Ad ances in Ma hema ics, 216 (2007) 647-676
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR
INTEGRALS, WITH APPLICATIONS TO THE HILBERT
TRANSFORM AND THE SARASON CONJECTURE
D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´
EREZ
Abs ac . We p o e wo-weigh no m inequali ies o Calde ´on-Zygmund singula
in eg als ha a e sha p o he Hilbe ans o m and o he Riesz ans o ms. In
addi ion, we gi e esul s o he dyadic squa e unc ion and o commu a o s o
singula in eg als. As an applica ion we gi e new esul s o he Sa ason conjec u e
on he p oduc o unbounded Toepli z ope a o s on Ha dy spaces
1. In oduc ion
1.1. Backg ound. A long-s anding p oblem in ha monic analysis has been o cha -
ac e ize he weigh s go e ning s ong- ype no m inequali ies o classical ope a o s.
To be p ecise: gi en an ope a o Tand p, 1 < p < ∞, de e mine su icien condi ions
on a pai o weigh s (i.e., non-nega i e, measu able unc ions) (u, ) such ha o all
∈Lp( p),
(1.1) ZRn
|u(x)T (x)|pdx ≤CZRn
| (x) (x)|pdx.
This p oblem was o iginally posed in he ea ly 1970’s o he Ha dy-Li lewood maxi-
mal ope a o and o he Hilbe ans o m on he eal line, bu i was soon expanded
o include a a ie y o ope a o s—singula in eg als, ac ional in eg als, and squa e
unc ions—on Rn. While a g ea deal o p og ess has been made, many ques ions
emain open e en o he Hilbe ans o m.
Fo many o hese p oblems, inequali y (1.1) is usually s a ed in an equi alen
o m:
(1.2) ZRn
|T (x)|pU(x)dx ≤CZRn
| (x)|pV(x)dx,
whe e U=upand V= p. Bu o ou pu poses (1.1) is a mo e sui able o m as i
makes he s a emen o ou main esul s mo e elegan .
2000 Ma hema ics Subjec Classi ica ion. 42B20,42B25,47B35.
Key wo ds and ph ases. Weigh s, Hilbe ans o m, singula in eg al ope a o s, Sa ason conjec-
u e, Toepli z ope a o , maximal unc ions, commu a o s.
The i s au ho is pa ially suppo ed by he S ewa acul y de elopmen und o T ini y Col-
lege; he would also like o hank his ad iso , Donald Sa ason, o posing he p oblem discussed in
Sec ion 1.4. The second au ho is pa ially suppo ed by Spanish Minis e io de Educaci´on y Ciencia
“P og ama Ram´on y Cajal, 2005” and by G an MTM2004-00678 om he same ins i u ion; he
hi d au ho is pa ially suppo ed by G an MTM2006-05622 om he same ins i u ion.
1
2 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´
EREZ
The pu pose o his pape is o gi e new wo-weigh no m inequali ies o singula
in eg als and o he ope a o s ha a e sha p o he Hilbe and Riesz ans o ms.
To pu ou esul s in o con ex , we will ske ch he ou lines o some ea lie wo k.
Fo mo e in o ma ion on he his o y o his p oblem, we e e he eade o Muck-
enhoup [28], Dynkin and Osilenke [15], Ga c´ıa-Cue a and Rubio de F ancia [18],
and Duoandikoe xea [13].
The ea lies weigh ed no m inequali ies we e o he one-weigh p oblem (i.e., when
u= ). Muckenhoup [27], and Hun , Muckenhoup and Wheeden [20] showed ha
o he maximal ope a o and o he Hilbe ans o m on he eal line, (1.1) held
i and only i upsa is ied he so-called Apcondi ion: he e exis s a ini e cons an C
such ha o all in e als Q,
(1.3) 1
|Q|ZQ
u(x)pdx1
p1
|Q|ZQ
u(x)−p0dx1
p0
≤C.
The p oo was simpli ied by Coi man and Fe e man [3] and ex ended o Calde ´on-
Zygmund singula in eg als on Rn(wi h in e als eplaced by cubes in (1.3)).
I was immedia ely conjec u ed ha in he wo-weigh case, he co esponding
wo-weigh Apcondi ion,
(1.4) 1
|Q|ZQ
u(x)pdx1
p1
|Q|ZQ
(x)−p0dx1
p0
≤C < ∞,
was necessa y and su icien o hese ope a o s o be bounded om Lp( p) o Lp(up).
Howe e , while his condi ion is necessa y o he maximal ope a o and o he
Hilbe ans o m, i is no su icien : see Muckenhopu and Wheeden [30]. Sawye
[44] ga e a necessa y and su icien condi ion o he maximal ope a o which in ol es
he ope a o i sel . Co la and Sadosky [4, 5] ga e a necessa y and su icien condi ion
o he Hilbe ans o m which is eminiscen o he Helson-Szeg¨o heo em and is
g ounded in ope a o heo y. Howe e , hei condi ion is di icul o check and does
no eadily ex end o highe dimensions and gene al singula in eg als.
Following hese esul s, a g ea deal o e o was de o ed o inding s onge con-
di ions ela ed o he mo e geome ic wo-weigh Apcondi ion and ha a e su icien
o (1.1) o hold o a a ie y o ope a o s, especially singula in eg als. In passing,
we no e he wo k o Muckenhoup and Wheeden [30], Fujii [17], Ka z and Pe ey a
[22], Leckband [25], Rako ond a simba [39, 40], Wilson [52], and P´e ez [34].
An impo an esul in his di ec ion is due o Neugebaue [32]: he showed ha
i he pai o weigh s (u, ) is such ha o some > 1 he pai (u , ) sa is ies
(1.4), hen (1.1) holds o singula in eg als. He did no p o e his di ec ly; a he ,
by applying he ideas on ac o iza ion o weigh s due o Rubio de F ancia, he showed
ha he e exis s w∈Apsuch ha c1u≤w≤c2 i and only i (u , )∈Ap o
some > 1. Two-weigh inequali ies o singula in eg als and o he ope a o s hen
ollow immedia ely om he one-weigh case.
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 3
We can es a e Neugebaue ’s esul as ollows. Gi en a cube Q, w i e
kukp,Q =1
|Q|ZQ
|u(x)|pdx1/p
o he no malized Lpno m on Q. The Apcondi ion is hen equi alen o
kukp,Qk −1kp0,Q ≤C < ∞,
and he condi ion ha (u , )∈Apcan be ew i en as
kuk p,Qk −1k p0,Q ≤C < ∞.
In o he wo ds, i we eplace he no malized Lpand Lp0no ms in he Apcondi ion
by la ge no ms (in he scale o Lebesgue spaces), hen we ge a su icien condi ion
o (1.1) o hold o singula in eg als and o he ope a o s. We e e o hese la ge
no ms as “powe bumps.”
