Two-weigh , Weak- ype No m Inequali ies o
F ac ional In eg als,
Calde ´
on-Zygmund Ope a o s and Commu a o s
D. CRUZ-URIBE,SFO,C.P´
EREZ
ABSTRACT.Wegi eAp- ype condi ions which a e sufficien
o he wo-weigh , weak- ype (p, p) inequali ies o ac ional
in eg al ope a o s, Calde ´
on-Zygmund ope a o s and commu a-
o s. Fo ac ionalin eg alope a o s, hissol esap oblemposed
by Sawye and Wheeden [28]. A he hea o all o ou p oo s
is an inequali y ela ing he Ha dy-Li lewood maximal unc ion
and he sha p maximal unc ion which is s ongly eminiscen o
he good-λinequali y o Feffe man and S ein [13].
.1INTRODUCTION
Le Mbe he Ha dy–Li lewood maximal ope a o . Gi en a pai o weigh s
(u, ) and p,1<p<∞, i is well known ha he weak- ype inequali y
u({x∈Rn:M (x) > })≤C
pRn| |p dx(1.1)
holds i and only i (u, ) ∈Ap: he e exis s a posi i e cons an Ksuch ha o
all cubes Q,
1
|Q|Qudx1
|Q|Q −p/p dxp/p
≤K.(1.2)
697
Indiana Uni e si y Ma hema ics Jou nal c
, Vol. 49, No. 2 (2000)
698 D. CRUZ-URIBE,SFO&C. P´
EREZ
Fo o he classical ope a o s, howe e , he Apcondi ion is no sufficien o he
weak (p, p) inequali y. In ac , o he ope a o s we a e in e es ed in, a necessa y
andsufficien condi ion o heweak(p, p) inequali yisknownonly o ac ional
in eg alope a o s. (SeeSawye [27].) This esul isin e es ingandimpo an , bu
i has he d awback ha he condi ion in ol es he ac ional in eg al ope a o .
Sufficien , Ap- ype condi ions can also be go en om sufficien condi ions
o he s ong (p, p) inequali y. Neugebaue [18] showed ha
1
|Q|Qu dx1/ p 1
|Q|Q − p/p dx1/ p
≤C, > 1,(1.3)
issufficien o hes ong(p, p) inequali y o hemaximalope a o , o Calde ´
on-
Zygmundope a o sandcommu a o s. Sawye andWheeden[28]showed ha o
0<α<n,
|Q|α/n 1
|Q|Qu dx1/ p 1
|Q|Q − p/p dx1/ p
≤C, > 1,(1.4)
is sufficien o he s ong- ype (p, p) inequali y o ac ional in eg al ope a o s.
(Addi ionalsufficien condi ionsa e oundin[20],[21],and[24]. Wegi ep ecise
de ini ions o hese ope a o s in Sec ion 2 below.)
In gene al, sufficien condi ions o he weak (p, p) inequali y which a e de-
i ed om s ong (p, p) condi ions a e no sha p. The pu pose o his pape is
o show ha o he ope a o s we conside , he e a e condi ions ha a e weake
han (1.3) and (1.4), which a e sufficien o he weak- ype inequali y. Roughly, i
suffices o s eng hen he Apcondi ion (1.2) by in oducing a “powe bump” on
he le -hand e m alone, a he han on bo h e ms as in (1.3) and (1.4).
Ou i s esul is o ac ional in eg al ope a o s. I sol es a p oblem posed
by Sawye and Wheeden [28].
Theo em 1.1. Gi en a pai o weigh s (u, ),p,1<p<∞, and α,0<α<
n, suppose ha o some >1and o all cubes Q,
|Q|α/n 1
|Q|Qu dx1/ p 1
|Q|Q −p/p dx1/p
≤C<∞.(1.5)
Then he ac ional in eg al ope a o Iαsa is ies he weak (p, p) inequali y
u({x∈Rn:|Iα (x)|> })≤C
pRn| |p dx.(1.6)
Ou second esul is o Calde ´
on-Zygmund ope a o s.
Two-weigh , Weak- ype No m Inequali ies 699
Theo em 1.2. Le Tbe a Calde ´
on-Zygmund ope a o . Gi en a pai o weigh s
(u, ) and p,1<p<∞, suppose ha o some >1and o all cubes Q,
1
|Q|Qu dx1/ p 1
|Q|Q −p/p dx1/p
≤C<∞.(1.7)
Then Tsa is ies he weak (p, p) inequali y
u({x∈Rn:|T (x)|> })≤C
pRn| |p dx.(1.8)
Rema k 1.3. Though o cla i y we ha e s a ed Theo em 1.2 o Calde ´
on-
Zygmund ope a o s, i is ue o amuchla ge classo ope a o s. Tobep ecise: i
he eexis ssomeδ,0<δ<1,andacons an Cδsuch ha o e e y ∈C∞
0(Rn),
M#(|T |δ)(x)1/δ ≤CδM (x),(1.9)
hen (1.7) implies (1.8).
Al a ez and P´
e ez [3] showed ha inequali y (1.9) holds o Calde ´
on-
Zygmund ope a o s. In his case i can be hough o as ex ending he classical
es ima e
M#(T )(x) ≤C M(| | )(x)1/ ,(1.10)
whe e Tis a egula singula in eg al ope a o and >1, (see Ga c´
ıa-Cue a and
Rubio de F ancia [14, p. 204]). In some sense, (1.9) con ains mo e in o ma ion
han (1.10) since he la e does no suffice o p o e Theo em 1.2.
Al a ez and P´
e ez also showed ha inequali y (1.9), and so Theo em 1.2,
hold o he ollowing ope a o s: weakly s ongly singula in eg al ope a o s (see
C. Feffe man [12]), some pseudo-diffe en ial ope a o s in he H¨
o mande class
(see H¨
o mande [15]), and a class o oscilla o y in eg al ope a o s ela ed o hose
in oduced by Phong and S ein [25]. They used (1.9) o gene alize Coi man’s
