei, 2007) extended Grigor’yan a
nique
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que
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ant
The
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Costes’
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The
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k07)
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2011,
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2011)
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011,
Tasena(Tasena,
2011)
extended
these
results
to
include
weight
functions
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ibility.
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is
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-skew
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mentioned
previously.
8the
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. can be applied
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. toNevertheless,
que
other
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surprising
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ngularities. set
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Poincare
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9
In
2011,
2011)
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these results to include weight func
4 extended these results to include weight functions
hclosed
unique
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011,
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2011)
with
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is10a stronger
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-accessible
ifanonincreasing
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are
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the
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and
is
a
ena,
,elts2011)
2011)
Let
Let
be
be
a
strongly
a
strongly
local
local
regular
regular
form
form
on
on
multiple
singularities.
He proved
Poincare
to include
functions
with
gularities.
Heweight
proved
Poincare
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under
the
that the
weightinequality under the assumptions that th
on
for
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and
for
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andmust
, the
setassumptions
onstant
function.
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singularity
setfunctions
must
also
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additional
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nstant
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also
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multiplication
between is
heatthekernels
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rgy
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measure
,
,
be
be
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k-accessible
a
k-accessible
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null
set,
set,
,
and
,
and
11
are
of
the
form
,
where
singularity
set initial
and is a noni
der
weightis the singularity set and is a nonincreasing
ecessibility.
ofthe
theassumptions
form
, the
where
4 44 This is athat
stronger
version
of
-skew
condition
mentioned
previously.
essibility.
This
is
a
stronger
version
of
-skew
condition
mentioned
previously.
Definition
8.
A
closed
subset
S
of
a
geodesic
space
(X,d)
e
12set remotely
constant
singularity
set of must
also satisfy
an additional as
heat The
profiles,
the behavior
heat kernels
will directly
ularity
set and The
is a singularity
nonincreasing
nstant
function.
must also
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an
additional
assumption
ed.
4
4
13
called
-accessibility.
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is
a
stronger
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-skew
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mentioned
previo
issubset
said
be
ifspace
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lso
an
additional
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essibility.
This
istoa of
stronger
version
-skew
mentioned
previously.
influence
the
of solutions of heat equations. On
8.AAsatisfy
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ofaka-accessible
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issaid
said
be
-accessible
itsatisfies
satisfies
8.
closed
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iscondition
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-accessible
ififitbehavior
14
. for
Then
Then local
extended
toset
a strongly
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asena,
2011)
Letsome kpreviously.
strongly
regular
Dirichlet
form
onlocal
for
’>and
kand
andbe
fora.any
any
and extended
, the
the
ndition
forsome
some
any
and
setto
dition
for
for
and
set
contrary,
heat
kernels
can be viewed
as solutions
of
15
Definition
8. ,A,the
closed
subset
ofifa itgeodesic
space
is said
to be -accessible
if
.
A
closed
subset
of
a
geodesic
space
is
said
to
be
-accessible
satisfies
form
, called
, called
weighted
weighted
Dirichlet
Dirichlet
form
form
,set,
on, on
. .
drm
energy measure
, thethe
be
a
k-accessible
null
,
and
heat equations
Dirac measures
16
-skew
condition
for some
and with
for any
andas initial ,heat
the profiles,
set
s, ifsaid to
-accessible
if itforsatisfies
dition
forbe
some
and
any
and
,doubling
thedoubling
set
satisfies
satisfies
Poincare
Poincare
inequality,
inequality,
satisfies
satisfies
property,
property,
andand
nected.
fine
ected.
is
path-connected.
, the set
soinequality
parabolic
Harnack
remotely
g remotely
constant
constant
function,
function,
then
then
satisfies
satisfies
Poincare
Poincare
inequality
if if inequality will force the heat kernels
17
is
path-connected.
(E,D(E))
be
a
strongly
Theorem
2.
(Tasena,
2011)
Let
ected.
2.(Tasena,
2011)
Let
be
a stronglylocal
localregular
regularDirichlet
Dirichlet
form
on estimates. The Poincare inequality and
Gaussian
(Tasena,
Let
be
form
on
easure
ure
is doubling.
is2011)
doubling.