P´e ez [35, 36] i s conside ed he ques ion o whe he powe bumps could be e-
placed by o he unc ion space no ms la ge han he Lpno m bu smalle han he
L p no m. He showed ha o he maximal ope a o and ac ional in eg als ce ain
no ms in he scale o O licz spaces, he so-called “O licz bumps”, a e su icien .
To s a e his esul s we need se e al de ini ions. Gi en a Young unc ion B:
[0,∞)→[0,∞), and a cube Q, de ine he no malized Luxembu g no m on Qby
kukB,Q = in λ > 0 : 1
|Q|ZQ
B|u(x)|
λdx ≤1.
I B( ) = p, hen kukB,Q =kukp,Q and he Luxembu g no m educes o he Lpno m.
When B( ) = plog(e+ )awe ge he no m on he Zygmund spaces Lp(log L)a. When
used o de ine an Ap ype condi ion, his no m is e e ed o as a “log bump.”
Gi en a Young unc ion B, le ¯
Bdeno e i s associa e unc ion: he Young unc ion
wi h he p ope y ha ≤B−1( )¯
B−1( )≤2 , > 0. I B( ) = p, hen ¯
B( ) = p0;
i B( ) = plog(e+ )a, hen ¯
B( )≈ p0log(e+ )−ap0/p.
The ollowing g ow h condi ion on Young unc ions plays an impo an ole in
de e mining sui able O licz bumps o gene alizing he Apcondi ion.
De ini ion 1.1. Gi en p,1< p < ∞, a Young unc ion Bsa is ies he Bpcondi ion
i o some c > 0,
(1.5) Z∞
c
B( )
p
d
<∞.
I B( ) = q, 1 < q < p, hen i is immedia e ha B∈Bp. Mo e in e es ing
examples a e gi en by he unc ions
B( ) = p
log(e+ )1+δ, δ > 0,
B( ) = p
log(e+ ) log log(ee+ )1+δ, δ > 0.
4 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´
EREZ
The Bpcondi ion was in oduced in [36] whe e i was used in o s a e and p o e
sha p wo-weigh no m inequali ies o he Ha dy-Li lewood maximal unc ion. I
Bis a Young unc ion such ha ¯
B∈Bp, and he pai o weigh s (u, ) is such ha
o e e y cube Q,
(1.6) kukp,Qk −1kB,Q ≤C < ∞,
hen (1.1) holds o he Ha dy-Li lewood maximal unc ion. Fu he mo e, he Bp
condi ion is necessa y: i (1.1) holds and (u, ) sa is y (1.6), hen ¯
B∈Bp. No e ha
unlike in he o iginal esul by Neugebaue , he e is no bump on he weigh u.
Via a disc e iza ion a gumen , he same echniques we e applied in [34] o p o e
weigh ed no m inequali ies o he ac ional in eg al ope a o s Iα, 0 < α < n. Le
Aand Bbe Young unc ions such ha ¯
A∈Bp0and ¯
B∈Bp. I (u, ) is a pai o
weigh s such ha
(1.7) `(Q)αkukA,Qk −1kB,Q ≤C < ∞,
hen ZRn
|u(x)Iα (x)|pdx ≤CZRn
| (x) (x)|pdx.
The condi ion (1.7) can be iewed as a wo-weigh e sion o he Chang-Wilson-Wol
condi ion [2] o Sch ¨odinge ope a o s which is an imp o emen o he well-known
Fe e man-Phong condi ion [16]. This esul o ac ional in eg als immedia ely sug-
ges ed he ollowing conjec u e:
Conjec u e. I Aand Ba e Young unc ions such ha ¯
A∈Bp0and ¯
B∈Bp, and
i he pai o weigh s (u, )is such ha o e e y cube Q,
(1.8) kukA,Qk −1kB,Q ≤C < ∞,
hen (1.1) holds o Calde ´on-Zygmund singula in eg als.
An impo an special case o his conjec u e is when Aand Ba e log bumps:
A( ) = plog(e+ )p−1+δ, B( ) = p0log(e+ )p0−1+δ, δ > 0.
Ou conjec u e is closely connec ed o an old conjec u e o Muckenhoup and Whee-
den [29]: i he pai (u, ) is such ha he maximal ope a o Msa is ies
(1.9) M:Lp( p)→Lp(up) and M:Lp0(u−p0)→Lp0( −p0),
hen he Hilbe ans o m is bounded om Lp( p) o Lp(up). By he esul s in
[36] desc ibed abo e, (1.8) is su icien o bo h inequali ies in (1.9) o hold, so ou
conjec u e is a special case o hei s.
Ou conjec u e is known o be ue in a numbe o special cases. When Aand
Ba e powe bumps—i.e., A( ) = p and B( ) = p0, > 1— hen ou conjec u e
educes o he heo em o Neugebaue s a ed abo e. His esul was imp o ed in [11],
whe e i was shown ha i is su icien o ake Aa powe bump and Bsuch ha
¯
B∈Bp. In [7] i was shown ha i Ais a la ge O licz bump, e.g., i
A( )≈ pexp[log(e+ p) ],0< < 1,
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 5
hen he conjec u e is ue. Howe e , i was also shown in his pape ha such
unc ions ep esen he bes ha can be go en using he echniques in [11]; hey
canno be used o p o e he ull conjec u e o e en he case when Ais a log bump.
A ela ed bu weake e sion o ou conjec u e was p o ed by T eil, Volbe g and
Zheng [48] o he pe iodic Hilbe ans o m (i.e., he conjuga e unc ion) on he
uni ci cle. Fo z∈D, le φzbe he M¨obius ans o m in he closed uni disk,
φz(w) = z−w
1−¯zw, w ∈¯
D.
I Aand Ba e Young unc ions such ha ¯
A∈Bp0and ¯
B∈Bp, and i (u, ) is a pai
o weigh s such ha
(1.10) sup
z∈D
ku◦φzkA,∂Dk −1◦φzkB,∂D<∞,
hen he pe iodic Hilbe ans o m is bounded om Lp( p, ∂D) o Lp(up, ∂D).
Ano he esul closely ela ed o ou conjec u e was p o ed in [9]. The e i was
shown ha i Ais he log bump A( ) = plog(e+ )p−1+δand i he pai o weigh s
(u, ) is such ha o e e y cube Q,
(1.11) kukA,Qk −1kp0,Q ≤C < ∞,
hen Calde ´on-Zygmund singula in eg als sa is y he weak (p, p) inequali y
(1.12) up({x∈Rn:|T (x)|> })≤C
pZRn
| (x) (x)|pdx.