heo em [7] ela ing he Lpno m o singula in eg al ope a o s and he maximal
unc ion.
Rema k 1.4. Fo Calde ´
on-Zygmund ope a o s we ha e been able o p o e
s onge esul s;see[11]. Bydiffe en me hodsweshowed ha wemay eplace he
“powe bump” in (1.7) by a “bump” in he scale o O licz spaces. Mo e p ecisely,
we eplace he L no m by he L(logL)p−1+δno m wi h δ>0. Howe e we a e
unable o ex end hese esul s o he b oade class o ope a o s discussed in he
p e ious ema k.
700 D. CRUZ-URIBE,SFO&C. P´
EREZ
Rema k 1.5. Condi ions(1.5)and(1.7)a esufficien o he ac ionalmax-
imal ope a o and he Ha dy-Li lewood maximal ope a o o be bounded om
Lp(u−p/p) o Lp( −p/p). (See [21], [22].) We conjec u e ha he bounded-
nesso heco espondingmaximalope a o isi sel sufficien o inequali ies(1.6)
and (1.8) o hold. In pa icula we belie e ha he O licz space condi ions gi en
in [21] and [22] a e sufficien .
Ou las esul is abou (linea ) commu a o s. These ope a o s a e de ined by
Ck
b (x)=(b(x) −b(y))kK(x,y) (y)dy,
whe e Kis a ke nel sa is ying he s anda d es ima es and bis a locally in eg able
unc ion. (See Sec ion 2 o a p ecise de ini ion.)
Sincecommu a o sha eag ea e deg eeo “singula i y” han heco espond-
ingCalde ´
on-Zygmundope a o s,weneedasligh lys onge condi ion. Roughly,
weneed o “bump” he igh -hand e mas well, bu i suffices odosoin hescale
o O licz spaces. Recall ha i Bis an inc easing Young unc ion and i Qis
any cube, we de ine he mean Luxembu g no m o a measu able unc ion wi h
espec o Bby
 B,Q =in λ>0: 1
|Q|QB| |
λdx ≤1.
(Fo mo e in o ma ion on O licz spaces, see Sec ion 2 below.)
Theo em 1.6. Le Tbe a Calde ´
on-Zygmund ope a o and ba unc ionin
BMO. Gi en a pai o weigh s (u, ),p,1<p<∞, and k≥0, suppose ha
o some >1and o all cubes Q,
1
|Q|Qu dx1/ p
 −1/pCk,Q ≤C<∞,(1.11)
whe e Ck( ) = plog(e + )kp. Then he commu a o Ck
bsa is ies he weak (p, p)
inequali y
u({x∈Rn:|Ck
b (x)|> })≤C
pRn| |p dx.(1.12)
When k=0, C0
b=T, and so in his case Theo em 1.6 educes o Theo em
1.2.
Two-weigh , Weak- ype No m Inequali ies 701
Rema k 1.7. As a co olla y o Theo em 1.6 we ge a new p oo o he one-
weigh , s ong (p, p) no m inequali y o commu a o s, which was i s p o ed
in a mo e gene al o m by Al a ez, Bagby, Ku z, and P´
e ez [2] and Sego ia and
To ea [29]. I w∈Ap, henwand w−p/p bo h sa is y he e e se H¨
olde
inequali y and so inequali y (1.11) holds o some >1and o p±ε.The
s ong- ype inequali y ollows by in e pola ion.
The p oo s o Theo ems 1.1, 1.2 and 1.6 all ollow he same ou line. Each
elies on ou so-called p incipal lemma, Theo em 3.4 below, which ela es he
Ha dy-Li lewood maximal ope a o and he Feffe man-S ein sha p maximal op-
e a o ia an inequali y s ongly eminiscen o a good-λinequali y. To apply
Theo em 3.4 we use h ee esul s which ela e he gi en ope a o , he sha p max-
imal ope a o and he maximal ope a o . Fo Calde ´
on-Zygmund ope a o s his
is inequali y (1.9). Simila inequali ies hold o ac ional in eg al ope a o s and
commu a o s: see Lemmas 4.4 and 6.1.
The emainde o his pape is o ganized as ollows: in Sec ion 2 we gi e
a numbe o de ini ions and lemmas needed in la e sec ions. The hea o he
pape is Sec ion 3, whe e we p o e Theo em 3.4. Finally, in Sec ions 4, 5, and 6
we p o e Theo ems 1.1, 1.2, and 1.6.
Th oughou his pape all no a ion is s anda d o will be de ined as needed.
All cubes a e assumed o ha e hei sides pa allel o he coo dina e axes. Gi en
acubeQ,#(Q) will deno e he leng h o i s sides and o any >0, Q will
deno e he cube wi h he same cen e as Qand such ha #( Q) = #(Q).We
will deno e he collec ion o all dyadic cubes by ∆and by ∆(Q) he collec ion o
all dyadic subcubes ela i e o he (no necessa ily dyadic) cube Q.Byweigh swe
will always mean non-nega i e, locally in eg able unc ions which a e posi i e on
a se o posi i e measu e. Gi en a Lebesgue measu able se Eand a weigh w,|E|
will deno e he Lebesgue measu e o Eand w(E) =Ewdx.Gi en1<p<∞,
p=p/(p −1)will deno e he conjuga e exponen o p. Finally, Cwill deno e a
posi i e cons an whose alue may change a each appea ance.
.2PRELIMINARY IDEAS
In his sec ion we gi e a numbe o de ini ions and lemmas needed in la e
sec ions.
Themainope a o s. Fi s wede ine heope a o sinTheo ems1.1,1.2,and
1.6.

