Ac a ma hema ica sinica, English Se ies, Se ies 18 (2002) 1, 1-20.
L1→LqPOINCAR´
E INEQUALITIES FOR
0< q < 1IMPLY REPRESENTATION FORMULAS
Guozhen Lu(∗)
Depa men o Ma hema ics
Wayne S a e Uni e si y
De oi , MI 48202, USA
E-mail: [email p o ec ed]ayne.edu
Ca los P´
e ez(∗)(∗∗)
Depa amen o de Ma em´a icas
Uni e sidad Au ´onoma de Mad id
28049 Mad id, Spain
E-mail: ca los.p[email p o ec ed]
Dedica ed o Dick Wheeden on he occasion o his 60 h bi hday wi h app ecia ion and admi a ion
1991 Ma hema ics Subjec Classi ica ion. 46E35, 41A10, 22E25.
Key wo ds and ph ases. Sobole spaces, ep esen a ion o mulas, high o de de i a i es, ec o
ields, me ic spaces, polynomials, doubling measu es, Poinca ´e inequali ies.
(*) The i s au ho was suppo ed pa ly by he U.S. Na ional Science Founda ion G an Nos.
DMS96-22996 and DMS99-70352. The second au ho was suppo ed pa ly by DGICYT g an PB940192,
Spain. Bo h au ho s we e suppo ed pa ly by NATO collabo a i e esea ch g an 972144.
(**) The main pa o his pape was comple ed du ing he second au ho ’s isi a W igh S a e
Uni e si y, Ohio in June, 1999. He wishes o hank he Depa men o Ma hema ics and S a is ics a
W igh S a e Uni e si y o i s hospi ali y and inancial suppo .
Typese by A
MS-T
EX
1
2 G. LU AND C. P´
EREZ
Abs ac . Gi en wo doubling measu es µand νin a me ic space (S, ρ) o homogeneous
ype and le B0⊂ S be a gi en ball. I has been a well-known esul by now (see [FLW],
[FW], [LW1], [LW2]) ha he alidi y o an L1→L1Poinca ´e inequali y o he ollowing
o m: ZB
| − B|dν ≤c (B)ZB
gdµ
o all me ic balls B⊂B0⊂ S implies a a ian o ep esen a ion o mula o ac ional
in e g al ype: o ν-a.e. x∈B0,
| (x)− B0| ≤ CZB0
g(y)ρ(x, y)
µ(B(x, ρ(x, y))) dµ(y) + C (B0)
µ(B0)ZB0
g(y)dµ(y).
One o he main esul s o his pape shows ha an L1 o LqPoinca ´e inequali y o some
0< q < 1, i.e.,
„ZB
| − B|qdν«1/q
≤c (B)ZB
gdµ
o all me ic balls B⊂B0will su ice o imply he abo e ep esen a ion o mula. As an
immedia e co olla y, we can show ha he weak ype condi ion
sup
λ>0
λν ({x∈B:| (x)− B|> λ})
ν(B)≤C (B)ZB
gdµ
also implies he same o mula.
Analogous heo ems ela ed o high o de Poinca ´e inequali ies and Sobole spaces in
me ic spaces a e also p o ed.
§1. In oduc ion
I is known ha L1→L1Poinca ´e inequali ies a e equi alen o he ac ional
in eg al es ima es in gene al me ic spaces o homogeneous ype (see [FLW2], [FW],
[LW1-2]). A na u al ques ion hus a ises: Is he L1→L1Poinca ´e inequali y he leas
we need o s a in o de o de i e such ep esen a ion o mulas? In his pape , we
s udy his issue and weaken he hypo hesis ha an L1→L1Poinca ´e inequali y has o
hold o ob ain any kind o ep esen a ion o mulas o ac ional ype. Mo e p ecisely,
we will show ha an L1→LqPoinca ´e inequali y o some 0 < q < 1 will su ice o
de i e he poin wise ac ional es ima es. On he o he hand i is well known ha he
ollowing Kolmogo o s inequali y holds o 0 < q < 1, any nonnega i e unc ion gand
a bi a y measu able se Ewi h ini e measu e
(1.1) µ1
µ(E)ZE
g(x)qdµ¶1/q
≤cq|g|L1,∞(E,µ).
See [GCRdF] p. 485 o ins ance. We will be using he ollowing no a ion o he local
a e age Ma cinkiewicz quasi-no m
|g|L1,∞(E,µ)= sup
λ>0
λµ({x∈E:| (x)|> λ})
µ(E).
Hence as an in e es ing co olla y o ou main esul , we p o e ha weak ype L1,∞→L1
Poinca ´e inequali y is su icien o imply ac ional ep esen a ion o mulas.