No e ha condi ion (1.11) is a special case o (1.6), and i is na u al o conjec-
u e ha (1.12) holds i Asuch ha ¯
A∈Bp0. This is a special case o ano he
conjec u e due o Muckenhoup and Wheeden [29]: i he maximal ope a o sa is ies
M:Lp0(u−p0)→Lp0( −p0), hen he Hilbe ans o m sa is ies (1.12).
1.2. Resul s o singula in eg als. Ou main esul s imp o e all p e ious wo k
by allowing us o ake A o be a log bump. Ou i s heo em is a sha p inequali y
o he Hilbe ans o m.
Theo em 1.2. Gi en p,1< p < ∞, suppose he pai o weigh s (u, )sa is ies
(1.13) kukA,Qk −1kB,Q ≤C < ∞,
whe e A( ) = plog(e+ )p−1+δ,δ > 0, and ¯
B∈Bp. Then
(1.14) ZR
|u(x)H (x)|pdx ≤CZR
| (x) (x)|pdx.
Fu he , his inequali y is sha p since i does no hold in gene al i we ake δ= 0 in
he de ini ion o A.
A coun e -example showing ha (1.14) need no hold i δ= 0 when p= 2 is gi en
in [9]. The example he e is a pai o weigh s o which (1.2) does no hold: (U, MΦU),
whe e Φ( ) = log(e+ ), and MΦis he O licz maximal ope a o
(1.15) MΦ (x) = sup
Q3x
k kΦ,Q.

6 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´
EREZ
(See Lemma 2.8 below.) By a change o a iables in he de ini ion o he Luxembu g
no m i is easy o see ha he pai o weigh s u=U1/2, = (MΦU)1/2sa is ies (1.13)
wi h A( ) = 2log(e+ ).
Theo em 1.2 is a special case o a mo e gene al esul which holds on Rn, p o ided
p>n. Recall ha a Calde ´on-Zygmund singula in eg al Tis a singula con olu ion
ope a o ,
T (x) = p. . ZRn
K(x−y) (y)dy,
whe e he ke nel Kis con inuously di e en iable on Rn {0}, has ze o a e age on
he uni sphe e, and o all x6= 0,
|K(x)| ≤ C
|x|nand |∇K(x)| ≤ C
|x|n+1 .
Mo e gene ally, we may assume ha Tis a Calde ´on-Zygmund ope a o . Fo a p ecise
de ini ion see Duoandikoe xea [13].
Theo em 1.3. Le Tbe a Calde ´on-Zygmund singula in eg al. Fix p,n<p<∞.
Suppose (u, )is a pai o weigh s such ha o all cubes Q,
(1.16) kukA,Qk −1kB,Q ≤C < ∞,
whe e A( ) = plog(e+ )p−1+δ,δ > 0, and ¯
B∈Bp. Then Tsa is ies he s ong (p, p)
inequali y
(1.17) ZRn
|u(x)T (x)|pdx ≤CZRn
| (x) (x)|pdx.
Fu he , his esul is sha p in he sense ha he e exis s a amily o pai s o weigh s
(u, )such ha (1.16) holds wi h δ= 0, bu (1.17) does no hold o all o he Riesz
ans o ms.
The sha pness o Theo em 1.3 comes om a necessa y condi ion p o ed in [34].
T ansla ed o ou se ing ( he esul s he e a e s a ed in e ms o inequali y (1.2)),
i shows ha i he pai s o weigh s (u, MAu) (which clea ly sa is y (1.16)) a e such
ha (1.17) holds o all no he Riesz ans o ms, hen δ > 0. By con aposi ion, i
δ= 0 hen (1.17) mus ail o a leas one o he Riesz ans o ms.
The es ic ion ha p > n in Theo em 1.3 seems unna u al, bu despi e epea ed
e o s we canno elimina e i . I n≥2, hen by duali y we ha e ha (1.17) holds o
1<p<n0o p > n i Aand Ba e bo h log bumps: A( ) = plog(e+ )p−1+δand
B( ) = p0log(e+ )p0−1+δ,δ > 0. Howe e , his s ill lea es he gap n0≤p≤n.
Ou nex esul shows ha we can ill his gap i we eplace Aby a la ge log bump.
Theo em 1.4. Le Tbe a Calde ´on-Zygmund singula in eg al. Gi en p,1< p < ∞,
suppose (u, )is a pai o weigh s such ha o all cubes Q,
(1.18) kukA,Qk −1kB,Q ≤C < ∞,
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 7
whe e A( ) = plog(e+ )2p−1+δ,δ > 0, and ¯
B∈Bp. Then Tsa is ies he s ong
(p, p)inequali y
(1.19) ZRn
|u(x)T (x)|pdx ≤CZRn
| (x) (x)|pdx.
The p oo s o bo h Theo ems 1.3 and 1.4 in ol e ca e ul disc e iza ion a gumen s
using he p ope ies o Calde ´on-Zygmund cubes. They a e e y simila in spi i ,
hough no in de ail, o he disc e iza ion a gumen used o p o e wo-weigh no m
inequali ies o ac ional in eg als in [35]. The p oblem wi h his app oach is ha
he e does no exis as good a echnique o disc e izing singula in eg als as exis s
o ac ional in eg als. Consequen ly, we need o a gue mo e obliquely using he
sha p maximal ope a o (explici ly in he p oo o Theo em 1.4 and in essence in he
p oo o Theo em 1.3). This leads di ec ly o he echnical obs acles which p e en
us om p o ing he ull conjec u e we desc ibed abo e.
In pa icula , in bo h p oo s we use he ollowing p ope y o log bumps: gi en
A( ) = plog(e+ )p−1+δ,δ > 0, hen ¯
A∈Bp0and he e exis s q, 0 < q < 1, such ha
i C( ) = A( 1/q), hen ¯
C∈B(p/q)0. This p ope y does no hold o a bi a y Young
unc ions: a coun e -example is gi en by A( ) = plog(e+ )p−1log log(ee+ )p−1+δ.
De ails a e le o he eade .
Key o he p oo o Theo em 1.4 is he poin wise inequali y [1]:
(1.20) M#
q(T )(x) = M#(|T |q)(x)1/q ≤CM (x),
o some 0 < q < 1, whe e M#is he sha p maximal ope a o o Fe e man-S ein.
Vec o - alued singula in eg als sa is y essen ially he same inequali y [38]: i 0 <
q < 1 and 1 < < ∞ he e exis s a cons an such ha
M#
qk{T j}jk` (x)≤C Mk{ j}jk` (x).
The e o e, as a co olla y o he p oo o Theo em 1.4 we ge wo-weigh es ima es o
ec o - alued singula in eg als. On he o he hand, i is no di icul o obse e ha
he p oo o Theo em 1.3 can be ca ied ou o ec o - alued singula in eg als and
hus we ge be e condi ions on (u, ) in he ange n < p < ∞. De ails a e le o
he eade .