702 D. CRUZ-URIBE,SFO&C. P´
EREZ
F ac ional in eg al ope a o s. Gi en α,0<α<n, de ine he ac ional
in eg al ope a o o o de αby
Iα (x)=Rn
(y)
|x−y|n−αdy.
Fo mo e in o ma ion, see S ein [31, pp. 117-120].
Calde on-Zygmund ope a o s. Gi en a ke nel Kon Rn×Rn—i.e. alocally
in eg able, complex- alued unc ion de ined off hediagonal—wesay ha i sa -
is ies he s anda d es ima es i he e exis δ,0<δ≤1, and C ini e such ha o
all dis inc poin s xand yin Rn,andallzsuch ha |x−z|<1
2|x−y|:
(1) |K(x,y)|≤C|x−y|−n;
(2) |K(x,y) −K(z,y)|≤C|x−z|δ|x−y|n+δ;
(3) |K(y,x) −K(y,z)|≤C|x−z|δ|x−y|n+δ.
A bounded linea ope a o T:C∞
0(Rn)→D
(Rn)(he e Dis he space o
dis ibu ions) is said o be associa ed wi h a ke nel Ki
T ,g=RnRnK(x,y)g(x) (y)dx dy
o all and gin C∞
0(Rn)wi h supp( ) ∩supp(g) =∅.Tis said o be a
Calde ´
on-Zygmundope a o i i sassocia edke nelsa is ies hes anda des ima es
and i ex ends o a bounded linea ope a o on L2. Fo mo e in o ma ion, see
Coi man and Meye [8] and Ch is [6].
Impo an examples o such ope a o s a e he Calde ´
on-Zygmund singula
in eg al ope a o s:
T (x)=p. .Rnk(x −y) (y)dy,
whe ek∈L1
loc(Rn {0})andK(x,y) =k(x−y)sa is ies hes anda des ima es.
Fo mo e in o ma ion see Ga c´
ıa-Cue a and Rubio de F ancia [14, p. 192].
Commu a o s. Gi en a Calde ´
on-Zygmund ope a o Tand a unc ion bin
BMO, le Mbdeno e mul iplica ion by b. We de ine he linea ope a o s Ck
bby
C0
b=T,C1
b=[Mb,T] =MbT−MbT,and o k>1, Ck
b=[Mb,Ck−1
b].I
∈C∞
0(Rn), hen
Ck
b (x)=(b(x) −b(y))kK(x,y) (y)dy, x ∈ supp( ).
Two-weigh , Weak- ype No m Inequali ies 703
Commu a o swe ein oducedbyCoi man,Rochbe gandWeiss[9],whoshowed
hey a e bounded on Lp,1<p<∞.
Maximal ope a o s. Key o he p oo s o ou esul s a e a numbe o maxi-
mal ope a o s. Fo comple eness we gi e hei de ini ions he e.
The maximal ope a o . Gi en a locally in eg able unc ion and α,0≤
α<n
,de ine
Mα (x)=sup
Qx
1
|Q|1−α/n Q| |dy.
I α=0 his is he Ha dy-Li lewood maximal ope a o and we w i e M o
M0 ;i 0<α<n his is he ac ional maximal ope a o o o de α.Weuse he
Ha dy-Li lewoodmaximalope a o ocon olCalde ´
on-Zygmundope a o sand
commu a o s, and he ac ional maximal ope a o o con ol ac ional in eg al
ope a o s. (See inequali y (1.9) and Lemmas 4.4 and 6.1.)
We de ine he dyadic maximal and ac ional maximal ope a o s Mdand Md
α
simila lyexcep hesup emumsa e es ic ed odyadiccubescon ainingx.Gi en
δ>0wede ine heδ-maximal ope a o by Mδ (x) =M(| |δ)(x)1/δ.We
de ine Md
δsimila ly. F om he con ex he e should be no con usion be ween he
ac ional maximal ope a o and he δ-maximal ope a o .
The sha p maximal ope a o . Gi en a locally in eg able unc ion and a
cube Q,le Qdeno e he a e age o o e Q:
Q=1
|Q|Q dx.
De ine he sha p maximal unc ion o by
M# (x)=sup
Qx
1
|Q|Q| (y)− Q|dy.
The sha p maximal unc ion was in oduced by Feffe man and S ein [13]. Again,
de ine he dyadic sha p maximal unc ion M#,d by es ic ing he sup emum o
dyadic cubes. Gi en δ>0, de ine he sha p δ-maximal unc ion by
M#
δ (x)=M#(| |δ)(x)1/δ,
and de ine M#,d
δsimila ly.
704 D. CRUZ-URIBE,SFO&C. P´
EREZ
O licz spaces. In Sec ion 6 we will need he ollowing ac s abou O licz
spaces. (Fo u he in o ma ion see Benne and Sha pley [4] o Rao and Ren
[26].) A unc ion B:[0,∞)→[0,∞)is a Young unc ion i i is con ex and
inc easing, and i B(0)=0andB( ) →∞as →∞.
Gi en a Young unc ion B, de ine he mean Luxembu g no m o on a cube
Qby
 B,Q =in λ>0: 1
|Q|QB| |
λdy ≤1.
When B( ) = p,1≤p<∞,
 B,Q =1
|Q|Q| |pdx1/p ;
ha is, he Luxembu g no m coincides wi h he (no malized) Lpno m. The e
is ano he cha ac e iza ion o he Luxembu g no m, due o K asnosel’ski˘
ıand
Ru icki˘
ı [17, p. 92] (also see Rao and Ren [26, p. 69]) which we will need:
 B,Q ≤in
s>0s+s
|Q|QB| |
sdx≤2 B,Q.(2.1)
Gi en h ee Young unc ions A,B,andCsuch ha o all >0,
A−1( )C−1( ) ≤B−1( ),(2.2)
henweha e he ollowinggene alizedH¨
olde ’sinequali ydue oO’Neil[19]: o
any cube Qand all unc ions and g,
 gB,Q ≤2 A,Q gC,Q.(2.3)
De ine he maximal ope a o MBby
MB (x)=sup
Qx
 B,Q.
The dyadic maximal ope a o Md
Bis de ined in simila ly, excep he sup emum
is es ic ed o dyadic cubes con aining x. I ollows om an inequali y due o
S ein [30] ha o k≥1, i Bk( ) = log(e + )k−1, henMk ≈MBk ,whe e
Mk=M·M···Mis he k- h i e a e o he maximal unc ion. (See Ca ozza and
Passa elli di Napoli [5] and he e e ences gi en he e.)
Two-weigh , Weak- ype No m Inequali ies 705
The Calde ´
on-Zygmund decomposi ion. Ou p oo s depend hea ily on he
Calde ´
on-Zygmunddecomposi ionandagene aliza iono i oO liczspaceno ms.
To be p ecise and o es ablish no a ion, we s a e he esul he e. Fo a p oo see
[22]; his is an adap a ion o he classical p oo gi en in Ga c´
ıa-Cue aandRubio
de F ancia [14, p. 137].
Lemma 2.1. Gi en a Young unc ion B, suppose is a non-nega i e unc ion
such ha  B,Q ends oze oas#(Q) ends o in ini y. Then o each >0 he e
exis s a disjoin collec ion o dyadic cubes {C
i}such ha o each i, < B,C
i≤
2n ,
{x∈Rn:Md
B (x)> }=

