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Admissibili y e sus Ap-Condi ions on Regula T ees
© 2020 Khanh Ngoc Nguyen and Zhuang Wang, published by De G uy e .
Published e sion
Nguyen, Khanh Ngoc; Wang, Zhuang
Nguyen, K. N., & Wang, Z. (2020). Admissibili y e sus Ap-Condi ions on Regula T ees. Analysis
and Geome y in Me ic Spaces, 8(1), 92-105. h ps://doi.o g/10.1515/agms-2020-0110
2020
Open Access. ©2020 Khanh Ngoc Nguyen and Zhuang Wang, published by De G uy e . This wo k is licensed unde he C ea i e
Commons A ibu ion alone 4.0 License.
Anal. Geom. Me . Spaces 2020; 8: 92–105
Resea ch A icle Open Access
Khanh Ngoc Nguyen and Zhuang Wang*
Admissibili y e sus Ap-Condi ions on
Regula T ees
h ps://doi.o g/10.1515/agms-2020-0110
Recei ed Decembe 30, 2019; accep ed May 20, 2020.
Abs ac : We show ha he combina ion o doubling and (1,p)-Poinca é inequali y is equi alen o a e sion
o he Ap-condi ion on oo ed K-a y ees.
Keywo ds: Ap-condi ion; doubling measu e; Poinca é inequali y; egula ee
MSC: 30L99, 31C45, 46E35
Dedica ed o P o esso Pekka Koskela on he occasion o his 59 h bi hday celeb a ion
1In oduc ion
The class o p-admissible weigh s o Sobole spaces and di e en ial equa ions on Rnwas in oduced in [12].
The de ini ion was ini ially based on ou condi ions, bu Theo em 2 in [10] and Theo em 5.2 in [13] educe
hem o he ollowing wo condi ions, see also [12, 2nd ed., Sec ion 20].
De ini ion 1.1. A measu e µon Rnis p-admissible,1≤p<∞, i i is doubling and suppo s a (1,p)-Poinca é
inequali y. I dµ =w dx, we also say ha he weigh wis p-admissible.
He e µsuppo s a (q,p)-Poinca é inequali y, 1≤q<∞,1≤p<∞, i he e is a cons an C>0such ha