Co olla y 1.5. Le Tbe a Calde ´on-Zygmund singula in eg al. Gi en p, wi h
1< p, < ∞, suppose (u, )sa is y (1.18). Then


X
j
|uT j| 1


Lp(Rn)≤C

X
j
| j| 1


Lp(Rn).
Mo eo e , he same es ima e holds i 1< < ∞,p>nand (u, )sa is y (1.16).
Rema k 1.6.O he ope a o s, including some pseudo-di e en ial ope a o s and squa e
unc ions, sa is y inequali y (1.20), and so simila weigh ed no m inequali ies hold
o hem. Fo examples see [1] and [11].
8 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´
EREZ
1.3. Resul s o o he ope a o s. The p oo s o Theo ems 1.3 and 1.4 can be
adap ed o gi e esul s o o he ope a o s. He e we conside wo: he dyadic squa e
unc ion and commu a o s o singula in eg als.
Dyadic squa e unc ions. We i s conside he dyadic squa e unc ion. Le ∆ deno e
he se o dyadic cubes in Rn, and o each m∈Z, le ∆m={Q∈∆ : `(Q) = 2m}.
Fo each Q∈∆, le b
Qdeno e he dyadic pa en o Q: i Q∈∆m, he unique cube
b
Q∈∆m+1 such ha Q⊂b
Q. Gi en a unc ion , le Q=|Q|−1RQ (x)dx. Fo each
, he dyadic squa e unc ion, Sd , is de ined by
Sd (x) = X
Q∈∆
| Q− b
Q|2χQ(x)1/2
.
Theo em 1.7. Gi en p,1< p < ∞, suppose (u, )is a pai o weigh s such ha o
all dyadic cubes Q,
(1.21) kukA,Qk −1kB,Q ≤C < ∞,
whe e A( ) = plog(e+ )p−1+δ,δ > 0, and ¯
B∈Bp. Then he dyadic squa e unc ion
sa is ies he s ong (p, p)inequali y
(1.22) ZRnu(x)Sd (x)pdx ≤CZRn
| (x) (x)|pdx.
The p oo o Theo em 1.7 is nea ly iden ical o ha o Theo em 1.3; he di e ence
is ha he squa e unc ion is su icien ly localized ha we can elimina e he es ic ion
on p. Gi en he close connec ion be ween squa e unc ions and singula in eg als, we
ake his esul as e idence ha he es ic ion on pin Theo em 1.3 is no necessa y.
Theo em 1.7 is ela ed o wo-weigh no m inequali ies o he dyadic squa e unc-
ion due o Uchiyama [49] and C uz-U ibe and P´e ez [10]. They showed ha o any
weigh u,ZRn
Sd (x)pu(x)dx ≤CZRn
| (x)|pMu(x)dx, 1< p ≤2,
ZRn
Sd (x)pu(x)dx ≤CZRn
| (x)|pMCu(x)dx, 2< p < ∞,
whe e C( ) = log(e+ )p/2−1+δ,δ > 0, and MCis he O licz maximal ope a o
(1.15). Simila bu weake inequali ies ollow om Theo em 1.7: i is s aigh o wa d
o see ha weigh s o he o m (u1/p,(MDu)1/p), whe e D( ) = log(e+ )p−1+δ,
δ > 0, sa is y (1.21). On he o he hand, one can also ind pai s o weigh s which
sa is y (1.21) which canno be w i en in his o m. I is emp ing o specula e ha
Theo em 1.7 can be imp o ed o include all o hese esul s as special cases.
1.3.1. Commu a o s. The second class o ope a o s we conside a e commu a o s o
singula in eg als. Gi en a Calde ´on-Zygmund singula in eg al Tand b∈BMO,
de ine he i s o de commu a o , [b, T], by
[b, T] (x) = b(x)T (x)−T(b )(x).
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 9
These ope a o s a e mo e singula han he associa ed singula in eg als, and so a
la ge log bump is equi ed on bo h weigh s.
Theo em 1.8. Le Tbe a Calde ´on-Zygmund singula in eg al and le b∈BMO.
Gi en p,1< p < ∞, suppose ha o all cubes Q he pai o weigh s (u, )sa is ies
(1.23) kukA,Qk −1kB,Q ≤C < ∞,
whe e A( ) = plog(e+ )3p−1+δ, and B( ) = p0log(e+ )2p0−1+δ,δ > 0. Then
(1.24) ZRnu(x)[b, T] (x)pdx ≤CZRn
| (x) (x)|pdx.
Theo em 1.8 imp o es a esul in [11], whe e he same inequali y was p o ed as-
suming ha Ais a powe bump: A( ) = p, > 1.
Rema k 1.9.An analogous esul holds o highe o de commu a o s Tk
b, wi h k≥2.
(Fo k= 1, T1
b= [b, T].) These a e de ined induc i ely by Tk
b= [b, Tk−1
b]. In his case
he condi ion imposed on he pai o weigh s (u, ) is (1.23) wi h A( ) = plog(e+
)(k+2) p−1+δand B( ) = p0log(e+ )(k+1) p0−1+δ,δ > 0. The p oo is essen ially he
same and some de ails a e gi en in Rema k 5.7 below.
Rema k 1.10.We conjec u e ha Theo em 1.8 can be imp o ed by aking A( ) =
plog(e+ )2p−1+δ— he commu a o should equi e one mo e log e m on each weigh
han he associa ed singula in eg al.
1.4. Applica ion o he Sa ason conjec u e. Theo em 1.2 has an applica ion o
an open p oblem in ope a o heo y on he uni disk. This p oblem was i s posed by
Sa ason (see Kha in and Nikol’ski˘ı [23]) and is e e ed o as he Sa ason conjec u e.
To s a e i we ecall some basic ac s abou ope a o heo y on he uni ci cle. (Fo
comple e in o ma ion, see Koosis [24].)
Gi en a unc ion ∈L1(∂D), we de ine he pe iodic Hilbe ans o m o , also
known as he conjuga e unc ion o , by
˜
(eiθ) = ˜
H (eiθ) = 1
πZπ
0
(ei(θ− ))− (ei(θ+ ))
2 an( /2) d .
The pe iodic Hilbe ans o m is a Calde ´on-Zygmund singula in eg al and so is
a bounded ope a o on L2(∂D). De ine he Riesz p ojec ion ope a o Pby
P (eiθ) = (eiθ) + ˜
H (eiθ) + ˆ
(0)
2.
Then Pis also bounded on L2(∂D), and in ac is he o hogonal p ojec ion om
L2(∂D) o he Ha dy space H2(∂D), he closu e o he analy ic polynomials in L2(∂D).