i
C
i,
{x∈Rn:MB (x)> 4n }⊂
i3C
i.
Mo eo e , hecubesa emaximal: i Qisadyadiccubesuch ha Q⊂{Md
B (x)> },
hen Q⊂C
i o some i.
To ecap u e he classical lemma, le B( ) = and no e ha i ∈Lq o
some q,1≤q<∞, hen
 B,Q =1
|Q|Q dx→0as|Q|→∞.
Mo e gene ally, o apply Lemma 2.1 i suffices o assume ha is bounded and
has compac suppo .
.3THE PRINCIPAL LEMMA
In hissec ion we p o e ou p incipal lemma: an inequali y linking he sha p
maximal unc ion and he Ha dy-Li lewood maximal unc ion. In spi i , hough
no inde ail i esembles he good-λinequali y o Feffe manand S ein[13]. (Also
see Ga c´
ıa-Cue a and Rubio de F ancia [14, pp. 161-3] and Jou n´
e [16, p. 41].)
To s a e he p incipal lemma we i s need a de ini ion and a lemma.
De ini ion 3.1. Gi en >1andaweigh u, de ine he se unc ion A
uon
measu able se s E⊂Rnby
A
u(E) =|E|1/ Eu dx1/
=|E|1
|E|Eu dx1/
.
(The second equali y holds p o ided |E|>0.)
712 D. CRUZ-URIBE,SFO&C. P´
EREZ
and (by he Lebesgue diffe en ia ion heo em)
sup
>0 pu({x∈Rn:|Id
α (x)|> })≤sup
>0 pu({x∈Rn:Md(Iα )(x) > })
≤Csup
>0 p
j
A
u(Q
j).
Fix ; hen by Lemma 4.4, o each j,
Q
j⊂{x∈Rn:Md
α (x)>εD−1
α }.
By an a gumen analogous o ha o he dyadic maximal ope a o (c . Lemma
2.1), we can w i e he igh -hand side as he union o disjoin dyadic cubes {P
k}
such ha o each k,
|P
k|α/n−1P
k
dx>εD
−1
α .
Fu he , he P
k’s a e maximal wi h his p ope y; in pa icula , o each j he e
exis s ksuch ha Q
j⊂P
k. The e o e, byLemma 3.2, Condi ion (3),
p
j
A
u(Q
j)
= p
k
Q
j⊂P
k
A
u(Q
j)≤ p
k
A
u(P
k)
≤(ε−1Dα)p
k
|P
k|1
|P
k|P
k
u dx1/ |P
k|α/n−1P
k
dx
p
.
By H¨
olde ’s inequali y and inequali y (1.5),
≤C
k
|P
k|αp/n 1
|P
k|P
k
u dx1/ 1
|P
k|P
k
−p/p dxp/pP
k
p dx
≤C
kP
k
p dx≤CRn p dx.
The cons an is independen o ,soi we ake hesup emumo e all >0
we ge inequali y (1.6).
❐
Rema k 4.5. A he cos o a mo e complex a gumen simila o ha o
Calde ´
on-Zygmund ope a o s (c . Lemma 5.1 below) we could dispense wi h
he dyadic ac ional in eg al ope a o and p o e Theo em 1.1 di ec ly o Iα.
The key inequali y is he non-dyadic analogue o Lemma 4.4 due o Adams [1]:
M#(Iα )(x) ≤CMα (x).

