Jou nal o ou ie analysis and applica ions (5) 9(2003), 511–540.
A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING
PROPERTIES OF POINCAR´
E INEQUALITIES IN METRIC SPACES
BRUNO FRANCHI, CARLOS P´
EREZ, AND RICHARD L. WHEEDEN
Abs ac . We de ine a class o summa ion ope a o s wi h applica ions o he sel -imp o ing
na u e o Poinca ´e–Sobole es ima es, in ai ly gene al quasime ic spaces o homogeneous ype.
We show ha hese sum ope a o s play he amilia ole o in eg al ope a o s o po en ial
ype (e.g., Riesz ac ional in eg als) in de i ing Poinca ´e–Sobole es ima es in cases when
ep esen a ions o unc ions by such in eg al ope a o s a e no eadily a ailable. In pa icula , we
de i e no m es ima es o sum ope a o s and use hese es ima es o ob ain imp o ed Poinca ´e–
Sobole esul s.
1. In oduc ion.
I is well-known ha Poinca ´e–Sobole es ima es in Euclidean space can be de i ed as co ol-
la ies o no m inequali ies o Riesz ac ional in eg al ope a o s. Fo example, he classical
es ima e
(1) ZB
| (x)− B|qdx1/q
≤cZB
|∇ (x)|pdx1/p
,1
q=1
p−1
n,1< p < n,
whe e Bis a Euclidean ball in Rnand B=1
|B|RB (x)dx, can be de i ed om he no m
inequali y
(2) ZRn
|I1 (x)|qdx1/q
≤cZRn
| (x)|pdx1/p
o he same alues o pand q, whe e cis independen o and
I1 (x) = ZRn
(y)
|x−y|n−1dy
is he Riesz ans o m o o o de 1. Simila ly, al hough (2) is alse in case p= 1 and
q=n/(n−1), he case p= 1 o (1) can be de i ed om he ollowing weak- ype analogue o
(2):
{x∈Rn:|I1 (x)|> λ}
(n−1)/n ≤c
λ|| ||L1(Rn), λ > 0,
wi h cindependen o λand , whe e |E|deno es he Lebesgue measu e o a se E.
The well-known poin wise ep esen a ion inequali y
| (x)− B| ≤ c I1(|∇ |χB)(x), x ∈B,
wi h cindependen o x, B and , makes i clea how (1) ollows om (2) in case p > 1, and a
a less ob ious a gumen based on unca ion can be used when p= 1 (see [23], [18], [28]).
In ac , no m es ima es o mo e gene al in eg al ans o ms ha e been used ecen ly o de i e
Poinca ´e–Sobole es ima es o ec o ields in ai ly gene al se ings, such as on mani olds and
g oups, and e en on abs ac me ic spaces in he sense o [6]. Fo example, le ρ(x, y) be a me ic
on Rn ha is induced by a collec ion Xo Ca no –Ca a h´eodo y ec o ields, and suppose ha
Lebesgue measu e is a doubling measu e o ρ-balls, i.e., ha |B(x, 2 )| ≤ C|B(x, )|wi h C
1991 Ma hema ics Subjec Classi ica ion. 46E35.
Key wo ds and ph ases. Poinca ´e–Sobole es ima es.
B.F. is suppo ed by Uni e si y o Bologna unds o selec ed esea ch opics, and by GNAMPA o INdAM,
I aly.
1
2 BRUNO FRANCHI, CARLOS P´
EREZ, AND RICHARD L. WHEEDEN
independen o xand , whe e B(x, ) deno es he ρ-ball wi h cen e xand adius . Then he
ope a o
I( )(x) = ZRn
(y)ρ(x, y)
|B(x, ρ(x, y))|dy
has known mapping p ope ies om Lp o Lqwi h p, q ela ed na u ally in e ms o he doubling
p ope y, and hese mapping p ope ies lead o Poinca ´e–Sobole es ima es o he o m
1
|B|ZB
| (x)− B|qdx1/q
≤c (B)1
|B|ZB
|X (x)|pdx1/p
,
whe e (B) is he adius o he ρ-ball B. The eason why Poinca ´e–Sobole es ima es ollow is
ha he e is a ep esen a ion inequali y o he o m
(3) | (x)− B| ≤ c I(|X |χB)(x), x ∈B,
wi h cindependen o x, B and ; see e.g. [11], [15], [20] o p ecise s a emen s o his ep esen-
a ion, and see e.g. [28] o he mapping p ope ies o he ope a o I.
On he o he hand, s a ing wi h wo k o Salo -Cos e [27], i is known ha Poinca ´e–Sobole
es ima es ha e a sel -imp o ing na u e, in he sense ha i is possible o de i e es ima es o
gene al p, q om pa icula special cases such as
(4) 1
|B|ZB
| (x)− B|dx ≤c (B)1
|B|ZB
|X |p0dx1/p0
o some p0, wi hou explici men ion o any in eg al ope a o a all.
A pa ial explana ion o he appa en mys e y abou he ole o in eg al ope a o s in he
sel -imp o ing echnique was gi en in [11] and wi h sha p cons an s in [21] (see also [15], [20],
[22], [25]). I was shown he e ha in case p0= 1, (4) is in ac equi alen o (3). In pa icula ,
by assuming (4) wi h p0= 1, we also ha e (3), and he mo e gene al Poinca ´e–Sobole es ima es
hen ollow om he co esponding no m es ima es o he in eg al ope a o I.
Howe e , in case p0>1, no sha p ep esen a ion analogous o (3) is known o ollow om
(4). When p0>1, he di icul y ha one encoun e s in ying o adap he a gumen s which
lead om (4) o (3) in case p0= 1 is ela ed o he p esence o he exponen 1/p0: he unc ional
a(B) de ined by
a(B) = (B)1
|B|ZB
|g|p0dx1/p0
(gand p0 ixed)
is no easy o add o e a class o non-o e lapping (o e en disjoin ) balls Bi p0>1. Thus,
s a ing om an es ima e o he ype
1
|B|ZB
| − B|dx ≤c a(B)
o all balls Bwi h a(B) as abo e and p0>1, o wi h an e en mo e gene al unc ional a(B), i
is no clea how o build an in eg al ope a o whose no m es ima es imply imp o ed Poinca ´e–
Sobole es ima es like
1
|B|ZB
| − B|qdx1/q
≤C a(B) o some q > 1.
The main pu pose o his pape is o show ha he amilia ole o in eg al ope a o s is ins ead
played by a sum ope a o T(x) which is o med by adding a(B) o e an app op ia e chain o
balls associa ed wi h a poin x:
T(x) = X
Bin a chain o x
a(B).