Gi en a unc ion h∈L2(∂D), de ine he Toepli z ope a o wi h symbol hby
Th (eiθ) = P(h )(eiθ).
The Toepli z ope a o This densely de ined on H2(∂D) and is a bounded ope a o on
H2(∂D) i and only i h∈L∞(∂D). Toepli z ope a o s ha e been in ensi ely s udied
and appea in many p oblems in ope a o heo y.
16 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´
EREZ
Le w=uqh; hen w∈L∞
csince h∈L∞
c(Rn) and u∈L∞(Rn). We will now o m
a kind o a omic decomposi ion o w ha is due o Le ne [26] and lies a he hea
o his p oo o Lemma 2.6. Fix a > 2nand m > 0 such ha kwkL∞≤am. Fo each
k≤m, le {Qk
j}be he Calde ´on-Zygmund cubes o wa heigh ak(Lemma 2.2).
Le wQk
j=|Qk
j|−1RQk
jw(x)dx, and o each kde ine he unc ions
bk(x) = X
j
(w(x)−wQk
j)χQk
j(x), gk(x) = w(x)−bk(x) = (wQk
jx∈Qk
j
w(x)x∈Rn Ωak.
Again by Lemma 2.2, o all kwe ha e gk(x)≤2nakand kgkk1=kwk1.
Since he se Ωamis emp y, bm= 0. The e o e, o e e y in ege l < 0, we ha e he
elescoping sequence
w(x) =
m−1
X
k=lbk(x)−bk+1(x)+gl(x).
By (2.1), wQk
j≤2nak. Since o each jand k,
(3.1) (bk(x)−bk+1(x))χQk
j(x)
= (w(x)−wQk
j)χQk
j(x)−X
Qk+1
i⊂Qk
j
(w(x)−wQk+1
i)χQk+1
i(x),
i ollows immedia ely ha o all x,
(3.2) |bk(x)−bk+1(x)| ≤ (1 + a) 2nak.
Fu he , by in eg a ing (3.1) we see ha
(3.3) ZQk
jbk(x)−bk+1(x)dx = 0.
We can now es ima e as ollows: o any l < 0,
(3.4) ZRn
|T (x)|qu(x)qh(x)dx =ZRn
|T (x)|qw(x)dx
=
m−1
X
k=lZRn
|T (x)|q(bk(x)−bk+1(x)) dx +ZRn
|T (x)|qgl(x)dx.
We now claim ha he las e m on he igh hand side ends o 0 as l→ −∞. This
ollows a once om H¨olde ’s inequali y, he ac ha Tis bounded on L2(Rn), and
ha and wa e bounded unc ions wi h compac suppo :
0≤ZRn
|T (x)|qgl(x)dx ≤ZRn
|T (x)|2dxq/2ZRn
gl(x)(2/q)0dx1/(2/q)0
≤Ck kq/2
2(2nal)(q/2)kglk1/(2/q)0
1=Ck kq/2
2(2nal)(q/2)kwk1/(2/q)0
1.

SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 17
As l→ −∞ he las e m ends o ze o. The e o e, aking he limi in (3.4) we ge
(3.5) ZRn
|T (x)|qu(x)qh(x)dx =
m−1
X
k=−∞ ZRn
|T (x)|q(bk(x)−bk+1(x)) dx.
We es ima e he igh hand side o (3.5) as ollows. Fo each j, k, le ck
jbe a cons an
whose alue will be speci ied below. Since q < 1, ||a|q− |b|q|≤|a−b|q. The e o e,
by (3.3) and (3.2),
m−1
X
k=−∞ ZRn
|T (x)|qbk(x)−bk+1(x)dx =X
k,j ZQk
j
|T (x)|qbk(x)−bk+1(x)dx
=X
k,j ZQk
j|T (x)|q− |ck
j|qbk(x)−bk+1(x)dx
≤CX
k,j
(1 + a) 2nakZQk
j|T (x)|q− |ck
j|qdx ≤CX
k,j
wQk
jZQk
j
|T (x)−ck
j|qdx
≤CX
k,j
wQk
jZQk
j
|T( χ2Qk
j)(x)|qdx +CX
k,j
wQk
jZQk
j
|T( χRn 2Qk
j)(x)−ck
j|qdx
=C(I1+I2).
We conside each e m sepa a ely. To es ima e I1we use Kolmogo o ’s inequali y
(since q < 1) and Lemmas 2.3 and 2.7:
I1≤CX
k,j
1
|2Qk
j|Z2Qk
j
w(x)dx1
|2Qk
j|Z2Qk
j
| (x)|dxq|Qk
j|
=CX
k,j
1
|2Qk
j|Z2Qk
j
u(x)qh(x)dx1
|2Qk
j|Z2Qk
j
(x)| (x)| (x)−1dxq|Qk
j|
≤CX
k,j
kuqkC,2Qk
jkhk¯
C,2Qk
jk kq
¯
B,2Qk
j
k −1kq
B,2Qk
j
|e
Qk
j|,
whe e C( ) = log(e+ ) −1+. Le Cq( ) = C( q) = plog(e+ q) −1+≈A( ).
The e o e, by a change o a iables in he de ini ion o he O licz no m,
kuqkC,2Qk
j=kukq
Cq,2Qk
j
≈ kukq
A,2Qk
j
.
Hence, (1.16), he ac ha he se s e
Qk
ja e disjoin , and H¨olde ’s inequali y yield
I1≤CX
k,j
khk¯
C,2Qk
jk kq
¯
B,2Qk
j
|e
Qk
j|
≤CX
k,j Ze
Qk
j
M¯
Ch(x)M¯
B( )(x)qdx
≤CZRn
M¯
Ch(x) 0dx1/ 0ZRn
M¯
B( )(x)pdxq/p
18 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´
EREZ
≤CZRn
h(x) 0dx1/ 0ZRn
| (x) (x)|pdxq/p
=CZRn
| (x) (x)|pdxq/p.
The las inequali y holds since ¯
C∈B 0and ¯
B∈Bp, and so by Lemma 2.8 M¯
Cis
bounded on L 0and M¯
Bis bounded on Lp. This comple es he es ima e o I1.
To es ima e I2we choose he alue o he cons an ck
j o be
ck
j=1
|Qk
j|ZQk
j
T( χRn 2Qk
j)(y)dy.