Z–
B(x, )
|u−uB(x, )|qdµ


1/q
≤C 

Z–
B(x, )
|∇u|pdµ


1/p
o e e y u∈C1(Rn), e e y x∈Rnand all >0.
In [12, Sec ion 15], i was shown ha Muckenhoup Ap-weigh s a e p-admissible, bu he con e se is no
ue in Rn,n≥2, see also [6]. Su p isingly, on he eal line R, any p-admissible measu e is ac ually gi en
by an Ap-weigh , see [7]. Ve y ecen ly, i was also shown in [5] ha a measu e on Ris locally p-admissible
i and only i i is gi en by a local Ap-weigh . Mo eo e , on Rn,p-admissible measu es can be cha ac e ized
by a s onge e sion o he Poinca é inequali y, he (q,p)-Poinca é inequali y wi h q>p. Unde doubling,
he (1,p)-Poinca é inequali y imp o es o a (q,p)-Poinca é inequali y wi h q>pby [10] and any measu e
sa is ying (q,p)-Poinca é inequali y wi h q>pis a doubling measu e, see [1] and [17].
In he ecen yea s, analysis on egula ees has been unde de elopmen , see [3, 18–21]. Gi en a K-
egula ee X(a oo ed K-a y ee), K≥1, we in oduce a me ic s uc u e on Xby conside ing each edge o
X o be an isome ic copy o he uni in e al. Then he dis ance be ween wo e ices is he numbe o edges
needed o connec hem and he e is a unique geodesic ha minimizes his numbe . Le us deno e he oo
Khanh Ngoc Nguyen, Uni e si y o Jy äskylä, Jy äskylä, Finland, E-mail: [email p o ec ed], [email protected]
*Co esponding Au ho : Zhuang Wang, Uni e si y o Jy äskylä, Jy äskylä, Finland, E-mail: [email p o ec ed]
Khanh Ngoc Nguyen and Zhuang Wang, Admissibili y e sus Ap-Condi ions on Regula T ees |93
by 0. I xis a e ex, we de ine |x| o be he dis ance be ween 0 and x. Since each edge is an isome ic copy o
he uni in e al, we may ex end his dis ance na u ally o any xbelonging o an edge.
W i e d|x| o he leng h elemen on Xand le µ: [0,∞)→(0,∞)be a locally in eg able unc ion. We
abuse no a ion and e e also o he measu e gene a ed ia dµ(x) = µ(|x|)d|x|by µ.Fu he , le λ: [0,∞)→
(0,∞)be locally in eg able and de ine a dis ance ia ds(x) = λ(|x|)d|x|by se ing d(z,y) = R[z,y]ds(x)when-
e e z,y∈Xand [z,y]is he unique geodesic be ween zand y. We abuse he no a ion and le µ(x)and λ(x)
deno e µ(|x|)and λ(|x|), espec i ely, o any x∈X, i he e is no dange o con usion. Th oughou his pape ,
we assume addi ionally ha he diame e o Xis in ini y.
Ou space (X,d,µ)is a me ic measu e space and hence one may de ine a New onian Sobole space
N1,p(X) := N1,p(X,d,µ)based on uppe g adien s [14] and [22]. I is hen na u al o ask i we can cha ac e ize
he p-admissibili y o a gi en µ, see Sec ion 2.2 o he de ini ions. To do so, we in oduce he ollowing Ap-
condi ions on egula ees.
Be o e con inuing, we i s in oduce some no a ions. Fo any x∈Xand >0, we deno e by ¯
x he poin
in [0,x]wi h d(¯
x ,x) = min{ ,d(0,x)}and deno e by x a poin in Xsuch ha x∈[0,x ]wi h d(x ,x) = .
Hence ¯
x is an ances o o xand x is a descendan o x, see Sec ion 2.1 o mo e ela ions be ween poin s on
egula ees. Also le
F(x, ) = {y∈X:x∈[0,y],d(x,y)< }
be he downwa d di ec ed “hal ball". I is pe haps wo h o men ion ha he no a ions ¯
x and F(x, )coincide
wi h he no a ion “z" and F(x, )in [3, Lemma 3.2], espec i ely.
Gi en 1<p<∞, we se
Ap(x, ) = µ(F(¯
x ,2 ))
2 ·