Two-weigh , Weak- ype No m Inequali ies 713
.5CALDER´
ON-ZYGMUND OPERATORS
In hissec ionwep o eTheo em1.2. Thep oo issimila o ha o Theo em
1.1, bu is complica ed by he ac ha we canno pass o an equi alen dyadic
ope a o . To compensa e we need he ollowing lemma which is also needed in
he p oo o Theo em 1.6.
Lemma 5.1. Le BbeaYoung unc ion. Suppose ha o some unc ion ∈Lq,
1≤q<∞, and o some >0 he e exis a cons an µ,0<µ≤1, and a collec ion
o dyadic cubes {Qj}such ha o each j,
|Qj∩{x∈Rn:MB (x)> }| ≥ µ|Qj|.
Then he e exis s a cons an ν>0, depending on nand µ, and a subcollec ion {Pk}
o he Calde ´
on-Zygmund decomposi ion wi h espec o Bo a heigh ν ,{Cν
i},
such ha o each j,Qj⊂3Pk o some k.
I we eplace MBby Md
Bin he hypo hesis, hen we can s eng hen he conclusion
by inding Pk’s such ha Qj⊂Pkand by le ing µ=ν.
P oo . We i s conside he non-dyadic case. By Lemma 2.1,
E ={x∈Rn:MB (x)> }⊂
i3Cγ
i,
whe e γ=4−n.I wehadQj⊂3Cγ
i o some iwe would be done, bu his need
no be he case, e en i µ=1. Howe e , o each j he e is a collec ion o indices
Ajsuch ha
Qj∩E ⊂
i∈Aj
3Cγ
iand 3Cγ
i∩Qj≠∅,i∈Aj.
The e a e wo possibili ies: i s , he e exis s i∈Ajsuch ha #(Qj)≤#(3Cγ
i).
Then Qj⊂9Cγ
iand by inequali y (2.1),
2 B,9Cγ
i≥in
s>0