Le C( ) be as in he es ima e o I1. Then, by a s anda d es ima e o Calde ´on-
Zygmund singula in eg als (see [13, 18]), since q < 1 and by Lemmas 2.3 and 2.7,
we ob ain
I2≤CX
k,j
1
|Qk
j|ZQk
j
u(x)qh(x)dx∞
X
i=1
2−i1
|2iQk
j|Z2iQk
j
| (x)|dxq|Qk
j|
≤CX
k,j
1
|Qk
j|ZQk
j
u(x)qh(x)dx
∞
X
i=1
2−iq 1
|2iQk
j|Z2iQk
j
(x) (x) (x)−1dxq|Qk
j|
≤CX
k,j
kuqkC,Qk
jkhk¯
C,Qk
j|e
Qk
j|
∞
X
i=1
2−iq k kq
¯
B,2iQk
j
k −1kq
B,2iQk
j
≤CX
k,j
kukq
A,Qk
j
khk¯
C,Qk
j|e
Qk
j|
∞
X
i=1
2−iq k kq
¯
B,2iQk
j
k −1kq
B,2iQk
j
.
Fo 0 < β < 1, A(β )≤βpA( ), so by he de ini ion o he Luxembu g no m, we
ha e ha kukA,Qk
j≤C2i n/p kukA,2iQk
j. Thus, by (1.16) and since p>ni ollows
I2≤CX
k,j
khk¯
C,Qk
j|e
Qk
j|
∞
X
i=1
2−iq 2inq/p kukq
A,2iQk
j
k kq
¯
B,2iQk
j
k −1kq
B,2iQk
j
≤CX
k,j
khk¯
C,Qk
j|e
Qk
j|in
x∈Qk
j
M¯
B( )(x)q
≤CX
k,j Ze
Qk
j
M¯
Ch(x)M¯
B( )(x)qdx.
We can now a gue as we did abo e o I1 o ob ain he desi ed es ima e o I2.
3.2. P oo o Theo em 1.7. The p oo is almos iden ical o he one jus gi en
and we only indica e he mino changes. We p oceed in he same manne wi h Sd
in place o T. We obse e ha Sdis bounded on L2(Rn) and so i su ices o ge
he app op ia e es ima es o I1and I2, whe e now in I1we w i e χQk
jin place o
χ2Qk
jand in I2we pu χRn Qk
jin place o χRn 2Qk
j. The es ima e o I1adap s
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 19
immedia ely o he dyadic squa e unc ion since Sdis o weak- ype (1,1) and hus
sa is ies Kolmogo o ’s inequali y.
Since he dyadic squa e unc ion is mo e localized han a singula in eg al, he
es ima e o I2is much easie . Fix a cube Qk
jand se
ck
j=X
Q∈∆
Q⊇Qk
j( χRn Qk
j)Q−( χRn Qk
j)b
Q21/2.
Then o any x∈Qk
jwe ha e ha Sd( χRn Qk
j)(x)≡ck
j; hus I2= 0 and we a e done.
4. P oo o Theo em 1.4
A he hea o he p oo o Theo em 1.4 is he ollowing lemma, whose p oo we
de e o he momen .
Lemma 4.1. Gi en pand (u, )as in he hypo heses o Theo em 1.4, he e exis s q,
0< q < 1, such ha o all , h ∈L∞
c(Rn),
(4.1)
ZRn
M (x)qM(uqh)(x)dx ≤CZRn
| (x) (x)|pdxq/p ZRn
|h(x)|(p/q)0dx1/(p/q)0
.
P oo o Theo em 1.4.Fix qas in Lemma 4.1 and le =p
q>1. Then by duali y,
ZRn
|u(x)T (x)|pdxq/p = sup ZRn
|T (x)|qu(x)qh(x)dx,
whe e he sup emum is aken o e all non-nega i e unc ions h∈L∞
c(Rn) such ha
khkL 0(Rn)= 1. Fix such a unc ion h. Then by Lemmas 2.6 and 4.1,
ZRn
|T (x)|qu(x)ph(x)dx ≤CZRn
M (x)qM(uqh)(x)dx
≤CZRn
| (x) (x)|pdxq/p ZRn
|h(x)| 0dx1/ 0
=CZRn
| (x) (x)|pdxq/p.
This comple es he p oo o Theo em 1.4. 
4.1. P oo o Lemma 4.1. Fix ; by a s anda d a gumen we may assume wi hou
loss o gene ali y ha ≥0. Fu he , as we no ed abo e, we may assume wi hou
loss o gene ali y ha ∈L∞
cand u, ∈L∞. Fix q, 0 < q < 1, su icien ly close o
1 ha he e exis s  > 0 such ha 2p−1 + δ= 2(p/q)−1 + . Le =p/q,w=uqh
and a= 4n>2n. Fo each j, k le
Ωkj ={ak−j−1< Mw(x)≤ak−j+1}∩{aj< M (x)q≤aj+1};
hen
ZRn
M (x)qMw(x)dx =X
k,j Z{ak<(M )qMw≤ak+1}∩{aj<(M )q≤aj+1}
M (x)qMw(x)dx
20 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´
EREZ
≤X
k,j ZΩkj
M (x)qMw(x)dx.
Fo each in ege l, m, le {R
l} be he CZ cubes o wa heigh al, and le {Ss
m}s
be he CZ cubes o a heigh am/q. Then by Lemma 2.4, o each pai (k, j),
{x:Mw(x)> ak−j−1} ⊂ [
3R
k−j−2,{x:M (x)q> aj} ⊂ [
s
3Ss
j−1.
I x∈Ωkj, he e exis s a leas one pai ( , s) such ha x∈3R
k−j−2∩3Ss
j−1.
Le E s
kj ={x∈Ωkj :x∈3R
k−j−2∩3Ss
j−1}. I he se E s
kj is no emp y, hen
3R
k−j−2∩3Ss
j−16=∅. The e o e, depending on hei ela i e sizes, we ei he ha e
3R
k−j−2⊂9Ss
j−1, o 3Ss
j−1⊂9R
k−j−2. I he i s inclusion holds we say ha
(k, j, , s)⊂Γ1; i he second holds we say ha (k, j, , s)∈Γ2. Hence,
ZRn
M (x)qMw(x)dx ≤X
k,j X
,s ZE s
kj
M (x)qMw(x)dx ≤X
k,j X
,s
ak−j+1aj+1|E s
kj|
≤X
(k,j, ,s)∈Γ1
ak−j+1aj+1|E s
kj|+X
(k,j, ,s)∈Γ2
ak−j+1aj+1|E s
kj|=I1+II2.
To comple e he p oo we will es ima e each e m sepa a ely. We conside i s I1.