18
. a strongly
Then2
extended
to to
a satisfy
strongly
local
iatedenergy
energylocal
measure
,
beaon
ak-accessible
k-accessible
nullset,
set,2011) Let
,and
and
regular ,Dirichlet
L (X,
ν) withnull
associated
ated
measure
be
, be
19
Theorem
a strongly local regular Dirichlet form on
________________________________________________________
et form 2011)
thebeform
weighted
Dirichlet
form
, on
. on
(Tasena,
Let , called
a strongly
local2.(Tasena,
regular
Dirichlet
form
Define
Sturm’s
Sturm’s
Result
Result
4
Define
energy
measure
⊂ inequality,
X
a k-accessible
set, measure
20
with
associated
energy
, and,S,and
be accessibility
a k-accessible
null connected
set,
al
regular
form
onΓ , S be
er,
ifenergyDirichlet
satisfies
satisfies
doubling
Inproperty,
case of singleton
fully
and relatively
ated
measure
, Poincare
a be
k-accessible
nullnull
set,
m,rm,
1994,
1994,
1995b,
1995b,
1996)
1996)
showed
showed
in21and
in
a series
a series
of of
articles
articles
that
that
Gaussian
Gaussian
estimates,
estimates,
.. Define
Define
annuli together
implies
le null
set, constant function, ,then
and
ing
remotely
satisfies Poincare
inequality
if k-accessibility. The converse is not true,
nequality,
kDefine
inequality,
Poincare
Poincare
inequality,
inequality,
andand
doubling
doubling
property
property
are
are
closely
closely
related.
related.
.
Then
extended
to
a
strongly
local
.
Then
extended
to
a
strongly
local
measure is doubling.
however.
richlet
form
,
called
the
weighted
Dirichlet
form
,
on
.
chlet
form
the
Dirichlet
form form
, on
. . .
22 weighted
for
any
. Then
extended to a stron
m,
1996)
1996)
LetLet
be,be
acalled
strongly
a strongly
local
regular
regular
Dirichlet
Dirichlet
form
on on
.local
Then
extended
to
a strongly
local
reover,ifif
satisfies
Poincare
inequality,
satisfies
doubling
property,
and
eover,
satisfies
Poincare
inequality,
satisfies
doubling
property,
and
Sturm’s
23Result
regularDirichlet
Dirichletform
form, on
, called the weighted Dirichlet form , on
.
extended
to
a strongly
local
gtatements
statements
areare
equivalent.
equivalent.
chlet
form
, called
the
weighted
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:
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175
creasing
reasing
remotely
constant
function,
then
satisfies
Poincare
inequality
if
easing
remotely
constant
function,
then
satisfies
Poincare
inequality
if
Sturm,
1994,
1995b,
1996)
showed
in
a
series
of
articles
that
Gaussian
estimates,
24Harnack
Moreover,
if doubling property,
satisfies Poincare
inequality, satisfies doubling prope
ichlet ifform , satisfies
onsatisfies
. Poincare
m
orm
parabolic
parabolic
Harnack
inequality.
inequality.
eover,
satisfies
inequality,
satisfies
and
themeasure
measure isPoincare
isdoubling.
doubling.
he
ack
inequality,
inequality,
and
doubling
property
are
closely
related.
25
is
a
nonincreasing
remotely
constant
function,
then
satisfies Poincare in
,m
satisfies
doubling
property,
and
orm remotely
satisfies
satisfies
Gaussian
Gaussian
estimates.
estimates.
easing
constant
function,
then
satisfies Poincare inequality if
26inequality
only
if the
the
measure
is doubling.
m
orm satisfies
satisfies
satisfies
Poincare
Poincare
inequality
and
the
volume
volume
measure
measure
satisfies
satisfies
Poincare
inequality
if and and
4
the doubling property, on the other hand, have almost
the weighted Dirichlet spaces satisfy the parabolic
99
nothing to do with heat equations. Yet these properties Harnack inequality. Even whether k-accessibility is the
directly
influence
both heat Harnack
kernels and
solutions of right condition to assume on singularity sets is unknown.
s asas initial
initial heat
heat profiles,
profiles, so
so parabolic
parabolicHarnack
inequality
inequality
9 9
heat equations.
AsPoincare
an example,
Tasena’s result
can be All these
require further investigation into the subject 9
Gaussian
Gaussian estimates.
estimates.
The
The
Poincare
inequality
inequality
and
and the
the
9
translated
asdo
follows.
so the author strongly believes this field will remain
have
ave almost
almostnothing
nothingtoto
do
with
withheat
heatequations.
equations. Yet
Yetthese
these
ac
Dirac
measures
measures
as1initial
as initial
heat
heat
profiles,
profiles,
sowith
so
Harnack
Harnack
kernels
kernels
and
andsolutions
solutions
heat
heat
equations.
equations.