A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 3
In case p0= 1, he sum ope a o becomes an in eg al ope a o , bu in any case, he Lp o Lq
mapping p ope ies o he sum ope a o can be de i ed in much he same ways as hose o
in eg al ans o ms o po en ial ype, and hese no m es ima es o Tlead o co espondingly
mo e gene al Poinca ´e es ima es. We will be able o ob ain such esul s o a ai ly gene al class
o unc ionals a(B) which includes he special choice
a(B) = (B)1
|B|ZB
|X |p0dx1/p0
.
See also [29], [17] and [8] o o he ypes o ope a o s which in ol e adding in eg al a e ages;
he sums in [17] in ol e in eg al a e ages o e annuli, while hose in [29] in ol e a e ages o e
po ions o dyadic “cubes”. On he o he hand, he sums in [8] in ol e concen ic balls cen e ed
a x ha hence ha e coun able o e lapping. In pa icula we imp o e some esul s ob ained in
[17].
We will use he sum ope a o o s eng hen se e al o he sel -imp o ing esul s ob ained in
[16], and also o de i e esul s o he weigh ed BV spaces de ined in [2]. Fo example, we will
p o e he ollowing esul , in which we use he no a ion |B|ω=RBdω o he ω-measu e o B.
Le p0>0 and Xbe a di e en ial ope a o on Rn o which
1
|B|ZB
| − B|dx ≤c (B)1
|B|ZB
|X |p0dx1/p0
o all ρ-balls Band all Lipschi z unc ions . I ωis a measu e which sa is ies he doubling
condi ion
|B|ω≤c (B)
(˜
B)N
|B|ω,˜
B⊂B,
o all ρ-balls ˜
B, B, hen we ha e
(5) 1
|B|ωZB
| − B|qdω1/q
≤C (B)1
|B|ZB
|X |pdx1/p
wi h p, q ela ed by
1
q=1
p−1
N, p0≤p < q < ∞,
and wi h Cindependen o and B. I was p o ed in [16] ha such a esul holds unde he
s onge assump ion ha ω∈A∞(dx) ( he de ini ion o A∞(dx) is gi en a e Co olla y 2.15),
bu we will be able o deduce i by assuming only he doubling condi ion. In ac , a mo e gene al
esul is p o ed in Co olla y 2.16 below, eplacing (5) by
(6) 1
|B|ωZB
| − B|qdω≤C (B)1
|B| dx ZB
|X |p dx1/p
.
In his e sion, Lebesgue measu e dx on he igh side o he conclusion is eplaced by a mo e
gene al measu e dx, p o ided ha p0≤p < q < ∞, ha we eplace ou assump ion abou he
doubling condi ion o o de Nby he balance condi ion
(7) (˜
B)
(B) |˜
B|ω
|B|ω!1/q
≤C |˜
B| dx
|B| dx !1/p
,˜
B⊂B,
and p o ided ∈Ap/p0(dx) (again, he de ini ion o Ap(dx) is gi en a e Co olla y 2.15).
No e ha he possibili y o choosing q=pis no add essed in he esul jus men ioned.
Howe e , in §3, we will show ha i ωis absolu ely con inuous wi h espec o Lebesgue measu e,
i is possible o ea he case q=p>p0≥1 by assuming a s onge e sion o he balance
condi ion, a e sion which we shall e e o as a Fe e man–Phong ype s eng hening o he
4 BRUNO FRANCHI, CARLOS P´
EREZ, AND RICHARD L. WHEEDEN
condi ion; i dω =w dx, his s eng hening in ol es eplacing |˜
B|ωin he nume a o on he le
side o (7) by he la ge quan i y
A (ω, ˜
B) = Z˜
B
w dx1/
|˜
B|1/ 0
o some > 1, 1/ + 1/ 0= 1. See §3 o he exac s a emen s.
We shall e e o inequali ies like (6) as wo-measu e (o wo-weigh ) inequali ies. To illus a e
he gene al in e es o wo-weigh inequali ies in applica ions, conside he pape [5], whe e he
au ho s p o e a Ha nack inequali y o aniso opic degene a e/singula ellip ic equa ions o he
o m di (A(x)Du) = 0 in an open se Ω, when
λ(x)|ξ|2≤ hA(x)ξ, ξi ≤ Λ(x)|ξ|2
o ξ∈Rnand a.e. x∈Ω. The e, a wo-weigh inequali y o he pai o measu es Λ dx and λ dx
is a key ool used in he p oo , and i is ob ained di ec ly om a balance condi ion akin o (7).
Clea ly, enla ging he class o weigh s o which (6) holds would yield Ha nack inequali ies o
mo e gene al classes o pde’s. Following he spi i o [5], he same condi ion is used in [14] o p o e
a compensa ed compac ness heo em and hen a homogeniza ion esul o nonlinea degene a e
ellip ic pde’s wi h oscilla ing coe icien s. Analogously, compac imbedding o weigh ed Sobole
and BV spaces can be deduced om wo-weigh Sobole -Poinca ´e inequali ies (see, e.g., [13]).
Bu wo-weigh inequali ies also a ise when dealing wi h iso opic equa ions o he o m
di (w(x)Du) = 0 in case wdoes no belong o he class A2( he si ua ion o w∈A2is
well-unde s ood due o [9]), bu when ne e heless ωcan be es ima ed om below and om
abo e by weigh s sa isying a wo-weigh Sobole -Poinca ´e inequali y. An assump ion o his
ype is much weake han he A2-condi ion which equi es mo e delica e con ol o he weigh
on e e y ball.
Finally, we no e ha ou mo i a ion o de i ing esul s in a he gene al quasime ic spaces
is ha he heo y hen wo ks in impo an non-Euclidean se ings like Ca no –Ca a h´eodo y
me ic spaces associa ed wi h subellip ic di e en ial ope a o s, g aphs and ac al se s (see e.g.
[17] o e e ences).
2. Main esul s and p oo s.
Th oughou he pape , we shall conside a ixed quasime ic space (S, ρ) endowed wi h a
doubling Bo el measu e µ ha makes (S, ρ, µ) a quasime ic space o homogeneous ype in he
sense ha he ollowing p ope ies hold:
(i) ρ(x, y)≥0 o all x, y ∈ S, and ρ(x, y) = 0 i x=y;
(ii) ρ(x, y) = ρ(y, x) o all x, y ∈ S;
(iii) ρ(x, y)≤Kρ(x, z) + ρ(z, y) o all x, y, z ∈ S.