Since E s
kj ⊂3R
k−j−2, by Lemma 2.3, |E s
kj| ≤ 3n|R
k−j−2| ≤ C|e
R
k−j−2|. On he o he
hand 3R
k−j−2⊂9Ss
j−1. Thus by Lemma 2.2,
I1≤a5X
(k,j, ,s)∈Γ11
|R
k−j−2|ZR
k−j−2
w(x)dx 1
|Ss
j−1|ZSs
j−1
(x)dxq
|E s
kj|
≤CX
j,s X
k, :
(k,j, ,s)∈Γ1
1
|R
k−j−2|ZR
k−j−2
w(x)dx · | e
R
k−j−2| 1
|9Ss
j−1|Z9Ss
j−1
(x)dxq
≤CX
j,s X
k, :
(k,j, ,s)∈Γ1
Ze
R
k−j−2
M(wχ9Ss
j−1)(x)dx 1
|9Ss
j−1|Z9Ss
j−1
(x)dxq
.
Since he se s e
R
k−j−2a e disjoin and con ained in 9Ss
j−1, we can apply Yano’s he-
o em (see Zygmund [56]) o ge
I1≤CX
j,s
1
|9Ss
j−1|Z9Ss
j−1
M(wχ9Ss
j−1)(x)dx 1
|9Ss
j−1|Z9Ss
j−1
(x)dx!q
|e
Ss
j−1|
≤CX
j,s
kwkΦ,9Ss
j−1 1
|9Ss
j−1|Z9Ss
j−1
(x)dx!q
|e
Ss
j−1|,
whe e Φ( ) = log(e+ ) and he cons an depends only on nand no on he cube
Ss
j−1. Recall ha 2p−1+δ= 2 −1+; hence, i we de ine D( ) = log(e+ )2 −1+,
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 21
hen D( q)≈A( ). Now de ine e
D( ) = 0log(e+ )−1−( 0−1) ∈B 0. Then we ha e
ha
Φ−1( )≈
log(e+ )= 1
log(e+ )2 −1+
× 1
0log(e+ )−1+ 2 −1+
≈D−1( )·e
D−1( ).
The e o e, ecalling ha w=uqh, we can apply Lemma 2.7 and (1.18) o ge
I1≤CX
j,s
kuqkD,9Ss
j−1khke
D,9Ss
j−1k kq
¯
B,9Ss
j−1
k −1kq
B,9Ss
j−1|e
Ss
j−1|
≤CX
j,s
kukq
A,9Ss
j−1khke
D,9Ss
j−1k kq
¯
B,9Ss
j−1
k −1kq
B,9Ss
j−1|e
Ss
j−1|
≤CX
j,s Ze
Ss
j−1
Me
Dh(x)M¯
B( )(x)qdx
≤CZRn
Me
Dh(x)M¯
B( )(x)qdx
≤CZRn
Me
Dh(x) 0dx1/ 0ZRn
M¯
B( )(x)pdxq/p
≤CZRn
|h(x)| 0dx1/ 0ZRn
( (x) (x))pdxq/p
,
whe e he hi d inequali y holds because he se s e
Ss
j−1a e disjoin , and he las
inequali y holds since by Lemma 2.8, ¯
B∈Bpso M¯
Bis bounded on Lp, and, as we
no ed abo e, e
D∈B 0, so Me
Dis bounded in L 0. Thus we ge he desi ed bound o
I1.
We will now es ima e I2. The ideas a e he same, excep ha a he key s ep we
will use Kolmogo o ’s inequali y ins ead o Yano’s heo em. Since E s
kj ⊂3Ss
j−1, by
Lemma 2.3, |E s
kj| ≤ C|e
Ss
j−1|. Fu he , Md (x)q> aj−1on Ss
j−1. Thus
I2≤a3X
(k,j, ,s)∈Γ2
aj+1|E s
kj|1
|R
k−j−2|ZR
k−j−2
w(x)dx
≤CX
(k,j, ,s)∈Γ2
aj+1|e
Ss
j−1|1
|9R
k−j−2|Z9R
k−j−2
w(x)dx
≤CX
(k,j, ,s)∈Γ2Ze
Ss
j−1
Md (x)qdx1
|9R
k−j−2|Z9R
k−j−2
w(x)dx
=CX
l, X
(k,j, ,s)∈Γ2
k−j−2=lZe
Ss
j−1
Md (x)qdx1
|9R
l|Z9R
l
w(x)dx.
Fo ixed land ,e
Ss
j−1⊂9R
l. Thus, by Lemma 2.2, o all x∈e
Ss
j−1,
(4.2) Md (x) = Md( χSs
j−1)(x)≤Md( χ9R
l)(x).

22 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´
EREZ
Since ≤Φ( ), kwkL1,9R
l≤ kwkΦ,9R
l(see [41]). The e o e, using his, (4.2),
Lemma 2.3, and Kolmogo o ’s inequali y, we ge ha
I2≤CX
l, X
(k,j, ,s)∈Γ2
k−j−2=lZe
Ss
j−1
Md( χ9R
l)(x)qdxkwkΦ,9R
l
≤CX
l,
1
|9R
l|Z9R
l
Md( χ9R
l)(x)qdx · kwkΦ,9R
l· | e
R
l|
≤CX
l, 1
|9R
l|Z9R
l
(x)dxq
kwkΦ,9R
l· | e
R
l|.
We can now a gue exac ly as we did in he es ima e o I1 o ge he desi ed bound
o I2. This comple es he p oo .
Rema k 4.2.The e m I2is less singula han he e m I1: i we did no eplace
kwkL1,9R
lby kwkΦ,9R
l, hen a sligh modi ica ion o ou a gumen would show ha
we ge he desi ed bound o I2assuming only he weake condi ion (1.16).
5. P oo o Theo em 1.8
The p oo o Theo em 1.8 is iden ical in basic idea and o ganiza ion o he p oo
o Theo em 1.4, di e ing only in de ails. The e o e, a he han gi e he comple e
a gumen , we will ou line he changes necessa y in he p oo o Theo em 1.4.
The key changes a e in he s a emen and p oo o Lemma 4.1. The new lemma is
he ollowing.
Lemma 5.1. Gi en pand (u, )as in he hypo heses o Theo em 1.8, he e exis s q,
0< q < 1, such ha o all , h ∈L∞
c(Rn),
ZRn
M2 (x)qM2(uqh)(x)dx ≤CZRn
| (x) (x)|pdxq/pZRn
|h(x)|(p/q)0dx1/(p/q)0
.
Gi en his inequali y, he p oo o Theo em 1.8 begins wi h he same duali y a gu-
men as he p oo o Theo em 1.4. Bu , ins ead o Lemma 2.5 we use he ollowing
poin wise es ima e om [37]: gi en 0 < q <  < 1,
(5.1) M#
q([b, T] )(x)≤C M(T )(x) + C M2 (x).
Thus by Lemma 2.6, we ha e o e e y weigh w ha
ZRn
|[b, T] (x)|qw(x)dx ≤CZRn
M#
q([b, T] )(x)qMw(x)dx
≤CZRn
M(T )qMw(x)dx +CZRn
(M2 )qMw(x)dx.