s+s
|9Cγ
i|9Cγ
i
B| |
sdx


≥9−nin
s>0

s+s
|Cγ
i|Cγ
i
B| |
sdx


=9−n B,Cγ
i>9−nγ .
714 D. CRUZ-URIBE,SFO&C. P´
EREZ
Al e na i ely, #(Qj)>#(3Cγ
i) o all i∈Aj.Bu hen o eachi∈Aj,3Cγ
i⊂
3Qj,andso
2|3Qj| B,3Qj≥in
s>0s|3Qj|+s3Qj
B| |
sdx
≥
i∈Aj
in
s>0s|Cγ
i|+sCγ
i
B| |
sdx
=
i∈Aj
|Cγ
i| B,Cγ
i>3−nγ 
i∈Aj
|3Cγ
i|
≥3−nγ |Qj∩E |≥9−nµγ |3Qj|.
So in ei he case, o each j he e exis s a cube ¯
Qjcon aining Qjsuch ha
 B, ¯
Qj>µγ
2·9n.
Now by he same a gumen ha is used o p o e he Calde ´
on-Zygmund
decomposi ion, Lemma 2.1, we can show ha he e exis s a subcollec ion {Pk}o
{Cν
i},ν=1
2µγ36−n=1
2µ144−n,such ha o eachj,Qj⊂¯
Qj⊂3Pk o some
k. This comple es he p oo o MB.
The p oo in he dyadic case is e y simila , bu is simpli ied conside ably by
he ac ha i wo dyadic cubes in e sec hen one is con ained in he o he .
❐
P oo . [P oo o Theo em 1.2] By a s anda d a gumen , we may assume ha
∈C∞(Rn)and has compac suppo . Fix p,1<p<∞; henT ∈Lq,whe e
q>1issuch ha p≥q/ . Hence, we may apply Theo em 3.4 o i . Fix δ<1.
Then he e exis s ε>0such ha o each >0 he e exis s a sequence o disjoin
dyadic cubes {Q
j}such ha