As
Asparabolic
an
anexample,
example,
of4.ofof
heat
equations
Dirac
measures
as inequality
initial
heat
profiles,
Theorem
Let
(E,D(E))
be
aparabolic
strongly
local
regularinequality
fruitful in
years
to come.so parabolic Harnack inequality
rnels
s
to
satisfy
to
satisfy
Gaussian
Gaussian
estimates.
estimates.
The
The
Poincare
Poincare
inequality
inequality
and
and
the
the
ows.
ows.
2
will
kernels
to satisfy
Harnack
inequality
Dirichlet
form force
on L2the
(X, νheat
) with
associated
energyGaussian estimates. The Poincare inequality and the
other
he
other
hand,
hand,
havehave
almost
almost
nothing
nothing
to do
to with
do
heatother
heat
equations.
equations.
YetYet
these
these
3the
doubling
property,
onwith
the
hand, have
almost
nothing to do with heat equations. Yet these
inequality
and
measure
Γ and
,
S solutions
⊂on
X isof aheat
ρ -accessible
null
set,
y
y
local
local
regular
regular
Dirichlet
Dirichlet
form
form
on
with
with
associated
associated
Acknowledgement
e
ence
both
both
heat
heat
kernels
kernels
and
solutions
of
heat
equations.
equations.
As
As
an
example,
an
example,
4
properties directly influence both heat kernels and solutions of heat equations. As an example,
quations. Yet these
and
ble
ible
null
null
set,
, ,and
and
..
ranslated
lated
asanset,
follows.
as
follows.
The author gratefully thanks Professor Sompong
5
Tasena’s result
can be translated
as follows. s.
As
example,
cHarnack
Harnackinequality,
inequality,
and
and is
isaanonincreasing
nonincreasing
remotely
remotely
6 If (E,D(E))
satisfies
parabolic Harnack
inequality, Dhompongsa for several helpful comments and
be abestrongly
a strongly
local
regular
regular
Dirichlet
Dirichlet
form
on onbe
withwith
associated
associated
7 local
Theorem
4.
Let form
a strongly
local
regular Dirichlet form on
with associated
Dirichlet
Dirichlet
space
space
satisfies
satisfies
parabolic
parabolic
Harnack
Harnack
and null
a is a nonincreasing
remotely
constant
function,. .suggestion.
saaaconstant
isconstant
-accessible
a
-accessible
null
set,
set,
,
and
,
and
8 such
energy
measure
is a -accessible null set,
, and
.
with associated
such
that
that
for
for any
any ,
and
hand h
then
the
weighted
Dirichlet
space
(E
,D(E
)) satisfiesremotely
es
isfies
parabolic
parabolic
Harnack
Harnack
inequality,
inequality,
and
and
is
a
is
nonincreasing
a
nonincreasing
remotely
9
If
satisfies
parabolic
Harnack
inequality,
and
is
a
nonincreasing
remotely
.
parabolic
Harnack
inequality
if
and
only
if
there
the
weighted
weighted
Dirichlet
Dirichlet
space
spacefunction, then
satisfies
satisfies
parabolic
parabolic
Harnack
Harnack
10
constant
the
weighted
Dirichlet
space
satisfies parabolic Harnack
References
ncreasing
remotely
exists
a
constant
such
that
for
any
and
here
if
there
exists
exists
a
constant
a
constant
such
such
that
that
for
for
any
any
and
and
11
inequality if and only if there exists a constant
such that
for for
anythe fundamental
and
parabolic
Harnack
Conclusion
Conclusion
Aronson, D.G. (1967).
Bounds
. .and
12
oduces
roduces
heat
heat equations
equations
on
on general
general metric
metric spaces
spaces.using
using
solution of a parabolic equation. Bulletin of the
generalization
eneralization of
of an
an approach
approach using
using Sobolev
Sobolev spaces
spaces and
and
13
American Mathematical Society, 73, 890-896.
Conclusion
Conclusion
author
uthor also
alsodiscusses
discusses
when
whenheat
heatkernels
kernelswill
willhave
haveaanice
nice
14
Conclusion
Conclusion
Aronson,
(1968). Non-negative
solutionsspaces
of linearusing
the
author
author
introduces
introduces
heatheat
equations
equations
on aaon
general
general
metric
metric
spaces
spaces
using
usingD.G.
kernel.
ernel.