L1→LqPOINCAR´
E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 3
Mo e p ecisely, gi en wo doubling measu es µand νin a me ic space (S, ρ) o
homogeneous ype and le B0⊂ S be a gi en ball. I has been a well-known esul
by now (see [FLW2], [FW], [LW1], [LW2]) ha he alidi y o an L1→L1Poinca ´e
inequali y o he ollowing o m:
(1.2) ZB
| − B|dν ≤c (B)ZB
g dµ
o all me ic balls B⊂B0⊂ S implies a a ian o ep esen a ion o mula o ac ional
in eg al ype: o ν-a.e. x∈B0,
| (x)− B0| ≤ CZB0
g(y)ρ(x, y)
µ(B(x, ρ(x, y))) dµ(y) + C (B0)
µ(B0)ZB0
g(y)dµ(y).
As usual we use he ollowing no a ion o he a e age o o e a ball B, B=
1
µ(B)RB dµ
Ou i s main esul o his pape demons a es ha an L1 o LqPoinca ´e inequali y
o some 0 < q < 1, i.e.,
µZB
| − B|qdν¶1/q
≤c (B)ZB
g dµ
o all me ic balls B⊂B0will su ice o imply he abo e ep esen a ion o mula. As a
by-p oduc o his, we de i e ou second main esul ha he weak ype condi ion
| − B|L1,∞(B,ν)≤C (B)ZB
g dµ
also implies he same poin wise es ima es.
We no e ha , simila o wha was i s shown in [LW1] and hen in [LW2], he
in eg als on he igh hand side is on he same ball B0, a he han on he enla ged ball
(see [FLW2], [FW]). We make emphasis on he ac ha he only assump ion we make
on he measu e µis he doubling p ope y. Indeed, i has ecen ly been shown in [LW2]
ha he e is no need o equi e addi ional assump ions o e e se doubling o o de 1+²
o 1 (see [FLW2], [FW]). Howe e , we need o add he second e m on he igh hand
side, which is no ha m ul a all as a as he Poinca ´e ype es ima es conce ned. I we
also assume ha he measu e µis o e e se doubling o de 1, hen his second e m can
be d opped (see also [FW] and [LW2]). We men ion ha he au ho s in [HK2] de i ed
independen ly om [LW2] a o mula wi hou he second e m wi hou he assump ion
ha µis doubling, bu wi h B0 eplaced by 1
2B0. I seems ha he passage om 1
2B0
o B0would also esul in he second e m in he o mula.
As applica ions, we p o ide weake , bu equi alen , de ini ions o Sobole spaces o
i s o de in me ic spaces han hose de ined in [H], and u he exploi ed in [FLW2],
[FHK] and [LW2]. The implica ions o L1→LqPoinca ´e inequali ies o high o de
o ep esen a ion o mulas also hold and imp o e hose in [LW2]. These also p o ide
us wi h weake , and also equi alen , de ini ions o high o de Sobole spaces in me ic
spaces de ined in [LLW1].
4 G. LU AND C. P´
EREZ
The me hods used in his pape a e ex ensions o se e al echniques adap ed om
[FLW2], [FW], [LW1], [LW2] and [LLW1]. In pa icula , we will use simila ideas om
[LW2]. Howe e , ou case is conce ned wi h he si ua ion q < 1, and he e a e some
sub le ies we ha e o o e come. Some inequali ies which hold o q≥1 ail o be ue
o q < 1. Thus, we ha e o p oceed wi h cau ion.
We ema k in passing ha he e has been ex ensi e esea ch o p o ing Lp→Lq
Poinca ´e inequali ies, i a ce ain ype o Lp→LpPoinca ´e inequali y is al eady known
o exis in he gi en se ing, see [SC], [HK1-2], [BM], [MSC], [GN], [BCSC], [FPW],
[MP1-2], [OP]. This is he so-called sel -imp o ing p ope y, which can be used o p o e
Poinca ´e inequali ies wi hou using ep esen a ion o mula. Thus, by combining wi h
Je ison’s esul o Poinca ´e inequali ies wi h q=p o H¨o mande ec o ields [J], his
a gumen will ecap u e he sha p Poinca ´e inequali ies o H¨o mande ec o ields i s
p o ed in [L2] ( o p > 1) and [FLW1] ( o p= 1) by using ep esen a ion o mulas.
Di ec p oo s o ep esen a ion o mulas o H¨o mande ec o ields o G ushin ec o
ields ha e been gi en in [F], [FL], [FSe], [L1], [FLW1], [FGW], [CDG], [LM].