I x∈ S and > 0, le B(x, ) deno e he ρ-ball cen ed a xo adius , i.e., B(x, ) = {y∈ S :
ρ(x, y)< }. I Bis a ρ-ball, we will o en call Bsimply ‘a ball’, and we will deno e i s adius
by (B) and i s µ-measu e by |B|µ. Mo eo e , i c > 0, we shall deno e by cB he ball wi h he
same cen e as Band such ha (cB) = c (B). Whene e we speak o a “measu e”, we mean
a nonnega i e Bo el measu e.
We always assume ha he ollowing doubling p ope y holds o µ:
(i ) The e exis s A > 0 such ha
|B(x, 2 )|µ≤A|B(x, )|µ
o all x∈ S and .
A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 5
De ini ion 2.1. We say ha a locally ini e Bo el measu e ωbelongs o he class D=D(S, ρ)
i he e is a cons an Aω>1 so ha ωsa is ies he doubling condi ion
(8) |B(x, 2 )|ω≤Aω|B(x, )|ω
o all x∈ S and > 0, whe e we deno e |E|ω=REdω o any measu able se E. In case
ωis absolu ely con inuous wi h espec o µ, i.e., i dω =w dµ o a nonnega i e unc ion
w∈Lloc(dµ), we w i e |E|ω=|E|wdµ and call wa weigh unc ion.
Rema k 2.2.I is easy o see ha (8) implies
(9) |B(x, )|ω≤Aω log2Aω|B(x, )|ω
o > 1, > 0 and x∈ S. We shall say ha ωsa is ies he doubling condi ion o o de Nand
w i e ω∈DN=DN(S, ρ) i
|B(x, )|ω≤C N|B(x, )|ω
o > 1, > 0 and x∈ S. I ω∈D, hen by [30], p. 269, assuming as we shall ha all annuli
B(x, R) B(x, ) wi h 0 < < R a e nonemp y, ωalso sa is ies a e e se doubling condi ion:
he e exis α, β > 1 depending on Aωsuch ha
(10) |B(x, α )|ω≥β|B(x, )|ω
o all > 0, and hence
(11) |B(x, )|ω≥c |B(x, )|ω
o all x∈ S and > 1, whe e and ca e posi i e cons an s depending on αand β. We will
usually be dealing only wi h he class o subballs o some ixed ball B0, and hen we only need
he condi ions abo e o such balls.
GEOMETRIC HYPOTHESES: Le B0be a ixed ball in (S, ρ). We suppose ha o each
x∈B0, he e exis s a chain o balls {Bj}={Bj(x)}∞
j=1 sa is ying
(H1) Bj⊂B0 o all j≥0;
(H2) (Bj)≈2−j (B0) o all j≥0;
(H3) ρ(Bj, x)≤c (Bj) o all j≥0,
whe e ρ(Bj, x) deno es he dis ance om x o Bj, and we assume ha he cons an s in (H2)
and (H3) a e independen o xand j. No e ha he balls Bj(x) may o may no con ain x, bu
he sequence {Bj(x)}depends on x.
F om now on, any posi i e cons an ha depends a mos on K,Aand he cons an s in (H2)
and (H3) will be called a geome ic cons an .
I ollows om (H2), (H3) and (iii) ha
(H4) I j < k hen Bk⊂CBj, whe e Cis a geome ic cons an .
Rema k 2.3.We know om [20] and [15] ha a chain o balls sa is ying (H1)–(H3), and so also
(H4), exis s in me ic spaces sa is ying he segmen (o geodesic) p ope y, i.e., in me ic spaces
such ha o e e y pai o poin s x, y ∈ S he e is a con inuous cu e γ: [0, T]→ S connec ing
xand ysuch ha ρ(γ( ), γ(s)) = | −s| o all s, ∈[0, T ]. In ac , we hen also ha e he ex a
p ope ies
(H5) Fo all j≥0, Bj∩Bj+1 con ains a ball Sjwi h (Sj)≈ (Bj);
(H6) ρ(Bj, x)≈ (Bj) o all j≥0;
(H7) {Bj}has bounded o e laps.
Mo eo e , he cons an s in (H5)–(H7) a e geome ic cons an s.
Typically, Ca no –Ca a h´eodo y and Riemannian me ics sa is y he segmen p ope y (see
Rema k 2.6 o [16] o e e ences).

6 BRUNO FRANCHI, CARLOS P´
EREZ, AND RICHARD L. WHEEDEN
De ini ion 2.4. Le a:B→a(B) be a nonnega i e unc ional de ined on balls B⊂B0. I
x∈B0, le
(12) T(x) =
∞
X
j=0
a(Bj(x)),
whe e {Bj(x)}∞
j=1 is a sequence o balls sa is ying (H1), (H2), and (H3), and B0(x) = B0 o all
x∈B0.
We call T(x) a sum ope a o associa ed wi h he unc ional a(B).
The signi icance o T(x) lies in he ollowing simple poin wise ep esen a ion o mula.
Theo em 2.5. Suppose (H1)–(H3) and (H5) hold. Le ∈L1(B0, µ)be such ha o any ball
B⊂B0,
(13) 1
|B|µZB
| − B|dµ ≤c a(B),
whe e B=1
|B|µRB dµ. Then o µ-a.e. x∈B0,
(14) | (x)− B0| ≤ C T(x),
whe e Cis a geome ic cons an which also depends on he cons an in (13).
Rema k 2.6.We hank P o esso G. Lu o poin ing ou ha he conclusion o Theo em 2.5
holds wi h a weake hypo hesis. In ac , by using he me hods o [19], he le -hand side o (13)
can be eplaced by
1
|B|µZB
| − B|dµ1/
o any  > 0. In pa icula , i can also be eplaced by he weak L1(B, µ) no m o − B, i.e.,
by
k − BkL1,∞(B,µ)= sup
λ>0
λ
|B|µ
|{x∈B:| − B|> λ}|µ.
The ac ha he weak no m can be subs i u ed ollows om Kolmogo o ’s inequali y: i 0 <
q < , hen o nonnega i e measu able unc ions g,
(15) 1
|B|µZB
g(x)qdµ1/q
≤
−q1/q
kgkL ,∞(B,µ),
whe e he no m on he igh is he weak L (B, µ) no m.
Mo eo e , he ole o he cons an s Band B0in (13) and (14) can ins ead be played by ap-
p op ia e polynomials, and in his way, ou main esul s ha e analogues o high o de Poinca ´e–
Sobole es ima es. We e e o [20] o he de ini ion and necessa y p ope ies o polynomials in
quasime ic spaces. Rema k 2.6 also applies he e.
We can now s a e one o ou main esul s, a weak ype es ima e o he ope a o T.