1
Z
[x,x ] Kj(w)−j(x)µ(w)
λ(w)!1
1−p
ds(w)


p−1
(1.1)
and we de ine
A1(x, ) = µ(F(¯
x ,2 ))
2 ·ess supw∈[x,x ]
λ(w)
Kj(w)−j(x)µ(w)(1.2)
whe e j(w)and j(x)a e he smalles in ege s such ha j(w)≥|w|and j(x)≥|x|, espec i ely. No ice ha
Ap(x, )is independen o he choice o x among he poin s ywi h x∈[0,y]and d(y,x) = .
De ini ion 1.2. Le 1≤p<∞and Xbe a K- egula ee wi h dis ance dand me ic µ. We say ha µsa is ies
he Ap-condi ion i
sup Ap(x, ) : x∈X, >0<∞.(1.3)
We say ha µsa is ies he Ap-condi ion a om 0i
sup Ap(x, ) : x∈X,0< ≤8d(0,x)<∞.(1.4)
I K= 1 and λ≡1, hen he 1- egula ee (X,d,µ)is isome ic o he hal line (R+,dx,µ dx)and ou Ap-
condi ion (1.3) is equi alen o µbeing a Muckenhoup Ap-weigh , see [5–7, 12] o mo e in o ma ion abou
Muckenhoup Ap-weigh s. Abo e, we call (1.4) “Ap-condi ion a om 0” since 0< ≤8d(0,x)is equi alen
o d(0,x)≥ /8>0, which means ha xhas o be “ a ” away om he oo 0in e ms o .
The main esul o his pape is he ollowing cha ac e iza ion o p-admissibili y on egula ees.
Theo em 1.3. Le 1≤p<∞and Xbe a K- egula ee wi h dis ance dand measu e µ. Then we ha e:
1. Fo K= 1,µis p-admissible i and only i µsa is ies he Ap-condi ion a om 0.
2. Fo K≥2,µis p-admissible i and only i µsa is ies he Ap-condi ion.
The cha ac e iza ions o K= 1 and K≥2a e di e en . Fo K≥2, a K- egula ee has a kind o symme y
p ope y wi h espec o he oo 0, since he oo has mo e han one b anch. Bu o K= 1, he oo 0beha es
like an end poin .
94 |Khanh Ngoc Nguyen and Zhuang Wang, Admissibili y e sus Ap-Condi ions on Regula T ees
Reade s who a e amilia wi h he esul s on he eal line Rmay ega d ou K- egula ee wi h K≥2as a
gene alized model o he eal line R. As a byp oduc , a sligh ly modi ied p oo o Theo em 1.3 o K≥2gi es
a new p oo o [7, Theo em 2]. On he o he hand, o K= 1, one may connec he esul on 1- egula ees
wi h he esul on bounded in e als (see [5, Theo em 4.6] o bounded in e als). Hence Theo em 1.3 is new
and in e es ing e en when K= 1 and λ≡1, since i gi es a ull cha ac e iza ion o p-admissibili y on he
hal line R+.
In [5, Example 4.7], one can ind a weigh ωon he in e al [0,1] which is 1-admissible bu no a Muck-
enhoup A1-weigh on (0,1). By a sui able cons an ex ension o ωon (1,∞), we ob ain a weigh ω0which is
1-admissible bu no a Muckenhoup A1-weigh on R+. As e idence owa ds Theo em 1.3 o K= 1, i is easy
o check ha he ex ended weigh ω0on R+sa is ies he A1-condi ion a om 0, i.e., condi ion (1.4) holds.
We e e o [5] and [8] o mo e de ails.
Le us close his in oduc ion by poin ing ou ha he cons an “8” in Ap-condi ion a om 0(1.4) is no
necessa y. Ac ually eplacing 8by any cons an ∞>c>1, Theo em 1.3 o K= 1 holds. He e he equi emen
o c>1is sha p in he sense ha he e exis s an example (R+,dx,µ dx)such ha (1.4) holds o any posi i e
cons an c0<1 eplacing 8, bu µis no e en doubling, see Rema k 4.5 and Example 4.6.
The pape is o ganized as ollows. In sec ion 2, we in oduce egula ees, p-admissibili y and New onian
spaces on ou ee. We gi e he p oo o Theo em 1.3 o K≥2in Sec ion 3 and he p oo o Theo em 1.3 o
K= 1 is gi en in Sec ion 4.
2P elimina ies
Th oughou his pape , he le e C(some imes wi h a subsc ip ) will deno e posi i e cons an s; i Cdepends
on a,b,. . ., we w i e C=C(a,b,. . .).
2.1 Regula ees and hei bounda ies
Ag aph Gis a pai (V,E), whe e Vis a se o e ices and Eis a se o edges. We call a pai o e ices x,y∈V
neighbo s i xis connec ed o yby an edge. The deg ee o a e ex is he numbe o i s neighbo s. The g aph