Fu he mo e, i is shown in he pape s [FPW], [MP1-2] and [OP] ha he Sobole -
Poinca ´e inequali ies a e special cases o a mo e gene al heo y ha includes, o in-
s ance, he classical heo em o John-Ni enbe g as well as he T udinge inequali y. The
idea he e is o eplace he exp ession on he igh hand side o (1.2) by a mo e gen-
e al “ unc ional” a(B) and o use he Calde on-Zygmund heo y, unde a ce ain mild
geome ic condi ion on a(see [P] o a su ey).
We men ion ha he sel -imp o ing p ope y by assuming he ini ial inequali y
µZB
| − B|qdν¶1/q
≤a(B)
o some 0 < q < 1 and some quan i y a(B) o hold has also been es ablished ecen ly
in [FLPW].
To make ou pape sel -con ained, and o he sake o cla i y o ou p esen a ion, we
ha e decided o ea he case o i s o de Poinca ´e inequali ies sepa a ely om he
ones o high o de . The plan o he pape is as ollows. In sec ion 2, we p o e ou esul s
o he i s o de in gene al me ic spaces. Sec ion 3 con ains new de ini ions o Sobole
spaces o i s o de in me ic spaces. Sec ion 4 deals wi h he implica ion o high o de
Poinca ´e inequali ies o ep esen a ion o mulas in me ic spaces and p o ides wi h new
de ini ions o Sobole spaces o any high o de in me ic spaces.
Acknowledgemen This wo k is an ou g ow h o join wo k wi h B uno F anchi
and Richa d Wheeden [FLW], [FW], [LW1], [LW2] and [LLW1]. We would like o
acknowledge he impo an con ibu ions hey ha e made in his di ec ion.
§2 Rep esen a ion o mulas o i s o de in me ic spaces
We begin wi h he de ini ion o “weak Boman chain domain” de ined in [LW2].
Boman chain domains in Euclidean spaces we e in oduced by Boman in his unpublished
wo k [Bom] and used o p o e Poinca ´e inequali ies on such domains (see [Boj], [Ch],
[IN]). Such a no ion in me ic space seems o be i s used in [FGW] and [L2] by sligh ly
modi ying he de ini ion in Euclidean spaces.
L1→LqPOINCAR´
E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 5
De ini ion 2.1 [LW2]. A domain (i.e., an open connec ed se ) Ωin Sis said o sa is y
he Boman chain condi ion o ype σ, M, o o be a membe o F(σ, M), i he e exis
cons an s σ > 1,M > 0, and a amily Fo me ic balls B⊂Ωsuch ha
(1) Ω = SB∈F B
(2) PB∈F χσB(x)≤M χΩ(x) o all x∈ S
(3) The e is a “cen al ball” B0∈ F such ha o each ball B∈ F, he e is a
posi i e in ege k=k(B)and a chain o balls {Bj}k
j=0 o which Bk=Band
each BjTBj+1 con ains a ball Djwi h BjSBj+1 ⊂MDj.
(4) B⊂MBj o all j= 0, . . . , k(B)
I we eplace he hypo hesis ha σ > 1by σ= 1, we say ha Ωsa is ies he weak
Boman chain condi ion.
We do no know i his weake de ini ion can ac ually be equi alen o he “Boman
chain domain”, whe e τhas o be aken bigge han 1. I will also be in e es ing
o know i he class o weake Boman chain domains is s ic ly la ge han he Boman
chain domains. We men ion ha Boman domain is equi alen o John domain as shown
independen ly in [BKL] and [GN].
We now s a e he ollowing ou hypo heses ha a e modi ica ions o hose gi en in
[LW1] and [LW2]. The c ucial di e ence is ha we ha e eplaced (H1) he e by ou
L1→LqPoinca ´e inequali y o some 0 < q < 1, a he han he L1→L1inequali y.
We no e ha no all ou hypo heses a e needed in e e y heo em. As always, (S, ρ) is
a me ic space. Le µand νbe doubling measu es wi h espec o me ic balls, and le
Ω be a domain in S.
(H1) is a unc ion sa is ying L1 o LqPoinca ´e inequali y o some 0 < q < 1, i.e.,
µZB
| − B|qdν¶1/q
≤c (B)ZB
gdµ
o me ic balls B⊂Ω.
(H2) The measu e µin (H1) sa is ies a e e se doubling condi ion o o de 1, i.e.,
he e is a cons an C > 0 such ha i Band ˜
Ba e balls wi h cen e s in Ω and wi h
B⊂˜
B, hen
µ(˜
B)≥CÃ (˜
B)
(B)!µ(B).