Theo em 2.7. Le 0< q < ∞and ω∈D. Suppose (H1)–(H3) hold, and ha he e exis
posi i e cons an s θand cso ha θ < 1and
(16) X
j
{a(Qj)q|Qj|ω}θ≤c{a(B0)q|B0|ω}θ
o all collec ions {Qj}o pai wise disjoin subballs o B0. Then
(17) sup
λ>0
λ|{x∈B0:T(x)> λ}|1/q
ω≤C a(B0)|B0|1/q
ω,
whe e Cis a geome ic cons an which also depends on he cons an in (16).
A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 7
Rema k 2.8.Since θ < 1, condi ion (16) implies ha aand ωalso sa is y he condi ion (called
Dqin [16])
X
j
a(Qj)q|Qj|ω≤c a(B0)q|B0|ω
o any amily {Qj}o pai wise disjoin subballs o B0. In ac , his condi ion is weake han
(16) and co esponds o he limi case θ= 1 (no allowed he e). Howe e , we s ess again ha
he p esen esul s, unlike hose in [16], do no equi e ha ω∈A∞(µ).
Rema k 2.9.In Theo ems 2.5 and 2.7, he chain {Bj(x)}is no equi ed o sa is y ei he (H6)
o (H7).
In P oposi ion 2.13 below, we will gi e impo an examples o unc ionals a(B) which sa is y
(16). He e we men ion he simple special case when
a(B) = (B)1
|B|µZB
gpdµ1/p
o 1 ≤p < ∞and a ixed unc ion g≥0 (e.g., g=|X | o some , whe e Xis a di e en ial
ope a o ). In ac , o his choice o a(B), we shall see ha i µ∈DN hen (16) is alid o
ω=µ, 1≤p < N, 1/q = 1/p −1/N and θ=p/q, i.e.,
X
j
a(Qj)p|Qj|p/q
µ≤c a(B0)p|B0|p/q
µ,1
q=1
p−1
N,1≤p < N
o any amily o pai wise disjoin subballs o B0.
As a i s consequence o Theo ems 2.5 and 2.7, we shall de i e he ollowing weak sel -
imp o ing p ope y o Poinca ´e’s inequali y in B0.
Theo em 2.10. Le (H1)–(H3) and (H5) hold. Suppose also ha ω∈Dand (16) holds o
some θ < 1and some 1< q < ∞. I is a eal- alued Bo el unc ion on B0 ha sa is ies
(18) 1
|B|µZB
| −cB|dµ ≤c a(B)
o e e y ball B⊂B0, whe e cBis a eal numbe depending on Band , hen
(19) sup
λ>0
λ|{x∈B0:| (x)− B0|> λ}|1/q
ω≤C a(B0)|B0|1/q
ω,
whe e B0=1
|B0|µRB0 dµ and Cis a geome ic cons an which also depends on he cons an s
in (16) and (18).
We now p o e Theo ems 2.5, 2.7 and 2.10, beginning wi h Theo em 2.7. Th oughou he
p oo s, we shall deno e by c, C di e en posi i e cons an s which may change om place o
place.
P oo o Theo em 2.7. Fo J o be chosen and x∈B0, w i e
T(x) =
∞
X
j=0
a(Bj(x)) =
J
X
j=0
+
∞
X
j=J+1
=I+II.
Then
I=
J
X
j=0
a(Bj(x)) |Bj(x)|1/q
ω· |Bj(x)|−1/q
ω
≤c a(B0)|B0|1/q
ω
J
X
j=0
|Bj(x)|−1/q
ω,
8 BRUNO FRANCHI, CARLOS P´
EREZ, AND RICHARD L. WHEEDEN
by he one- e m e sion o (16), namely
a(B)|B|1/q
ω≤c a(B0)|B0|1/q
ω, B ⊂B0.
By (H4), BJ(x)⊂CBj(x) i j≤J, and hen by e e se doubling (11), (H2) and (9),
|BJ(x)|ω≤c (BJ(x))
(Bj(x)) 
|CBj(x)|ωi j≤J
≤c2(j−J)|Bj(x)|ω.
Thus
J
X
j=0
|Bj(x)|−1/q
ω≤c
J
X
j=0
2(j−J)/q|BJ(x)|−1/q
ω≤c|BJ(x)|−1/q
ω,
and so
I≤c a(B0)|B0|1/q
ω|BJ(x)|−1/q
ω.
No ice now by (H3) ha he e exis s a geome ic cons an α > 1 such ha x∈αBj(x) o all
j≥0. Thus, we can w i e
II =
∞
X
j=J+1
a(Bj(x)) =
∞
X
j=J+1 a(Bj(x)) |Bj(x)|
1
q−1
θq
ω|Bj(x)|
1
θq −1
q
ω
≤"sup
B:B⊂B0,x∈αB
a(B)|B|
1
q−1
θq
ω#∞
X
j=J+1
|Bj(x)|
1
θq −1
q
ω.
I j≥J+ 1, hen Bj(x)⊂CBJ(x) by (H4), and consequen ly by (11),
|Bj(x)|ω≤c (Bj(x))
(BJ(x))
|CBJ(x)|ωi j≥J+ 1.
Since 1/(θq)−1/q > 0, i ollows by using (H2) ha
∞
X
j=J+1
|Bj(x)|
1
θq −1
q
ω≤c|BJ(x)|
1
θq −1
q
ω.
Le ing S(x) be de ined by
(20) S(x) = sup
B:B⊂B0,x∈αB
a(B)|B|
1
q−1
θq
ω,
we ob ain
II ≤c S(x)|BJ(x)|
1
θq −1
q
ω.
Hence
(21) T(x) = I+II ≤ca(B0)|B0|1/q
ω|BJ(x)|−1/q
ω+S(x)|BJ(x)|
1
θq −1
q
ω.
We claim ha
(22) T(x)≤c S(x)θha(B0)|B0|1/q
ωi1−θ.
I S(x) is in ini e, (22) is ob ious. I S(x) is ini e, pick Jsuch ha he wo e ms on he igh
side o (21) a e compa able, i.e., so ha
|BJ(x)|1/(θq)
ω≈a(B0)|B0|1/q
ω
S(x).
A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 9
Indeed, o see ha his choice is possible, le M=a(B0)|B0|1/q
ω/S(x). Then
0< M ≤a(B0)|B0|1/q
ω
a(B0)|B0|
1
q−1
θq
ω
=|B0|1/(θq)
ω,
o equi alen ly |B0|ω≥Mθq. By e e se doubling, |Bj(x)|ω→0 as j→ ∞. Hence, he e exis s
Jsuch ha
|BJ+1(x)|ω< Mθq and |BJ(x)|ω≥Mθq.