1
|Q
j|Q
j
|T |δ−(|T |δ)Q
j
dx

1/δ
>ε
1/δ
and
sup
>0 pu({x∈Rn:|T (x)|> })≤sup
>0 pu({x∈Rn:Md
δ(T )(x) > })
≤Csup
>0 p
j
A
u(Q
j).
Two-weigh , Weak- ype No m Inequali ies 715
As we no ed in he In oduc ion, Tsa is ies inequali y (1.9). The e o e, o each
j,
Q
j⊂{x∈Rn:M#
δ(T )(x) > ε1/δ }⊂{x∈Rn:M (x) > β },
whe e β=C−1
δε1/δ.
By Lemma 5.1 (wi h µ=1), o each >0 he eexis s asequence o disjoin
dyadic cubes {P
k}such ha o each j,Q
j⊂3P
k o some k,andsuch ha
1
|P
k|P
k
| |dx > ρ ,
whe e ρ>0 depends only on βand n. Then by Lemma 3.2, Condi ion (3), o
each >0,
p
j
A
u(Q
j)
= p
k
Q
j⊂3P
k
A
u(Q
j)
≤ p
k
A
u(3P
k)
≤ρ−p
k
|3P
k|1
|3P
k|3P
k
u dx1/ 1
|P
k|P
k
| |dxp
.
By H¨
olde ’s inequali y and inequali y (1.7),
≤C
k1
|3P
k|3P
k
u dx1/ 1
|3P
k|3P
k
−p/p dxp/pP
k
| |p dx
≤C
kP
k
| |p dx
≤CRn| |p dx.
The cons an is independen o ,soi we ake hesup emumo e all >0
we ge inequali y (1.8). This comple es ou p oo .
❐
716 D. CRUZ-URIBE,SFO&C. P´
EREZ
.6COMMUTATORS
In his sec ion we p o e Theo em 1.6. The p oo depends on Theo em 1.2
and he ollowing analogue o inequali y(1.9) o commu a o s.
Lemma 6.1. Gi enaCalde
´
on-Zygmund ope a o T, a unc ion bin BMO,
cons an s δ0and δ1,0<δ
0<δ
1<1, and k≥1, he e exis s a cons an K,
depending on he BMO no m o b, such ha o e e y unc ion ∈C∞
0(Rn)and
any x∈Rn,
M#,d
δ0(Ck
b )(x) ≤K
k−1

i=0
Md
δ1(Ci
b )(x) +KMk+1 (x).
This esul is ound in [23, 24]. As gi en he e, he non-dyadic maximal
ope a o appea s in he i s e m on he igh -hand side, bu i is immedia e om
he p oo ha i is s ill ue wi h he dyadic maximal ope a o he e.
P oo . [P oo o Theo em 1.6] When k=0, Theo em 1.6 educes o The-
o em 1.2, so we may ix k≥1. By a s anda d a gumen we may assume ha
∈C∞(Rn)andhas compac suppo . Fixp,1<p<∞; henCi
b ∈Lq,whe e
0≤i≤kand q>1issuch ha p≥q/ . Hence, we may apply Theo em 3.4
o Ck
b .Fixδ0and δ1,0<δ
0<δ
1<1. Then he e exis s ε>0such ha o
each >0 he e exis s a sequence o disjoin dyadic cubes {Q
j}such ha

1
|Q
j|Q
j
|Ck
b |δ0−(|Ck
b |δ0)Q
j
dx

1/δ0
>ε
1/δ0
and
sup
>0 pu({x∈Rn:|Ck
b (x)|> })≤sup
>0 pu({x∈Rn:Md
δ0(Ck
b )(x) > })
≤Csup
>0 p
j
A
u(Q
j).
By Lemma 6.1, o each jand ,
Two-weigh , Weak- ype No m Inequali ies 717
Q
j⊂
k−1

i=1
{x∈Rn:Md
δ1(Ci
b )(x) > β }
∪{x∈Rn:Md
δ1(T )(x) > β }
∪{x∈Rn:Mk+1 (x)>β }
≡k−1

i=1
Fβ
i∪Fβ
0∪Fβ
k,
whe eβ=ε1/δ0K−1(k +1)−1.Fo eachjand we canno ha e ha |Q
j∩Fβ
i|<
(k +1)−1|Q
j| o all i.Hence, o somei,|Q
j∩Fβ
i|≥(k +1)−1|Q
j|;i hisis
he case, we w i e Q
j∈F
β
i.Thus,
sup
>0 p
j
A
u(Q
j)≤
k

i=0sup
>0 p
Q
j∈Fβ
i
A
u(Q
j).
To comple e he p oo we will show ha each e m o he ou e sum on he
igh -hand side is domina ed by
CRn| |p dx.
The e a e h ee cases.
9Case 1: Cubes in Fβ
k.As we no ed in Sec ion 2, he e exis s a cons an
β>0such ha
{x∈Rn:Mk+1 (x)>β }⊂{x∈Rn:MB (x)>β },
whe e B( ) = log(e + )k. The e o e, by Lemma 5.1 (wi h µ=(k +1)−1),
he e exis s a cons an ν>0such ha , o each >0 he e exis s a collec ion o
disjoin dyadic cubes {P
#}such ha o each j,Q
j⊂3P
# o some #and such
ha  B,P
#>ν . We now p oceed exac ly as we did a he end o he p oo
o Theo em 1.2. Since Ck( ) = plog(e + )kp,C−1
k( ) ≈ 1/plog(e + )−k,