s uc u e gi es ise o a na u al connec i i y s uc u e. A ee is a connec ed g aph wi hou cycles. A g aph
(o ee) is made in o a me ic g aph by conside ing each edge as a geodesic o leng h one.
We call a ee Xa oo ed ee i i has a dis inguished e ex called he oo , which we will deno e by 0.
The neighbo s o a e ex x∈Xa e o wo ypes: he neighbo s ha a e close o he oo a e called pa en s o
xand all o he neighbo s a e called child en o x. Each e ex has a unique pa en , excep o he oo i sel
ha has none.
AK-a y ee is a oo ed ee such ha each e ex has exac ly Kchild en. Then all e ices excep he oo
o a K-a y ee ha e deg ee K+ 1, and he oo has deg ee K. In his pape we say ha a ee is egula i i is a
K-a y ee o some K≥1.
Fo x∈X, le |x|be he dis ance om he oo 0 o x, ha is, he leng h o he geodesic om 0 o x,
whe e he leng h o e e y edge is 1and we conside each edge o be an isome ic copy o he uni in e al.
The geodesic connec ing wo poin s x,y∈Vis deno ed by [x,y], and i s leng h is deno ed |x−y|. I |x|<|y|
and xlies on he geodesic connec ing 0 o y, we w i e x<yand call ya descendan o he poin x. Mo e
gene ally, we w i e x≤yi he geodesic om 0 o ypasses h ough x, and in his case |x−y|=|y|−|x|.
On ou K- egula ee X, we de ine he me ic ds and measu e dµ by se ing
dµ =µ(|x|)d|x|,ds(x) = λ(|x|)d|x|,
whe e λ,µ: [0,∞)→(0,∞)wi h λ,µ∈L1
loc([0,∞)). He e d|x|is he measu e which gi es each edge
Lebesgue measu e 1, as we conside each edge o be an isome ic copy o he uni in e al and he e ices
Khanh Ngoc Nguyen and Zhuang Wang, Admissibili y e sus Ap-Condi ions on Regula T ees |95
a e he end poin s o his in e al. Hence o any wo poin s z,y∈X, he dis ance be ween hem is
d(z,y) = Z
[z,y]
ds(x) = Z
[z,y]
λ(|x|)d|x|,
whe e [z,y]is he unique geodesic om z o yin X.
We abuse he no a ion and le µ(x)and λ(x)deno e µ(|x|)and λ(|x|), espec i ely, o any x∈X, i he e
is no dange o con usion.
Th oughou he pape , we le
B(x, ) = {y∈X:d(x,y)< }
deno e he (open) ball in Xwi h cen e xand adius , and le σB(x, ) = B(x,σ ). Also
F(x, ) = {y∈X:x∈[0,y],d(x,y)< }
is he downwa d di ec ed hal ball. Fo any x∈Xand >0, we deno e by ¯
x he poin in [0,x]wi h d(¯
x ,x) =
min{ ,d(0,x)}and deno e by x a poin in Xsuch ha x∈[0,x ]wi h d(x ,x) = . Hence ¯
x is he ances o
o any poin y∈B(x, ). Usually, he choice o x is no unique, bu we will no speci y i since he esul s and
p oo s in his pape a e independen o he choice o x .
2.2 Admissibili y
Le u∈L1
loc(X). We say ha a Bo el unc ion g:X→[0,∞]is an uppe g adien o ui
|u(z)−u(y)|≤Z
γ
g ds (2.1)
whene e z,y∈Xand γis he geodesic om z o y. In he se ing o a ee any ec i iable cu e wi h end
poin s zand ycon ains he geodesic connec ing zand y, and he e o e he uppe g adien de ined abo e is
equi alen o he de ini ion which equi es ha inequali y (2.1) holds o all ec i iable cu es wi h end poin s
zand y. In [9, 15], he no ion o a p-weak uppe g adien is gi en. A Bo el unc ion g:X→[0,∞]is called a
p-weak uppe g adien o ui (2.1) holds on p-a.e. cu e. He e we say ha a p ope y holds o p-a.e. cu e i i
ails only o a ec i iable cu e amily Γwi h ze o p-modulus, i.e., he e is Bo el unc ion 0≤ρ∈Lp(X)such
ha Rγρ ds =∞ o e e y cu e γ∈Γ. We e e o [9, 15] o mo e in o ma ion abou p-weak uppe g adien s.
The no ion o uppe g adien s is due o Heinonen and Koskela [14]; we e e in e es ed eade s o [2, 9,
15, 22] o a mo e de ailed discussion on uppe g adien s.
The New onian space N1,p(X), o 1≤p<∞, is de ined as he collec ion o he unc ions o which he
gi en no m
kukN1,p(X):= 
Z
X
|u|pdµ + in
gZ
X
|g|pdµ