Thus |BJ+1(x)|ω<|BJ(x)|ω. On he o he hand, |BJ+1(x)|ω≈ |BJ(x)|ωsince by (H4),
(23) BJ+1(x)⊂CBJ(x),
and hen we ha e
|BJ(x)|ω≤ |CBJ(x)|ω≤ (CBJ(x))
(BJ+1)N
|BJ+1(x)|ω≤c|BJ+1(x)|ω,
whe e he nex - o-las inequali y ollows om (23) and doubling applied o he balls BJ+1(x)
and CBJ(x), and he las inequali y ollows om (H2). Hence |BJ(x)|ω≈Mθq, as desi ed, and
we hen ob ain (22) by di ec compu a ion.
I T(x)> λ, hen (22) implies ha
λ < c S(x)θha(B0)|B0|1/q
ωi1−θ.
Hence, by de ini ion o S(x), he e exis s a ball Bxwi h Bx⊂B0,x∈αBxand
λ<c a(Bx)|Bx|
1
q−1
θq
ωθha(B0)|B0|1/q
ωi1−θ,
so ha
(24) λq|Bx|ω≤c a(Bx)θq|Bx|θ
ωha(B0)|B0|1/q
ωiq(1−θ).
Since he collec ion o balls C={αBx:x∈B0and T(x)> λ}co e s {x∈B0:T(x)> λ}, an
a gumen o Vi ali ype shows ha he e is a disjoin coun able sub amily {αBk}∞
k=1 o C( hus
(24) holds o each Bk) and a geome ic cons an α1>1 such ha
{x∈B0:T(x)> λ} ⊂
∞
[
k=1
α1αBk.
Hence
|{x∈B0:T(x)> λ}|ω≤
∞
X
k=1
|α1αBk|ω
≤c
∞
X
k=1
|Bk|ωby doubling
≤c
λq
∞
X
k=1
a(Bk)θq|Bk|θ
ωha(B0)|B0|1/q
ωiq(1−θ),
by (24). Since he balls Bka e disjoin and lie in B0, we ob ain om (16) ha
|{x∈B0:T(x)> λ}|ω≤c
λqa(B0)θq|B0|θ
ωha(B0)|B0|1/q
ωiq(1−θ)
=c
λqa(B0)q|B0|ω.
16 BRUNO FRANCHI, CARLOS P´
EREZ, AND RICHARD L. WHEEDEN
Theo em 2.22. Le (S, ρ, µ)be a me ic space o homogeneous ype which is also o Poinca ´e
ype (i.e., (43) holds o all ∈Liploc(S,R)), and suppose ha (H1)–(H3) and (H5) hold. Le
w dµ ∈Dbe such ha he e exis s q > 1so ha
(B)
(B0)|B|ωdµ
|B0|ωdµ 1/q
≤C|Q|µ
|B0|µ
o all balls B⊂B0. Then
(44) 1
|B|ωdµ ZB
| − B|qω dµ1/q
≤C (B)
|B|µ
kD kS(B)
o all ∈BVS(B0)and B⊂B0.
Rema k 2.23.I we assume in addi ion ha (S, ρ) enjoys he segmen p ope y, hen bo h
Theo ems 2.19 and 2.22 could be p o ed al e na i ely h ough a ep esen a ion o mula and
an Lp, Lqcon inui y esul o in eg al ope a o s o po en ial ype in spaces o homogeneous
ype. See [20] o he o m o his ep esen a ion, and see e.g. [10] o he Lp, Lqcon inui y
esul . Simila ep esen a ion o mulas we e in oduced ea lie in [11] and [15] in case a s onge
assump ion is sa is ied by he measu es in ol ed. In he case o Theo em 2.19 o example, he
assump ion equi es he exis ence o c > 0 such ha o all balls B,˜
Bwi h ˜
B⊂B⊂B0,
|B|wdx
|˜
B|wdx
≥c (B)
(˜
B),
which ails o hold o gene al A∗
1weigh s ( hink o ins ance o w(x) = |x|−n+ o 0 <  < 1).
Howe e , by [20], his s onge condi ion is no equi ed i he ep esen a ion o mula in [11],
[15] is al e ed sligh ly by adding an innocuous cons an e m o he igh -hand side.
3. The case p=q.
In his sec ion, we shall conside he special case when he unc ional a(B) is gi en by
(45) a(B) = (B)1
|B|µZB
gp0dµ1/p0
, B ⊂B0,
whe e p0≥1, µ∈D, and g≥0. We will no need o assume ha gis a de i a i e, bu we will
assume ha ωis absolu ely con inuous wi h espec o µ:
dω =w dµ.
Ou goal is o de i e an analogue o Co olla y 2.16 in which qis allowed o equal pi p > p0,
i.e., o p o e ha o app op ia e wand , he es ima e
1
|B0|wdµ ZB0
| − B0|pw dµ1/p
≤C (B0)1
|B0| dµ ZB0
gp dµ1/p
wi h p > p0can be deduced om an ini ial assump ion o he o m
1
|B|µZB
| − B|dµ ≤C (B)1
|B|µZB
gp0dµ1/p0
o all balls B⊂B0. The exac s a emen is gi en in Theo em 3.1 below. I will be con enien
o assume as we may ha supp g ⊂B0, and hen o de ine a(B) by he same o mula o all
B⊂ S.
Le
A (w, B) = ZB
w dµ1/
|B|1/ 0
µ, > 1,1
+1
0= 1.
No e ha |B|wdµ ≤ A (w, B) o any wby H¨olde ’s inequali y, and ha i w∈A∞(dµ), hen
|B|wdµ ≈ A (w, B) uni o mly in Bi is su icien ly close o 1. In his sec ion, in o de o p o e

A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 17
a di ec s ong ype es ima e o he sum ope a o T(x), we will assume a di e en o m o he
balance condi ion. We will assume ha o a gi en p > p0, he e exis s > 1 so ha o all
B⊂cB0(c > 1 is an app op ia e geome ic cons an ),
(46)  (B)
(B0)p0A (w, B)
|cB0|wdµ p0
pA (σ, B)
|cB0|σdµ 1−p0
p≤C|B|µ
|B0|µ
, σ = −1
(p/p0)−1.
We e e o his condi ion as a s eng hened balance condi ion in he Fe e man–Phong sense.
In case ∈Ap/p0(dµ), i is easy o check ha (46) amoun s o he balance condi ion
(47) (B)
(B0)A (w, B)
|cB0|wdµ 1/p
≤C|B| dµ
|B0| dµ 1/p
, B ⊂cB0.
Mo eo e , i w∈A∞(dµ), (47) is equi alen o (36) in case p=q(and ν=µ):
(B)
(B0)|B|ωdµ
|B0|ωdµ 1/p
≤c|B| dµ
|B0| dµ 1/p
, B ⊂B0.