718 D. CRUZ-URIBE,SFO&C. P´
EREZ
and so 1/pC−1
k( ) ≤B−1( ). Then, by Lemma 3.2, Condi ions (2) and (3), he
gene alized H¨
olde ’s inequali y (2.3) and inequali y (1.11),
sup
>0 p
Q
j∈Fβ
k
A
u(Q
j)
≤sup
>0 p
#
A
u(3P
#)
≤Csup
>0
#
|3P
#|1
|3P
#|3P
#
u dx1/ 
 
p
B,P
#
≤Csup
>0
#1
|3P
#|3P
#
u dx1/ 
 −1/p
p
Ck,P
#P
#
| |p dx
≤Csup
>0
#1
|3P
#|3P
#
u dx1/ 
 −1/p
p
Ck,3P
#P
#
| |p dx
≤Csup
>0
#P
#
| |p dx
≤CRn| |p dx.
9Case 2: Cubes in Fβ
0.Gi en >0, le s=(β )δ1. Again by Lemma
5.1 ( he dyadic case), i Q
j∈F
β
0, hen o somei,Q
j⊂Cs
i,whe e{Cs
i}is he
Calde ´
on-Zygmund decomposi ion o |T |δ1a heigh s. Hence, by Lemma 3.2,
Condi ions (2) and (3),
sup
>0 p
Q
j∈Fβ
0
A
u(Q
j)≤sup
>0 p
i
A
u(Cs
i).
By Co olla y 3.5, he e exis ε>0 and a subcollec ion {¯
Q
j}o {Cs
i}such ha i
x∈¯
Q
j, henM#,d
δ1(T )(x) > β ,whe eβ =ε1/δ1β,andsuch ha
sup
>0 p
i
A
u(Cs
i)≤Csup
>0 p
j
A
u(¯
Q
j).
We can now a gue exac ly aswe did in he p oo o Theo em 1.2 o ge
sup
>0 p
Q
j∈Fβ
0
A
u(Q
j)≤CRn| |p dx.
Two-weigh , Weak- ype No m Inequali ies 719
9Case 3: Cubes in Fβ
i,1≤i≤k−1.Fix i; hena guingexac ly as wedid
in Case 2, by Co olla y 3.5 he e exis ε>0 and a collec ion o disjoin dyadic
cubes {¯
Q
j}such ha i x∈¯
Q
j, henM#
δ1(Ci
b )(x) > β ,whe eβ =ε1/δ1β,
and such ha
sup
>0 p
Q
j∈Fβ
i
A
u(Q
j)≤Csup
>0 p
j
A
u(¯
Q
j).
We now apply Lemma 6.1 and epea he a gumen a he beginning o his
p oo . When we do so we educe he deg ee o he highes o de commu a o
appea ing omi oi−1. The e o e,a e epea ingou a gumen a ini enumbe
o imes, we will educe o collec ions o cubes sa is ying condi ions such as hose
in Case 1 and Case 2. Repea ing hose a gumen s will hen gi e us he desi ed
inequali y.
❐
Acknowledgemen . We would like o hank E. Sawye o sha ing wi h he
second au ho an unpublished manusc ip which sugges ed ou app oach. We
wouldalsolike o hank he e e ee o poin ingou ane o inTheo em1.6. The
i s au ho was suppo ed by a Fo d Founda ion ellowship; he second au ho
by DGICYT G an PB40192, Spain. The second au ho is also g a e ul o he
in i a ion and hospi ali y o he Cen e de Rece ca Ma em`
a ica, Ba celona, whe e
his wo k was comple ed.
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D. CRUZ-URIBE,SFO
Depa men o Ma hema ics, T ini y College
Ha o d, CT 06106-3100, U. S. A.
EMAIL: [email p o ec ed]
Two-weigh , Weak- ype No m Inequali ies 721
C. P´
EREZ
Depa imen o de Ma em´
a icas
Uni e sidad Au ´
onoma de Mad id
28049 Mad id, Spain
EMAIL:ca los.pe [email protected]
SUBJECT CLASSIFICATION: 42B20, 42B25
KEYWORDS: weigh s, weak- ype inequali ies, ac ional in eg al ope a o s, Calde ´
on-Zygmund op-
e a o s, commu a o s
Submi ed: Ap il 23 d, 1999, e ised: Ma ch 3 d, 2000.