1/p
is ini e, whe e he in imum is aken o e all p-weak uppe g adien s go u.
A measu e µis doubling i he e exis s a posi i e cons an Cdsuch ha o all balls B(x, )wi h x∈Xand
>0,
µ(B(x,2 )) ≤Cdµ(B(x, )),(2.2)
whe e he cons an Cdis called he doubling cons an .
(X,d,µ)suppo s a (1,p)-Poinca é inequali y i he e exis posi i e cons an s CP>0and σ≥1such ha
o all balls B(x, )wi h x∈Xand >0, e e y in eg able unc ion uon σB(x, )and all uppe g adien s g,
Z–
B(x, )
|u−uB(x, )|dµ ≤CP 

Z–
σB(x, )
gpdµ


1/p
(2.3)

96 |Khanh Ngoc Nguyen and Zhuang Wang, Admissibili y e sus Ap-Condi ions on Regula T ees
whe e uB:=R
−Bu dµ =1
µ(B)RBu dµ. We say ha µis p-admissible i µis a doubling measu e and (X,d,µ)
suppo s a (1,p)-Poinca é inequali y.
The doubling p ope y (2.2) and (1,p)-Poinca é inequali y (2.3) can be de ined on gene al me ic measu e
spaces. In pa icula , on Rn, in iew o [16, Theo em 2] o [15, Theo em 8.4.2], he (1,p)-Poinca é inequali y
(2.3) is equi alen o he (1,p)-Poinca é inequali y gi en in he In oduc ion. I pe haps wo h o poin ou
ha , since ou K- egula ees a e geodesic spaces, i µis p-admissible, he dila ion cons an σin (2.3) can be
aken o 1, see [10] and [11].
3P oo o Theo em 1.3 o K≥2
In his sec ion, we gi e he p oo o Theo em 1.3 o K≥2. To do so, we es ablish he ollowing lemmas.
Lemma 3.1. Le 1≤p<∞and Xbe a K- egula ee wi h dis ance dand measu e µwhe e K≥1. Assume ha
µsa is ies he Ap-condi ion. Then µis p-admissible.
P oo . Fo 1≤p<∞, le
CA:= sup Ap(x, ) : x∈X, >0.
Since µsa is ies he Ap-condi ion, 0<CA<∞.
Case p= 1:We i s show ha µis a doubling measu e. Le x∈Xand >0be a bi a y. No ice ha
A1(x,2 )≤CA. Then i ollows om (1.2) ha
ess supw∈[x,x2 ]
λ(w)
Kj(w)−j(x)µ(w)≤4 CA
µ(F(¯
x2 ,4 )) .
Hence
=Z
[x,x ]
ds =Z
[x,x ] Kj(w)−j(x)µ(w)
λ(w)!λ(w)
Kj(w)−j(x)µ(w)ds(w)
≤

Z
[x,x ]
Kj(w)−j(x)µ(w)
λ(w)ds(w)

4 CA
µ(F(¯
x2 ,4 )).(3.1)
No ice ha
Z
[x,x ]
Kj(w)−j(x)µ(w)
λ(w)ds(w) = µ(F(x, )) ≤µ(B(x, ))
and ha
µ(F(¯
x2 ,4 )) ≥µ(B(x,2 )).
I ollows om es ima e (3.1) ha
≤4CA µ(B(x, ))
µ(B(x,2 )) ,
which p o es ha µis a doubling measu e wi h doubling cons an 4CAsince >0and he pai (x, )is
a bi a y.
Nex we p o e ha (X,d,µ)suppo s a (1,1)-Poinca é inequali y. Conside an a bi a y ball B(x, )wi h
x∈Xand >0. By he iangle inequali y, we ob ain ha
Z–
B(x, )
|u−uB(x, )|dµ ≤2Z–
B(x, )
|u(y)−u(¯
x )|dµ(y)(3.2)
Khanh Ngoc Nguyen and Zhuang Wang, Admissibili y e sus Ap-Condi ions on Regula T ees |97
o he le -hand side o ou Poinca é inequali y. By he de ini ion o uppe g adien s and he Fubini heo em,
o any uppe g adien guo u, he igh -hand side o (3.2) ew i es as
2Z–
B(x, )
|u(y)−u(¯
x )|dµ(y)≤2Z–
B(x, )Z
[¯
x ,y]
gu(w)ds(w)dµ(y)
= 2 Z–
B(x, )
gu(w)λ(w)
µ(w)

Z
B(x, )
χ[¯
x ,y](w)dµ(y)