We will p o e he analogue o Co olla y 2.16 gi en in he nex heo em.
Theo em 3.1. Assume ha (H1)–(H3) and (H5) hold o a ball B0in a space (S, ρ, µ)o
homogeneous ype. Le be a unc ion which sa is ies
1
|B|µZB
| − B|dµ ≤C (B)1
|B|µZB
gp0dµ1/p0
, B ⊂B0,
o some p0≥1and some unc ion g≥0. I wand a e a pai o weigh s so ha he balance
condi ion (47) holds o some p > p0, > 1and all B⊂cB0, and i ∈Ap/p0(dµ), hen
ZB0
| − B0|pw dµ1/p
≤C|cB0|1/p
wdµ (B0)1
|B0| dµ ZB0
gp dµ1/p
.
No e ha he unc ion gabo e is no assumed o be a de i a i e. No e also ha w dµ is no
assumed o be a doubling measu e; i w dµ is doubling hen we may ake c= 1 in he conclusion.
We will use he ollowing esul abou sum ope a o s as a basis o de i ing Theo em 3.1.
Theo em 3.2. Assume ha (H1)–(H3) hold o a ball B0in a space (S, ρ, µ)o homogeneous
ype. Le Tbe he sum ope a o o med by using he unc ional a(B)in (45) o some p0≥1.
Le wand be weigh s which sa is y (46) o some p > p0and all B⊂cB0. Then
(48) ZB0
Tpw dµ1/p
≤CB0ZB0
gp dµ1/p
wi h
CB0=C (B0)|cB0|
1
p
wdµ|cB0|
1
p0−1
p
σdµ
|B0|
1
p0
µ
.
Rema k 3.3.As always, µis assumed o be a doubling measu e bu none o w, o σis assumed
o be a doubling weigh . I ∈Ap/p0(dµ), hen (48) means simply ha
kTkLp
wdµ(B0)≤C (B0)|cB0|1/p
wdµ 1
|B0| dµ ZB0
gp dµ1/p
,
since i ∈Ap/p0(dµ) hen
|cB0|
1
p0−1
p
σdµ
|B0|
1
p0
µ
≤C
|B0|
1
p
dµ
.
18 BRUNO FRANCHI, CARLOS P´
EREZ, AND RICHARD L. WHEEDEN
Finally, no e ha Theo em 3.2 is a s ong ype esul , as opposed o ou ea lie weak ype
esul abou T(x).
P oo o Theo em 3.2. To p o e he heo em, we will use a g id o dyadic se s in Swhich
a e “almos balls”, as cons uc ed in [28]. In ac , he ollowing has been p o ed he e:
I τ= 8K5(whe e Kis he quasime ic cons an o ρ), hen o any (la ge
nega i e) in ege m, he e a e poin s {xk
j}and a amily Dm={Dk
j}o se s o
k=m, m + 1,· · · and j= 1,2,· · · such ha
•B(xk
j, τk)⊂Dk
j⊂B(xk
j, τk+1)
•Fo each k=m, m + 1,· · · , he amily {Dk
j}is pai wise disjoin in j, and
S=∪jDk
j.
•I m≤k < l, hen ei he Dk
j∩Dl
i=∅o Dk
j⊂Dl
i.
We call he amily D=∪m∈ZDma dyadic cube decomposi ion o Sand e e o he se s in Das
dyadic cubes. A dyadic cube will usually be deno ed by Q, and B(Q) will deno e he con aining
ball desc ibed abo e wi h 1
τB(Q)⊂Q⊂B(Q); hus, i Q=Dk
j hen B(Q) = B(xk
j, τk+1). We
se `(Q) = (B(Q))/τ and call `(Q) he “sideleng h” o Q. We no e ha while he cubes in
each Dmha e he dyadic p ope ies lis ed abo e, he e may be no nes edness p ope ies o he
cubes in Dm1 ela i e o he cubes in Dm2i m1, m2a e di e en .
Since supp g ⊂B0, he heo em will ollow by p o ing (48) wi h in eg a ion on he igh -hand
side ex ended o e S. Le x∈B0. By de ini ion,
T(x) = Xa(B),
whe e he sum is o e all balls Bin a chain o x. De ine
Tm(x) = X
B: (B)≥τm
a(B),
whe e he sum is only o e hose balls in he same chain whose adius is a leas τm. Since
Tm(x) inc eases o T(x) as m→ −∞, i is enough o p o e (48) wi h T eplaced by Tm o he
same cons an CB0(independen o m).
Fix m. I Bbelongs o he chain o xand (B)≥τm, hen i (B)≈2−n (B0), n≥0, we
can choose pai wise disjoin dyadic cubes Qn
`∈ Dm,`= 1, . . . , N, o compa able size o B(i.e.,
wi h `(Qn
`)≈ (B)) such ha B⊂SN
`=1 Qn
`. In ac , Ncan be chosen o be independen o B.
I Qis a dyadic cube, le
a(Q) = `(Q)1
|Q|µZQ
gp0dµ1/p0
.
Since (B)≈`(Qn
`), i ollows om doubling ha |B|µ≈ |Qn
`|µ. Thus, since he Qn
`a e disjoin
in `, he e is a geome ic cons an cdepending possibly also on Nand p0so ha
a(B) = (B)1
|B|µZB
gp0dµ1/p0
≤cX
`
`(Qn
`) 1
|Qn
`|µZQn
`
gp0dµ!1/p0
=cX
`
a(Qn
`).
Since ρ(x, B)≤c (B) (by (H3)), hen ρ(x, Qn
`)≤c `(Qn
`) o all `. Hence,
(49) Tm(x)≤cX
`, n :Qn
`∈Dm
ρ(x,Qn
`)≤c `(Qn
`)
a(Qn
`).
A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 19
By duali y,
kTmkLp
wdµ(B0)= sup
h≥0, supp h⊂B0
khkLp0
dµ(B0)=1 ZTmh w 1
pdµ, 1
p+1
p0= 1.
Since B⊂B0, i is easy o see ha each Qn
`is con ained in cB0, and consequen ly we ob ain
ZTmh w1
pdµ ≤cX
Q∈Dm;Q⊂cB0
a(Q)ZcB(Q)
h w1
pdµ
=cX
Q∈Dm;Q⊂cB0
`(Q)1
|Q|µZQ
gp0dµ1
p0ZcB(Q)
h w1
pdµ := c S.
To p o e Theo em 3.2, i is enough o show ha
(50) S≤CB0Zgp dµ1
pZhp0dµ1
p0
.