(H3) (S, ρ) has he segmen (o geodesic) p ope y ha o each pai o poin s x, y ∈
S, he e is a con inuous cu e γconnec ing xand ysuch ha ρ(γ( ), γ(s)) = | −s|.
(H4) Ω is a weak Boman chain domain.
The main esul s o his sec ion a e imp o emen s o hose in [LW2] whe e L1→L1
Poinca ´e inequali ies ha e o be assumed.
Rema k. Since weak L1implies locally s ong L1 o 0 < q < 1 as men ioned in he
in oduc ion, hus ou heo ems below s ill emain o be ue i we eplace (H1) abo e
by

6 G. LU AND C. P´
EREZ
(WH1) is a unc ion sa is ying weak L1 o L1Poinca ´e inequali y, i.e.,
| − B|L1,∞(B,ν)≤C (B)ZB
g dµ
o me ic balls B⊂Ω.
F om Kolmogo o ’s inequali y (1.1) we see ha (WH1) implies (H1) o all 0 < q < 1.
Theo em 2.2. Le ν, µ be doubling measu es on a me ic space (S, ρ). Le B0be a ball
and suppose ha (H1) and (H3) hold wi h Ω = B0and B=RB (y)dν(y). Then o
ν-a.e. x∈B0,
| (x)− B0|
≤CZB0
g(y)ρ(x, y)
µ(B(x, ρ(x, y))) dµ(y) + C (B0)
µ(B0)ZB0
g(y)dµ(y),
whe e Cdepends only on ν, µ and he cons an s in (H1).
I in addi ion we impose he e e se doubling condi ion (H2) in Theo em 2.2, hen
we ha e
Theo em 2.3. Le ν, µ be doubling measu es on a me ic space (S, ρ). Le B0be a ball
and suppose ha (H1), (H2) and (H3) hold wi h Ω = B0. Then o ν-a.e. x∈B0,
| (x)− B0| ≤ CZB0
g(y)ρ(x, y)
µ(B(x, ρ(x, y))) dµ(y),
whe e Cdepends only on ν, µ and he cons an s in (H1), (H2).
The nex heo em is a gene aliza ion o Theo em 2.3 o any weak Boman chain
domain Ω.
Theo em 2.4. Suppose ha νand µa e doubling measu es on a me ic space (S, ρ)
and ha hypo heses (H1)–(H4) hold o a domain Ω⊂ S. Then o ν-a.e. x∈Ω,
| (x)− B0| ≤ CZΩ
g(y)ρ(x, y)
µ(B(x, ρ(x, y))) dµ(y),
whe e B0is he cen al ball in Ω, , g, ν and µin (H1), and Cdepends only on ν, µ and
he cons an s in (H1), (H2) and (H4).
As is well-known, unde he segmen hypo hesis (H3), any me ic ball is a Boman
chain domain (see [FGW], [L2]), and hus Theo em 2.3 is a special case o Theo em 2.4
.
The p oo o Theo em 2.2 elies on he cons uc ion o he ollowing chain o me ic
balls gi en in [LW2], assuming he segmen hypo hesis (H3). A simila cons uc ion was
gi en in [FW], bu he ollowing one enables us o selec all balls in he chain lying inside
en i ely he gi en ball B0. The chain o balls will allow us o p o e he ep esen a ion
o mulas on he same ball on bo h sides di ec ly (see [LW2]), a he han using he
o mula on he enla ged ball o ge he co esponding one on he same ball (see [LW1]).
A somewha di e en chain o ini e leng h is gi en independen ly in [HK2].
L1→LqPOINCAR´
E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 7
Theo em 2.5 [LW2]. Le (S, ρ)be a me ic space in which he segmen p ope y (H3)
holds. Le B0be a ball in S. Gi en x∈B0, he e exis s a chain {Bk}k≥1o balls wi h
he ollowing p ope ies:
(1) Bk⊂B0and ρ(Bk, x)→0as k→ ∞.
(2) (B1)≈ (B0)and (Bk)→0as k→ ∞.
(3) I y∈Bk, hen ρ(y, x)≈ (Bk).
(4) BkTBk−1con ains a ball Skwi h (Sk)≈ (Bk)≈ (Bk+1)≈2−k (B0).
(5) I j < k, hen Bk⊂cBj.
(6) {Bk}k≥1has bounded o e laps, i.e., PkχBk(y)≤c o all y.
The cons an s o equi alence in (2), (3) and (4) and he cons an s cin (5) and (6)
a e independen o x, k, j and B0, bu he chain {Bk}depends on x.