This
This
isis15
ofofcourse
course
by
byInno
nothis
means
means
complete
complete
list
list
ofof
article,
the author
introduces
heat
equations
on general metric
a using
In this
article,
the
author
introduces
heat
equations
h.
Thishowever,
This
is aisgeneralization
generalization
of an
of
approach
an approach.
approach
using
using
Sobolev
Sobolev
spaces
spaces
andand
parabolic
equations.using
Annali
della spaces
Scuola and
will,
will,
however,
provide
provide
an
an
introduction
introduction
toto
this
this
field.
field.
16
Dirichlet
form
This
is a generalization
of an approach
Sobolev
etric
spaces
unctions.
ions.Dirichlet
The
The
author
author
also
discusses
discusses
when
when
heat
heat
kernels
kernels
will
will
have
have
a nice
a nice
bed
rbed
Dirichlet
forms.
Here
Here
the
the
author
author
introduces
introduces
weighted
weighted
onforms.
general
metric
spaces
using
Dirichlet
form
approach.
17
weakly
differentiable
functions.
The
author
also
discusses
when diheat
will have
obolev
spaces
andalso
Normale
Superiore
Pisa,kernels
22(3), 607-694,
1968.a nice
ssical
classical
heat
heat
kernel.
kernel.
This
This
is
of
is
course
of
course
by
no
by
means
no
means
a
complete
a
complete
list
list
of
of
fls
Nash
Nash
(Nash,
(Nash,
1958),
1958),
Moser
Moser
(Moser,
(Moser,
1961,
1961,
1964,
1964,
1967),
1967),
18
behavior
similar
to
the
classical
heat
kernel.
This
is
of
course
by
no
means
a
complete
list of
will haveThis
a nice
is a generalization of an approach using Sobolev Aronson, D.G. (1971). Addendum: “non-negative solutions
hopes
es
that
that
this
this
will,
will,
however,
however,
provide
provide
an
introduction
an
introduction
to
this
to
this
field.
field.
971)
71)
can
canalso
alsobe
be
considered
results
results
ininauthor
this
thiscategory.
category.
Over
Over
19considered
references.
The
hopes that
this
will, however, provide an introduction to this field.
a complete
list
ofand
spaces
weakly
differentiable
functions.
The
author
weighted
of linear
parabolic
Scuola. Norm.
ays
loways
to
perturbed
to
perturbed
Dirichlet
Dirichlet
forms.
forms.
Here
Here
the
the
author
author
introduces
introduces
weighted
been
been
made,
made,
many
many
assumptions
assumptions
have
have
been
been
weaken.
weaken.
Yet,
Yet,
20
There
are
several
ways
to
perturbed
Dirichlet
forms.
Here equations”
the author(Ann.
introduces
weighted
this field.
also
discusses
when
heat
kernels
will
have
a
nice
behavior
Sup.
Pisa
(3)22(1968),
607-694).
Annali
della
Scuola
original
inal
work
work
of
Nash
of
Nash
(Nash,
(Nash,
1958),
1958),
Moser
Moser
(Moser,
(Moser,
1961,
1961,
1964,
1964,
1967),
1967),
For
For examples,
examples,
no one
one knows
knows
sufficient
sufficient
and
necessary
necessary
21no
Dirichlet
spaces.
Theand
original
work of Nash (Nash, 1958), Moser (Moser, 1961, 1964, 1967),
ntroduces
weighted
similar
tocan
the
classical
heat
kernel.
This
isinto
oftothis
course
by1971)
no can
67,
1967,
1968,
1968,
1971)
1971)
can
also
also
be considered
beDirichlet
considered
results
results
in
this
category.
category.
Over
be
be
doubling,
doubling,
or
or
for
weighted
weighted
Dirichlet
spaces
spaces
satisfy
satisfy
22for
and
Aronson
(Aronson,
1967,
1968,
also
be considered
in this
category.
Over
Normale
Superiore diresults
Pisa, 25(3),
221-228,
1971. Over
1961,
1964,
1967),
ments
vements
have
have
been
been
made,
made,
many
many
assumptions
assumptions
have
have
been
been
weaken.
weaken.
Yet,
Yet,
ngularity
ngularity
set
setmeans
isisOver
k-accessible,
k-accessible,
no
nomany
one
one
knows
knows
the
the
necessary
23
the years,
improvements
have
been Bouleau,
made, many
assumptions
been Forms
weaken.
a complete
list
of
references.