We may assume wi hou loss o gene ali y ha
1
|B0|µZB0
gp0dµ1
p0= 1.
Fo γ > 1 o be chosen and k∈Z, le
(51) Ck={Q∈ Dm:Q⊂cB0;γk<1
|Q|µZQ
gp0dµ1
p0≤γk+1}.
Then
S=X
Q∈Dm;Q⊂cB0
`(Q)1
|Q|µZQ
gp0dµ1
p0ZcB(Q)
h w1
pdµ
=X
kX
Q∈Ck
`(Q)1
|Q|µZQ
gp0dµ1
p0ZcB(Q)
h w1
pdµ
=X
k≤0
+X
k≥1
:= S1+S2.
Le us i s es ima e S1. We ha e
S1≤X
k≤0X
Q∈Ck
`(Q)γk+1 ZcB(Q)
h w1
pdµ
≤X
k≤0
γk+1 X
Q∈Dm:Q⊂cB0
`(Q)ZcB(Q)
h w1
pdµ.
We claim ha i Bis any ball, hen
(52) X
Q∈Dm;Q⊂cB
`(Q)ZcB(Q)
h w1
pdµ ≤c (B)ZcB
h w1
pdµ.
To p o e (52), no e ha he le -hand side o (52) is a mos
X
`:τ`≤c (B)X
Q∈Dm, Q⊂cB
`(Q)=τ`
τ`ZcB(Q)
h w1
pdµ
20 BRUNO FRANCHI, CARLOS P´
EREZ, AND RICHARD L. WHEEDEN
≤X
`:τ`≤c (B)
τ`ZcB 
X
Q∈Dm:`(Q)=τ`
χcB(Q)
h w1
pdµ := I.
Bu
X
Q∈Dm:`(Q)=τ`
χcB(Q)(y)≤C
uni o mly wi h espec o `(c . (59) o [26]). Thus since
X
`:τ`≤c (B)
τ`≤c (B),
we ob ain
I≤C (B)ZcB
h w1
pdµ.
which p o es (52).
Going back o S1, we ob ain om (52) ha
S1≤c (B0)ZcB0
h w1
pdµ.
Thus, since 1
|B0|µRB0gp0dµ = 1, we may w i e
S1≤c (B0)1
|B0|µZB0
gp0dµ1
p0ZcB0
h w1
pdµ
=c (B0)1
|B0|µZB0
gp0 1
s −1
sdµ1
p0ZcB0
h w1
pdµ
wi h s=p/p0. By H¨olde ’s inequali y,
S1≤c (B0)1
|B0|p0
µZB0
gp dµ1
pZB0
−s0
sdµ1
p0s0ZcB0
hp0dµ1
p0
|cB0|
1
p
wdµ
≤CB0ZB0
gp dµ1
pZcB0
hp0dµ1
p0
.
This comple es ou es ima ion o S1.
To es ima e S2, le {Qk
j}jbe he maximal dyadic cubes in Dmwi h
1
|Q|µZQ
gp0dµ1
p0> γk.
The Qk
ja e disjoin in jby maximali y. We do no assume Qk
j⊂cB0, bu i k≥1, his mus
be so o a sui ably la ge geome ic cons an cp o ided γis la ge, as we now show. In ac , i
Qk
jis no con ained in cB0and cis su icien ly la ge depending on he quasime ic cons an K,
hen `(Qk
j) is a leas compa able o (B0) since Qk
jmus in e sec B0(due o he suppo o
g). Consequen ly, we mus ha e |B0|µ≤c1|Qk
j|µby doubling, wi h c1depending on c, and hen
1 = 1
|B0|µZB0
gp0dµ1
p0=1
|B0|µZB0
gp0dµ1
p0
≥c−1
p0
1 1
|Qk
j|µZQk
j
gp0dµ!1
p0
≥c−1
p0
1γk,
which is impossible o k≥1 i γis su icien ly la ge.
A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 21
Thus Qk
j⊂cB0i k≥1. By maximali y and since µis doubling we ha e
(53) γk< 1
|Qk
j|µZQk
j
gp0dµ!1
p0
< cγk≤γk+1,
i γis la ge, so ha Qk
j∈ Ckwhen k≥1. On he o he hand, again by maximali y, any cube
Q∈ Ckis con ained in a cube Qk
j o some j. Then
S2≤cX
k
γk+1 X
jX
Q∈Dm:Q⊂Qk
j
`(Q)ZcB(Q)
h w1
pdµ.
I we w i e Bk
j=B(Qk
j) and apply (52), we ob ain ha S2is bounded by
cX
k
γk+1 X
j
`(Qk
j)ZcBk
j
h w1
pdµ
≤cγ X
k,j
`(Qk
j) 1
|Qk
j|µZQk
j
gp0dµ!1
p0ZcBk
j
h w1
pdµ
=cγ X
k,j
a(Qk
j)ZcBk
j
h w1
pdµ.(54)
By H¨olde inequali y wi h exponen s (p )0,p ,
ZcBk
j
h w1
pdµ ≤ ZcBk
j
h(p )0
dµ!1
(p )0 ZcBk
j
w dµ!1
p
≤ ZcBk
j
h(p )0
dµ!1
(p )0
A (w, cBk
j)1
p|cBk
j|−1
p 0
µ.
Then, by H¨olde ’s inequali y o p, p0, (54) and so also S2is bounded by
(55) cγ 
X
k,j
a(Qk
j)pA (w, cBk
j)

1
p

X
k,j ZcBk
j
h(p )0
dµ!p
0
(p )0
|Qk
j|−p
0
p 0
µ


1
p0
.
We s ess he ac ha Qk
j, Bk
j⊂cB0, as we p o ed abo e. No e ha −p0/p 0=−p0/(p )0+ 1,
so he second ac o in (55) is


X
k,j 1
|Qk
j|µZcBk
j
h(p )0
dµ!p
0
(p )0
|Qk
j|µ


1
p0
.
Le
Ωk={x: sup
Q∈Dm:x∈Q1
|Q|µZQ
gp0dµ1
p0> γk}.
Then Ωk=SjQk
j. Le
Ek
j=Qk
j Ωk+1.
No e Ek
j⊂Ωk Ωk+1, and he e o e he se s {Ek
j}a e disjoin in bo h kand j. We claim ha
(56) |Qk
j|µ≤2|Ek
j|µ.

22 BRUNO FRANCHI, CARLOS P´
EREZ, AND RICHARD L. WHEEDEN
I so, he second ac o in (55) is bounded by


X
k,j 1
|Qk
j|µZcBk
j
h(p )0
dµ!p0/(p )0
2|Ek
j|µ


1/p0
.