The ollowing ema k is in o de . The a gumen gi en he e is simila o he p oo
o Theo em A in [LW2]. Howe e , since ou case is o q < 1 and hen he Minkowski’s
inequali y ails. Thus, ou si ua ion becomes mo e delica e han he case o q= 1. In
pa icula , we will use he inequali y
(2.6) µZE
( +g)qdν¶1/q
≤2q"µZE
q¶1/q
+µZE
gq¶1/q#.
Howe e , his inequali y does no hold when we ha e in ini ely many e ms in he
in eg and unlike he case o q≥1, namely, we do no ha e
ÃZEÃ∞
X
i=1
i!q
dν!1/q
≤C(q)
∞
X
i=1 µZE
q
idν¶1/q
.
The e o e we ha e o p oceed wi h cau ion, see he es ima e o I2below.
P oo o Theo em 2.2. We will use Theo em 2.5 o p o e Theo em 2.2. Le B0be a
ball in Sand suppose ha (H1) and he segmen p ope y (H3) hold o B0. Gi en
x∈B0, le {Bk}k≥1be a sequence o balls wi h he p ope ies gua an eed by Theo em
2.5. Then
(2.7) | (x)− B0| ≤ | (x)− B1|+| B1− B0|.
Fo he second e m on he igh in (2.7), we ge o ν−a.e. x∈B0 ha
| B1− B0|=µZB1
| B1− B0|qdν¶1/q
≤CµZB1
| (y)− B1|qdν(y)¶1/q
+CµZB1
| (y)− B0|qdν(y)¶1/q
≤CµZB1
| (y)− B1|qdν(y)¶1/q
+CµZB0
| (y)− B0|qdν(y)¶1/q
since ν(B1)≈ν(B0) and νis doubling
≤C (B1)
µ(B1)ZB1
g dµ +C (B0)
µ(B0)ZB0
g dµ by he Poinca ´e inequali y (H1)
≤C (B0)
µ(B0)ZB0
g dµ
8 G. LU AND C. P´
EREZ
since B1⊂B0, (B1)≈ (B0) and µ(B1)≈µ(B0).
Assuming as we may ha xis a Lebesgue poin o bo h | − B1|qand gwi h
espec o νand using p ope ies (1)–(3) om Theo em 2.5 and he inequali y (2.6), we
ha e o he i s e m on he igh in (2.7) ha
| (x)− B1|= lim
k→∞ µZBk
| (y)− B1|qdν(y)¶1/q
≤2qlim sup
k→∞ µZBk
| (y)− Bk|qdν(y)¶1/q
+ 2qlim sup
k→∞ µZBk
| Bk− B1|qdν(y)¶1/q
=I1+I2,
whe e I1and I2a e de ined by he las equali y.
I is easy o show ha I1= 0 o e e y Lebesgue poin xo g. This can be seen by
he Poinca ´e inequali y (H1):
I1= lim sup
k→∞ µZBk
| (y)− Bk|qdν(y)¶1/q
≤Clim sup
k→∞
(Bk)
µ(Bk)ZBk
g(y)dµ(y) = 0 ·g(x) = 0.
We now es ima e I2. By obse ing ha Bj+1 − Bjis a cons an unc ion, we ha e
I2= lim sup
k→∞
| Bk− B1|
≤lim sup
k→∞
k−1
X
j=1
| Bj+1 − Bj|
= lim sup
k→∞
k−1
X
j=1 ÃZSj
| Bj+1 − Bj|qdν!1/q
≤2q
∞
X
j=1 ÃZSj
| Bj+1 − |qdν!1/q
+ 2q
∞
X
j=1 ÃZSj
| Bj− |qdν!1/q
≤2q
∞
X
j=1 ÃZBj+1
| Bj+1 − |qdν!1/q
+ 2q
∞
X
j=1 ÃZBj
| Bj− |qdν!1/q
since Sj⊂Bj∩Bj+1 and ν(Sj)≈ν(Bj)≈ν(Bj+1) by Theo em 2.5. Combining
es ima es and applying (H1) o he e ms o each o he las wo sums, we ob ain
I2≤C
∞
X
j=1
(Bj)ZBj
g(y)dµ(y).
L1→LqPOINCAR´
E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 9
Now, as a guing in [LW2], i y∈Bj, hen
(Bj)
µ(Bj)≈ρ(x, y)
µ(B(y, ρ(x, y))) ≈ρ(x, y)
µ(B(x, ρ(x, y)))
by pa (3) o Theo em 2.5 and he ac ha µis a doubling measu e. Thus we ob ain
I2≤C
∞
X
j=1 ZBj
g(y)ρ(x, y)
µ(B(x, ρ(x, y))) dµ(y)
≤CZB0
g(y)ρ(x, y)
µ(B(x, ρ(x, y))) dµ(y)
by p ope ies (6) and (1) o Theo em 2.5. This comple es he p oo o Theo em 2.2 by
combining es ima es o I1and I2.