Thenecessary
author
hopes
his
category.
N. & Hirsch,
F. (1991).have
Dirichlet
andYet,
nnanswered.
unanswered.
For
For
examples,
examples,
no
no
one
one
knows
knows
sufficient
sufficient
and
and
necessary
necessary
functions
functions
for
for
which
which
the
the
weighted
weighted
Dirichlet
Dirichlet
spaces
spaces
satisfy
satisfy
24
many
questions
remain
unanswered.
For
examples,
no
one
knows
sufficient
and
necessary
been
weaken.
Yet,
that this will, however, provide an introduction to this Analysis on Wiener Space. Berlin; New York: Walter
measures
be
to 25
doubling,
be doubling,
or
for
or right
for
weighted
weighted
Dirichlet
Dirichlet
spaces
spaces
satisfy
to satisfyor for weighted Dirichlet spaces to satisfy
ndient
neasures
whether
whether
k-accessibility
k-accessibility
is
isthe
the
right
condition
condition
totoassume
assume
conditions
for
weighted
measures
to
beto doubling,
and tonecessary
field.
de is
Gruyter.
ven
when
when
the to
the
singularity
singularity
set is
setk-accessible,
isinequality.
k-accessible,
no
one
no when
one
knows
knows
the
the
necessary
necessary
hese
hese
require
require
further
further
investigation
investigation
into
into the
the
subject
subject
so
so
the
26
Poincare
Even
the
singularity
set
k-accessible, no one knows the necessary
et
spaces
satisfy
There
are
several
ways
to
perturbed
Dirichlet
Chen,
Z.Q.
&
Fukushima,
M.(2012).Dirichlet
Symmetric
Markov
fnows
semain
the
of the
weight
weight
functions
functions
for
for
which
which
the
the
weighted
weighted
Dirichlet
Dirichlet
spaces
spaces
satisfy
satisfy
emain
fruitful
fruitful
in
in
years
years
to
to
come.
come.
27
and
sufficient
conditions
of
the
weight
functions
for
which
the weighted
spaces
satisfy
the necessary
nequality.
uality.spaces
Even
Even
whether
k-accessibility
is the
is inequality.
the
right
right
condition
condition
assume
to assume
forms.
Here
the
author Harnack
introduces
weighted
Dirichlet
28 whether
thek-accessibility
parabolic
Eventowhether
k-accessibility
is the right
to assume
chlet
satisfy
Processes,
Time Change,
and condition
Boundary Theory.
nknown.
own.
All
All
these
these
require
require
further
further
investigation
investigation
into
into
the
the
subject
subject
so
the
so
the
Acknowledgement
Acknowledgement
29
on
singularity
sets
is
unknown.
All
these
require
further
investigation
into
the
subject
so the
ondition to assume
spaces. The original work of Nash (Nash, 1958), Moser Princeton University Press.
sProfessor
this
field
field
will
will
remain
remain
fruitful
fruitful
in
years
in
years
to
come.
to
come.
Professor
Sompong
Dhompongsa
Dhompongsa
for
forAronson
several
several
helpful
helpful
author
strongly
believes
this(Aronson,
field
will
remain fruitful in years to come.
the
subjectSompong
so30the1961,
(Moser,
1964,
1967), and
1967,
Grigor’yan, A. (2009). Heat Kernel and Analysis on
31
1968,
1971)
can
also
be
considered
results
in
this
category.
Manifolds, volume 47 of AMS.IP Studies in
Acknowledgement
Acknowledgement
32
Acknowledgement
Over
the
years,
many
improvements
have
been
made,
ully
tefully
thanks
thanks
Professor
Sompong
Sompong
Dhompongsa
Dhompongsa
for
several
several
helpful
helpful
Advanced
Mathematics.
American
References
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33Professor
The author
gratefullyforthanks
Professor
Sompong
Dhompongsa
for Mathematical
several helpful
on.
fundamental
fundamental
solution
solution
of
of
a
a
parabolic
parabolic
equation.
equation.
Bulletin
Bulletin
of
of
34 assumptions
comments and
many
havesuggestion.
been weaken. Yet, many Society. International Press.
or several helpful
y,y,73,
73,890–896.
890–896.
35
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ichlet
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พฤติกรรมแบบเกาส์ของเคอร์เนลความร้อน Gaussian

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