On he o he hand, i x∈Ek
j hen x∈Qk
j, so ha i we deno e by M he Ha dy-Li lewood
maximal unc ion de ined by
M( )(x) = sup
B:x∈B
1
|B|µZB
| |dµ,
we ob ain om he doubling o µ ha
1
|Qk
j|µZcBk
j
h(p )0
dµ ≤cM h(p )0(x) i x∈Ek
j.
Hence, he second ac o in (55) is bounded by
c
X
j,k ZEk
j
Mh(p )0p0
(p )0dµ

1
p0
(57) ≤c"ZMh(p )0p0
(p )0dµ#1
p0
≤cZhp0
dµ1
p0
since p0/(p )0>1.
To p o e (56), i is enough o show ha
|Qk
j∩Ωk+1|µ≤1
2µ(Qk
j).
W i e
|Qk
j∩Ωk+1|µ=|Qk
j∩[
i
Qk+1
i|µ
=X
i
|Qk
j∩Qk+1
i|µ.
I Qk
j∩Qk+1
i6=∅, hen ei he Qk+1
i⊂Qk
jo Qk
j⊂Qk+1
iand Qk
j6=Qk+1
i.Bu he las is
impossible, since by maximali y o Qk
ji would imply ha
γk≥ 1
|Qk+1
i|µZQk+1
i
gp0dµ!1
p0
,
A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 23
which is alse since he igh -hand side exceeds γk+1. Thus Qk+1
i⊂Qk
ji he wo in e sec , so
ha
|Qk
j∩Ωk+1|µ=X
i:Qk+1
i⊂Qk
j
|Qk+1
i|µ
≤X
i:Qk+1
i⊂Qk
j
1
γ(k+1)p0ZQk+1
i
gp0dµ
≤1
γ(k+1)p0ZQk
j
gp0dµ since he Qk+1
ia e disjoin in i
≤1
γ(k+1)p0(cγk)p0|Qk
j|µby (57)
=c
γp0
|Qk
j|µ≤1
2µ(Qk
j)
i γis chosen su icien ly la ge. Thus, ou claim (56) is p o ed.
We now wan o es ima e he i s ac o in (55). Recall ha
a(Q) = `(Q)1
|Q|µZQ
gp0dµ1
p0
and p > p0≥1. Then, w i ing again Bk
j=B(Qk
j) and se ing s=p/p0, we ha e
X
k,j
a(Qk
j)pA (w, cBk
j)≤cX
k,j
(Bk
j)p 1
|Bk
j|µZBk
j
gp0 1
s −1
sdµ!s
A (w, cBk
j).
By H¨olde ’s inequali y wi h exponen s (s0 )0, s0 , he las sum is bounded by
cX
k,j
(Bk
j)p|Bk
j|−s
µ ZBk
j
gp0(s0 )0 (s0 )0
sdµ!s
(s0 )0 ZBk
j
−s0
sdµ!s
s0
A (w, cBk
j).(58)
Remembe ha by de ini ion o A ,
ZBk
j
−s0
sdµ!s
s0
=A  −s0
s, Bk
js
s0
|Bk
j|−s
s0 0
µ.
In addi ion, since s/s0=s−1 = p/p0−1, we ha e by (46) ha i Bis any subball o cB0, hen
 (B)
(B0)pA (w, B)
|cB0|wdµ 

A ( −s0
s, B)
RcB0 −s0
sdµ

s
s0
≤c|B|µ
|B0|µp
p0.
Applying his wi h B=cBk
j, ecalling ha µis doubling, and w i ing cin place o c2as
necessa y, we ob ain ha (58) is bounded by
c (B0)p|cB0|wdµ RcB0 −s0
sdµs
s0
|B0|
p
p0
µ
(59)
·X
j,k
|Bk
j|
p
p0−s−s
s0 0
µ ZBk
j
gp0(s0 )0 (s0 )0
sdµ!s
(s0 )0
.
24 BRUNO FRANCHI, CARLOS P´
EREZ, AND RICHARD L. WHEEDEN
The i s ac o in (59) is p ecisely he scaling ac o Cp
B0appea ing in (48). To es ima e he
second ac o in (59) (i.e., he sum), no e ha
p
p0
−s−s
s0 0=−s
s0 0= 1 −s
(s0 )0
since
s
(s0 )0−s
s0 0=s1
(s0 )0−1
s0 0
=s1−1
s0 −1
s0 0=s1−1
s0=s1
s= 1.
The e o e, he sum in (59) equals
X
j,k 1
|Bk
j|µZBk
j
gp0(s0 )0 (s0 )0
sdµ !s
(s0 )0
|Bk
j|µ,
which as be o e (using |Bk
j|µ≈ |Qk
j|µ≤c|Ek
j|µ) is bounded by
cZMgp0(s0 )0 (s0 )0
ss
(s0 )0
dµ.
Since s
(s0 )0>1, he las in eg al is a mos
cZgp0(s0 )0 (s0 )0
ss
(s0 )0
dµ =cZgp0s dµ =cZgp dµ.
Combining es ima es and aking he p- h oo shows ha he i s ac o in (55) is bounded by
CB0Rgp dµ1/p. Using his oge he wi h he es ima e (57) o he second ac o in (55), we
see ha ha (55), and so also S2, is bounded by
CB0Zgp dµ1/p Zhp0dµ1/p0
.
We ha e al eady shown ha S1has he same bound, and he e o e so does S, i.e., (50) holds,
and he p oo is comple e. 
P oo o Theo em 3.1.
The hypo hesis o Theo em 3.1 oge he wi h Theo em 2.5 gi es | (x)− B0| ≤ cT(x) o
µ-a.e. x∈B0, whe e Tis he sum ope a o o med by using he unc ional
a(B) = (B)1
|B|µZB
gp0dµ1/p0
.
By hypo hesis, ∈Ap/p0(dµ) and he balance condi ion (47) holds. Applying Theo em 3.2 (see
Rema k 3.3 in pa icula ), Theo em 3.1 ollows immedia ely. 
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B uno F anchi: Dipa imen o di Ma ema ica, Uni e si `
a di Bologna, Piazza di po a San Do-
na o, 5, 40126 Bologna, I aly.
E-mail add ess:[email p o ec ed]
Ca los P´
e ez: Depa men o de An´
alisis Ma em´
a ico, Facul ad de Ma em´
a icas, Uni e sidad de
Se illa, 41080 Se illa, Spain.
E-mail add ess:[email p o ec ed]
Richa d. L. Wheeden: Depa men o Ma hema ics, Ru ge s Uni e si y, New B unswick, New
Je sey 08903, USA.
E-mail add ess:[email p o ec ed]