P oo o Theo em 2.3. By obse ing ha i x, y ∈B0, hen ρ(x, y)≤2 (B0) and
consequen ly by (H2), we ha e
(B0)
µ(B0)≤Cρ(x, y)
µ(B(x, ρ(x, y))) i x, y ∈B0.
Thus, he second e m on he igh in he conclusion o Theo em 2.2 is bounded by he
i s e m.
P oo o Theo em 2.4. Le x∈Ω. By he de ini ion o weak Boman chain domain, we
may selec B∗wi h x∈B∗and a chain {Bj}k
j=0 connec ing B∗=Bk o he cen al
ball B0. We ha e
(2.8) | (x)− B0| ≤ | (x)− B∗|+| B∗− B0|.
Fo he i s e m on he igh side o (2.8), we ha e by Theo em B ha
| (x)− B∗| ≤ CZB∗
g(y)ρ(x, y)
µ(B(x, ρ(x, y))) dµ(y).
This holds o ν−a.e. poin o B∗, and we may assume i holds o ou ixed xby ini ially
excluding om Ω he se o measu e ze o o med by aking he union o he excep ional
se s o measu e ze o in each Boman ball. Since B∗⊂Ω, we ob ain he desi ed es ima e
| (x)− B∗| ≤ CZΩ
g(y)ρ(x, y)
µ(B(x, ρ(x, y))) dµ(y).
Thus we only need o es ima e | B∗− B0|.By using he chain {Bj}connec ing B0and
Bk=B∗, we ha e
| B∗− B0| ≤
k
X
j=1
| Bj− Bj−1|.
16 G. LU AND C. P´
EREZ
by pa (3) o Theo em 2.5 and he ac ha µis a doubling measu e. Thus, we ge
I2≤C
∞
X
j=1 ZBj
g(y)ρ(x, y)m
µ(B(x, ρ(x, y))) dµ(y)
≤CZB0
g(y)ρ(x, y)m
µ(B(x, ρ(x, y))) dµ(y)
by p ope ies (6) and (1) o Theo em 2.5. The p oo o Theo em 2.2 now is comple e.
Rema k. We omi he p oo o Theo em 4.3 since is simila o ha o Theo em 2.3 by
using (A2) ins ead o (H2).
P oo o Theo em 4.4. Le x∈Ω. By he de ini ion o weak Boman chain domain, we
may selec B∗wi h x∈B∗and a chain {Bj}k
j=0 connec ing B∗=Bk o he cen al
ball B0. We ha e
(4.6) | (x)−Pm(B0, )(x)| ≤| (x)−Pm(B∗, )(x)|
+|Pm(B∗, )(x)−Pm(B0, )(x)|.
Fo he i s e m on he igh side o (4.6), we ha e by Theo em B ha
| (x)−Pm(B∗, )(x)| ≤ CZB∗
g(y)ρ(x, y)m
µ(B(x, ρ(x, y))) dµ(y).
This holds o ν−a.e. poin o B∗, and we may assume i holds o ou ixed xby
ini ially excluding om Ω he se o measu e ze o o med by aking he union o he
excep ional se s o measu e ze o in each Boman ball. Since B∗⊂Ω, we ob ain
| (x)−Pm(B∗, )(x)| ≤ CZΩ
g(y)ρ(x, y)m
µ(B(x, ρ(x, y))) dµ(y).
We now es ima e |Pm(B∗, )(x)−Pm(B0, )(x)|.By using he chain {Bj}connec ing
B0and Bk=B∗and no icing ha B∗⊂MBjand x∈B∗, we ha e
|Pm(B∗, )(x)−Pm(B0, )(x)|
≤
k
X
j=1
|Pm(Bj, )(x)−Pm(Bj−1, )(x)|
≤
k
X
j=1
||Pm(Bj, )−Pm(Bj−1, )||L∞
ν(B∗)
≤
k
X
j=1
||Pm(Bj, )−Pm(Bj−1, )||L∞
ν(MBj).