dµ(w)
= 2 Z–
B(x, )
gu(w)λ(w)
µ(w)µ({y∈B(x, ) : w∈[0,y]})dµ(w).(3.3)
He e he las equali y holds since χ[¯
x ,y](w)is no ze o only i w∈[0,y].
Since he measu e µsa is ies he A1-condi ion, A1(¯
x ,2 )<CA. I ollows om (1.2) ha
µ(F(¯
x3 ,4 ))
4 ·ess supw∈[¯
x ,x ]
λ(w)
Kj(w)−j(¯
x )µ(w)≤CA.
Combining wi h he ac ha Kj(¯
x3 )≤Kj(¯
x ), we ob ain ha
λ(w)
µ(w)µ({y∈B(x, ) : w∈[0,y]}) = λ(w)
µ(w)Z
{y∈[w,w ]∩B(x, )}
Kj(y)−j(w)µ(y)
λ(y)ds(y)
≤λ(w)Kj(¯
x3 )
µ(w)Kj(w)Z
[¯
x3 ,x ]
Kj(y)−j(¯
x3 )µ(y)
λ(y)ds(y)
≤λ(w)
µ(w)Kj(w)−j(¯
x )µ(F(¯
x3 ,4 )) ≤4CA (3.4)
o any w∈B(x, ). Combining (3.2)-(3.4), yields
Z–
B(x, )
|u−uB(x, )|dµ ≤8CA Z–
B(x, )
gudµ
o all balls B(x, ).
Case p>1:Le us i s p o e ha µis a doubling measu e. Le B(x, )be an a bi a y ball in X. Since µ
sa is ies he Ap-condi ion, we ha e Ap(x,2 )≤CA, and hence
µ(F(¯
x2 ,4 ))
4 ·


1
2 Z
[x,x2 ] Kj(w)−j(x)µ(w)
λ(w)!1
1−p
ds(w)


p−1
≤CA.(3.5)
A simple calcula ion using he Hölde inequali y shows ha
=Z
[x,x ] Kj(w)−j(x)µ(w)
λ(w)!1/p Kj(w)−j(x)µ(w)
λ(w)!−1/p
ds(w)
≤

Z
[x,x ]
Kj(w)−j(x)µ(w)
λ(w)ds(w)


1/p

Z
[x,x ] Kj(w)−j(x)µ(w)
λ(w)!1
1−p
ds(w)


p−1
p
≤µ(F(x, ))1/p(2 )p−1
p


1
2 Z
[x,x2 ] Kj(w)−j(x)µ(w)
λ(w)!1
1−p
ds(w)


p−1
p
.
98 |Khanh Ngoc Nguyen and Zhuang Wang, Admissibili y e sus Ap-Condi ions on Regula T ees
Inse ing (3.5) in o he abo e es ima e yields
≤(2 )p−1
pµ(F(x, ))1/pµ(F(¯
x2 ,4 ))
4 CA−1
p
=CA1/p2p+1
p µ(F(x, ))
µ(F(¯
x2 ,4 ))1/p
.(3.6)
No e ha µ(F(x, )) ≤µ(B(x, )) and µ(F(¯
x2 ,4 )) ≥µ(B(x,2 )). Then he es ima e (3.6) implies ha
≤CA1/p2p+1
p µ(B(x, ))
µ(B(x,2 ))1/p
,
which gi es ha µis a doubling measu e wi h doubling cons an CA2p+1, since >0and B(x, )is a bi a y.
Nex we show ha (X,d,µ)suppo s a (1,p)-Poinca é inequali y. Suppose B(x, )is an a bi a y ball wi h
cen e x∈Xand adius >0. Since he measu e µsa is ies he Ap-condi ion, hen Ap(¯
x ,2 )<CA. I ollows
om (1.1) ha
µ(F(¯
x3 ,4 ))
4 ·


1
2 Z
[¯
x ,x ] Kj(w)−j(¯
x )µ(w)
λ(w)!1
1−p
ds(w)


p−1
≤CA.(3.7)
Recall ha he le -hand side o ou Poinca é inequali y can be es ima ed by (3.3). A simple calcula ion
shows ha
λ(w)
µ(w)µ({y∈B(x, ) : w∈[0,y]}) = λ(w)
µ(w)Z
{y∈[w,w ]∩B(x, )}
Kj(y)−j(w)µ(y)
λ(y)ds(y)
≤λ(w)
µ(w)Kj(w)−j(¯
x )Z
[¯
x ,x ]
Kj(y)−j(¯
x )µ(y)
λ(y)ds(y)
=λ(w)
µ(w)Kj(w)−j(¯
x )µ(F(¯
x ,2 )) (3.8)
o any poin w∈B(x, ). Inse ing he es ima e (3.8) in o (3.3) yields ha
Z–
B(x, )
|u−uB(x, )|dµ ≤2

Z–
B(x, )
gu(w)λ(w)
µ(w)Kj(w)−j(¯
x )dµ(w)

µ(F(¯
x ,2 )).
Applying he Hölde inequali y o he igh -hand side o he abo e inequali y, i ollows ha
Z–
B(x, )
|u−uB(x, )|dµ ≤2