L1→LqPOINCAR´
E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 17
I Djis a ball wi h Dj⊂Bj∩Bj−1⊂MBjand (Dj)≈ (Bj)≈ (Bj−1), hen by
(P1) and (P2), he las sum is majo ized by
C
k
X
j=1
||Pm(Bj, )−Pm(Bj−1, )||L∞
ν(Dj)
≤C
k
X
j=1 ÃZDj
|Pm(Bj, )−Pm(Bj−1, )|qdν!1/q
,
which by he inequali y (2.6) and doubling is bounded by
C2q
k
X
j=1 ÃZDj
|Pm(Bj, )(y)− (y)|qdν(y)!1/q
+C2q
k
X
j=1 ÃZDj
|Pm(Bj−1, )(y)− (y)|qdν(y)!1/q
≤C2q
k
X
j=1 ÃZBj
|Pm(Bj, )(y)− (y)|qdν(y)!1/q
+C2q
k
X
j=1 ÃZBj−1
|Pm(Bj−1, )(y)− (y)|qdν(y)!1/q
≤C
k
X
j=0 Ã1
ν(Bj)ZBj
|Pm(Bj, )(y)− (y)|qdν(y)!1/q
By Poinca ´e’s inequali y, he las exp ession abo e is a mos
C
k
X
j=0
(Bj)m
µ(Bj)ZBj
g(y)dµ(y)
=CZΩ


k
X
j=0
(Bj)m
µ(Bj)χBj(y)


g(y)dµ(y).
As shown in [LW2], he sum abo e in cu ly b acke s is bounded by a ixed mul iple o
ρ(x, y)m/µ(B(x, ρ(x, y))) o each y∈Ω. Thus, we ha e comple ed he p oo .
By using Theo em (4.2), we will be able o weaken he hypo heses in de ining high
o de Sobole spaces in me ic spaces gi en in [LLW1] (see also [LLW2]).
De ini ion 4.7. Gi en a posi i e in ege mand 1< p < ∞, we de ine he Sobole class
Am,p(Ω) o be he se o unc ions ∈Lp(Ω) so ha o each k= 1,··· , m, he e exis
kwi h 1≤ k< p and qkwi h 0< qk<1, unc ions gk(x)wi h 0≤gk∈Lp(Ω), and
polynomials Pk(B, )wi h
(4.8) µZB
| (x)−Pk(B, )(x)|qkdµ(x)¶1
qk≤ (B)kµZB
g k
k(x)dµ(x)¶1
k
18 G. LU AND C. P´
EREZ
o e e y ball B⊂Ω. The polynomials Pk(B, )a e assumed o belong o a linea class
which sa is ies (P1) and (P2) wi h cons an s depending only on k, γ, µ. I ∈Am,p(Ω),
we de ine
|| ||Am,p(Ω) =|| ||Lp(Ω) + in
{gk}
m
X
k=1
||gk||Lp(Ω),
whe e he in imum is aken o e all sequences such ha (4.8) holds o o k=
1, . . . , m.
I is easy o see ha Am,p(Ω) is a linea space.
The eason we can impose he L kno m a he han he L1no m is because we can
show ha de ini ion (4.7) is equi alen o he ollowing de ini ions (4.9) and (4.11) gi en
in [LLW1]. The p oo o equi alence ollows om ou Theo em (4.2) in his sec ion by
combining he p oo s gi en in [LLW1]. We shall omi he de ails he e.
De ini ion 4.9 [LLW1]. Gi en a posi i e in ege mand 1< p < ∞, we de ine he
Sobole class Bm,p(Ω) o be he se o unc ions ∈Lp(Ω) so ha o each k= 1,· · · , m,
he e exis unc ions 0≤gk∈Lp(Ω) and polynomials Pk(B, )such ha
(4.10) | (x)−Pk(B, )(x)| ≤ ZB
ρ(x, y)kgk(y)
µ(B(x, ρ(x, y)))dµ(y) + (B)kZB
gk(y)dµ(y)
o µ−a.e. x∈B o e e y ball B⊂Ω. The polynomials Pk(B, )a e assumed o
belong o a linea class which sa is ies (P1) and (P2) wi h cons an s depending only on
k, γ, µ. I ∈Bm,p(Ω), we de ine
|| ||Bm,p(Ω) =|| ||Lp(Ω) + in
{gk}
m
X
k=1
||gk||Lp(Ω),
whe e he in imum is aken o e all sequences such ha (4.10) holds o o k=
1,··· , m.
The class Bm,p(Ω) is clea ly a Banach space wi h no m || · ||Bm,p(Ω).