Z–
B(x, )
gupdµ


1/p

Z–
B(x, )λ(w)
Kj(w)−j(¯
x )µ(w)p
p−1
dµ(w)


p−1
p
µ(F(¯
x ,2 )).(3.9)
By using he es ima e (3.7), we ob ain ha


Z–
B(x, )λ(w)
Kj(w)−j(¯
x )µ(w)p
p−1
dµ(w)


p−1
p
µ(F(¯
x ,2 )) ≤µ(F(¯
x ,2 ))
µ(B(x, ))p−1
p

Z
F(¯
x ,2 )λ(w)
Kj(w)−j(¯
x )µ(w)p
p−1
dµ(w)


p−1
p
≤µ(F(¯
x ,2 ))
µ(B(x, ))p−1
p
(2 )p−1
p


1
2 Z
[¯
x ,x ] Kj(w)−j(¯
x )µ(w)
λ(w)!1
1−p
ds(w)


p−1
p
≤µ(F(¯
x ,2 ))
µ(B(x, ))p−1
p
(2 )p−1
pµ(F(¯
x3 ,4 ))
4 CA−1
p
=CA1/p2p+1
p µ(F(¯
x ,2 ))
µ(B(x, ))p−1
pµ(F(¯
x3 ,4 ))1/p
.(3.10)
Khanh Ngoc Nguyen and Zhuang Wang, Admissibili y e sus Ap-Condi ions on Regula T ees |99
No e ha F(¯
x ,2 )) ⊂B(x,4 )and ha B(x, )⊂F(¯
x3 ,4 ). Since µis a doubling measu e wi h doubling
cons an CA2p+1, we ha e ha
µ(F(¯
x ,2 ))
µ(B(x, ))p−1
pµ(F(¯
x3 ,4 ))1/p
≤µ(B(x,4 ))
µ(B(x, )) ≤(CA2p+1)2.
Inse ing he abo e es ima e in o he es ima e (3.10), we ha e


Z–
B(x, )λ(w)
Kj(w)−j(¯
x )µ(w)p
p−1
dµ(w)


p−1
p
µ(F(¯
x ,2 )) ≤CA2+ 1
p2p+1
p+2(p+1) .(3.11)
Thanks o he es ima es (3.9) and (3.11), we ob ain
Z–
B(x, )
|u−uB(x, )|dµ ≤CA2+ 1
p21
p+2p+4 

Z–
B(x, )
gupdµ


1
p
o all balls B(x, ).
Lemma 3.2. Le 1≤p<∞and Xbe a K- egula ee wi h dis ance dand measu e µwhe e K≥2. Suppose
ha µis p-admissible. Then µsa is ies he Ap-condi ion.
P oo . Le x∈Xand >0be a bi a y. Le εbe an a bi a y posi i e numbe . Le x1∈Xbe a closes e ex
o xwi h |x1|>|x|. Then we de ine
Tx1:= {y∈X:x1∈[0,y]}and T1:= [x,x1]∪Tx1
Since µis p-admissible, we may assume ha µsa is ies he doubling condi ion (2.2) and he (1,p)-Poinca é
inequali y (2.3).
Case p= 1:Le
m=ess in w∈[x,x
2]
Kj(w)−j(x)µ(w)
λ(w).
In o de o es he (1,1)-Poinca é inequali y (2.3), we de ine
u(y) = 






0i y∈X T1,
R[x,y]χEε(w)ds(w)i y∈F(x, /2) ∩T1,
ao he wise
whe e Eε:= nw∈F(x,
2) : Kj(w)−j(x)µ(w)
λ(w)<m+εoand a=R[x,x
2]χEε(w)ds(w). No e ha Eεis a non-emp y se
by he de ini ion o mand ha
>a=Z
[x,x
2]
χEε(w)ds(w)>0.
By he de ini ion o u, we ob ain ha gu:= χEεis an uppe g adien o u. Hence he igh -hand side o he
(1,1)-Poinca é inequali y (2.3) is
CP Z–
σB(x, )
gudµ =CP Z–
σB(x, )
χEε(w)dµ(w)
=CP
µ(σB(x, )) Z
F(x, /2)
χEε(w)dµ(w)
=CP
µ(σB(x, )) Z
[x,x
2]
χEε(w)Kj(w)−j(x)µ(w)
λ(w)ds(w).