De ini ion 4.11 [LLW1]. Gi en a posi i e in ege mand 1< p < ∞, we de ine he
Sobole class Cm,p(Ω) o be he se o unc ions ∈Lp(Ω) so ha o each k= 1,· · · , m
he e exis unc ions 0≤gk∈Lp(Ω) and polynomials Pk(B, )such ha
(4.12) | (x)−Pk(B, )(x)| ≤ (B)kgk(x)
o µ−a.e. x∈B o e e y me ic ball B⊂Ω. The polynomials Pk(B, )a e assumed
o belong o a linea class which sa is ies (P1) and (P2) wi h cons an s depending only
on k, γ, µ. I ∈Cm,p(Ω), le
|| ||Cm,p(Ω) =|| ||Lp(Ω) + in
{gk}
m
X
k=1
||gk||Lp(Ω).
The class Cm,p(Ω) is a Banach space wi h no m || · ||Cm,p .
To show ha de ini ions (4.7), (4.9) and (4.11) a e all equi alen , we will need he
ollowing heo em.
L1→LqPOINCAR´
E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 19
Theo em 4.13. Le 1≤ < ∞,mbe a posi i e in ege , B0⊂Ωbe a ixed ball,
and suppose ha he segmen p ope y (H3) holds o B0. Le be a locally in eg able
unc ion in Ω o which he e exis a unc ion 0≤g∈L (Ω) and polynomials Pm(B, ),
and 0< q < 1such ha he Poinca ´e inequali y
µZB
| (x)−Pm(B, )(x)|qdµ(x)¶1/q
≤c (B)mµZB
|g(x)| dµ(x)¶1/
holds o e e y ball B⊂Ω. The polynomials Pm(B, )a e assumed o belong o a linea
class which sa is ies (P1) and (P2) wi h cons an s depending only on m, γ, µ. Then o
µ−a.e. x∈B0,
| (x)−Pm(B0, )(x)| ≤ C (B0)mM(g )(x)1/
wi h Cindependen o x.
P oo o Theo em 4.13. Le x∈B0. We will use he chain o subballs {Bj}o B0
cons uc ed om Theo em 2.5. The chain depends on x. We may assume wi hou loss
o gene ali y ha xis a Lebesgue poin o bo h | −Pm(B0, )|qand |g| wi h espec
o µ. Then by p ope ies (1), (2) and (3) o he chain,
| (x)−Pm(B0, )(x)|= lim
j→∞ ÃZBj
| (y)−Pm(B0, )(y)|qdµ(y)!1/q
≤lim sup
j→∞
2qÃZBj
| (y)−Pm(Bj, )(y)|qdµ(y)!1/q
+ lim sup
j→∞
2qÃZBj
|Pm(Bj, )(y)−Pm(B0, )(y)|qdµ(y)!1/q
=I1+I2.
By he Poinca ´e inequali y, o e e y Lebesgue poin xo |g|
I1≤clim sup
j→∞
(Bj)mÃZBj
|g(y)| dµ(y)!1/
= 0 · |g(x)|= 0
by p ope ies (1), (2) and (3) o he chain.
20 G. LU AND C. P´
EREZ
We ha e o I2
I2≤lim sup
j→∞
||Pm(Bj, )(y)−Pm(B0, )(y)||L∞
µ(Bj)
≤lim sup
j→∞
j−1
X
`=0
||Pm(B`+1, )−Pm(B`, )||L∞
µ(Bj)
≤lim sup
j→∞
j−1
X
`=0
||Pm(B`+1, )−Pm(B`, )||L∞
µ(cB`)by (5)
≤C
∞
X
`=0
||Pm(B`+1, )−Pm(B`, )||L∞
µ(S`)by (4) and (P2)
≤C
∞
X
`=0 µZSl
|Pm(B`+1, )−Pm(B`, )|qdµ¶1/q
by (P1)
≤C
∞
X
`=0
2qµZB`
|Pm(B`, )(y)− (y)|qdµ(y)¶1/q
+C
∞
X
`=0
2qÃZB`+1
|Pm(B`+1, )(y)− (y)|qdµ(y)!1/q
by (4)
≤C
∞
X
`=0
(B`)mµZB`
|g(y)|qdµ(y)¶1/q
≤C
∞
X
`=0
(B`)mM(|g| )(x)1/ by (3)
=C
∞
X
`=0
2−`m (B0)mM(|g| )(x)1/ by (2)
≤C (B0)mM(|g| )(x)1/ .
This comple es he p oo o Theo em 4.13.
By using Theo em 4.13, and a guing simila ly as in [LLW1], we will be able o show
he equi alence o de ini ions (4.7), (4.9) and (4.11). We shall omi he de ails he e and
e e he eade o [LLW1].
We end his sec ion by men ioning ha abo e de ini ions o Sobole spaces o highe
o de coincide wi h hose o classical Sobole spaces in Euclidean space and non-iso opic
Sobole spaces on s a i ied nilpo en Lie g oups (see [LLW1] o de ailed p oo ).
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