The second in eg al in he las e m is dominan . To see his we use he ac ha
q/ < 1, and Lemmas 2.6 and 2.5 o ge
ZRn
M(T )(x)qMw(x)dx =ZRn
M(|T |)(x)q
Mw(x)dx
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 23
≤CZRn
M#(|T |)(x)q
M2w(x)dx =CZRn
M#
(T )(x)qM2w(x)dx
≤CZRn
M (x)qM2w(x)dx ≤CZRn
M2 (x)qM2w(x)dx.
Hence, we ha e shown ha
(5.2) ZRn
M(T )(x)qMw(x)dx ≤CZRn
M2 (x)qM2w(x)dx.
Fix qas in Lemma 5.1 and le =p
q>1. Then by duali y,
ZRn
|u(x) [b, T] (x)|pdxq/p = sup ZRn
|[b, T] (x)|qu(x)qh(x)dx,
whe e he sup emum is aken o e all non-nega i e unc ions h∈L∞
c(Rn) such ha
khkL 0(Rn)= 1. Fix such a unc ion h. By (5.2) and Lemma 5.1 i ollows ha
ZRn
|[b, T] (x)|qu(x)qh(x)dx ≤CZRn
M2 (x)qM2(uqh)(x)dx
≤CZRn
| (x) (x)|pdxq
pZRn
|h(x)| 0dx1
0=CZRn
| (x) (x)|pdxq/p
.
This comple es he p oo o Theo em 1.8.
5.1. P oo o Lemma 5.1. De ine, as be o e, Φ( ) = log(e+ ). I is well known
(see, o ins ance, [45, 12]) ha M2 ≈MΦ , so in he desi ed es ima e we can
eplace M2by he maximal unc ion MΦ. We ollow he same s eps as in he p oo o
Lemma 4.1, bu we eplace he Calde ´on-Zygmund decomposi ion by a mo e gene al
decomposi ion based on he O licz maximal ope a o MΦ. I s essen ial p ope ies a e
exac ly he same and a e cap u ed in he ollowing de ini ion and lemmas.
De ini ion 5.2. Gi en a Young unc ion Φ, a non-nega i e unc ion ∈L1(Rn)
(e.g., ∈L∞
c(Rn)), and λ > 0, we de ine he CZ cubes o a heigh λwi h espec
o Φ o be he maximal disjoin dyadic subcubes o he se {x∈Rn:Md,Φ (x)> λ}.
Lemma 5.3. Gi en λ > 0and ∈L1(Rn)non-nega i e, le {Qj}be he se o CZ
cubes o wi h espec o Φ. Then o all j, we ha e λ < k kΦ,Qj≤2nλ. Fu he ,
Md,Φ (x) = Md,Φ( χQj)(x) o all x∈Qj.
Lemma 5.4. Le ∈L1(Rn). Fix a > 2n, and o k∈Z, le {Qk
j}be he CZ cubes
o a heigh akwi h espec o Φ. Then he e exis se s {e
Qk
j},e
Qk
j⊂Qk
j, which a e
pai wise disjoin o all jand k, and such ha he e exis s α > 1depending only on
aand nsuch ha |Qk
j| ≤ α|e
Qk
j|.
Lemma 5.5. Le ∈L1(Rn)and ix λ > 0. Le {Qj}be he se o CZ cubes o a
heigh λ/4nwi h espec o Φ. Then {x∈Rn:MΦ (x)> λ} ⊂ ∪j3Qj.
The p oo o each o hese lemmas is gi en in [9] excep o he iden i y in Lemma
5.3, whose p oo is iden ical o he p oo o (2.2).
24 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´
EREZ
We now ob ain he desi ed es ima e wi h MΦin place o M2. P oceed as in Lemma
4.1 bu and w=uqha e decomposed wi h espec o MΦ(in place o M). We
es ima e I1and I2using he p e ious lemmas and epea ing he compu a ions in
Lemma 4.1. We ge ha
I1≤CX
j,s
1
|9Ss
j−1|Z9Ss
j−1
MΦ(wχ9Ss
j−1)(x)dx · k kq
Φ,9Ss
j−1|e
Ss
j−1|,
I2≤CX
l,
1
|9R
l|Z9R
l
Md,Φ( χ9R
l)(x)qdx · kwkΦ,9R
l|e
R
l|.
Fo k≥0 de ine Φk( ) = log(e+ )k; hen Φ = Φ1. We ha e he ollowing
auxilia y esul : he i s inequali y gene alizes Yano’s heo em and is well known
(see o ins ance [12]), and he p oo o he second is gi en below.
Lemma 5.6. Le k≥0and 0< q < 1. Then he e exis s a cons an Csuch ha o
any cube Q
1
|Q|ZQ
MΦk(gχQ)(x)dx ≤CkgkΦk+1,Q,1
|Q|ZQ
MΦk(gχQ)(x)qdx ≤Ckgkq
Φk,Q.
I we apply Lemma 5.6 o he es ima es o I1and I2we ge
I1≤CX
j,s
kwkΦ2,9Ss
j−1k kq
Φ,9Ss
j−1|e
Ss
j−1|
I2≤CX
l,
k kq
Φ,9R
lkwkΦ,9R
l|e
R
l| ≤ CX
l,
k kq
Φ,9R
lkwkΦ2,9R
l|e
R
l|.
Hence, bo h hese es ima es can be handled in he same way. Le
D( ) = p0log(e+ )2p0−1+δ,e
D( ) = plog(e+ )−1−δ(p−1) ∈Bp,
E( ) = log(e+ )3 −1+,e
E( ) = 0log(e+ )−1−( 0−1) ∈B 0.
Then Φ−1( )≈D−1( )·e
D−1( ), Φ−1
2( )≈E−1( )·e
E−1( ), D( ) = B( ) and E( q)≈
A( ). The e o e, by Lemma 2.7 we ha e o e e y cube Q ha
k kΦ,Q ≤Ck ke
D,Q k −1kB,Q,kwkΦ2,Q ≤CkuqkE,Q khke
E,Q ≤Ckukq
A,Q khke
E,Q.
Subs i u e hese alues in o he abo e es ima es o I1,I2; since e
D∈Bp,e
E∈B 0,
he p oo can now be comple ed exac ly as in he p oo o Lemma 4.1.
Rema k 5.7.As we no ed in Rema k 1.9, he p oo o Theo em 1.8 can be adap ed
o ea he highe o de commu a o s Tk
b,k≥2. The ideas a e essen ially he same.
Beginning wi h he duali y a gumen and applying he analog o (5.1) o highe
o de commu a o s (also ound in [37]), i is no di icul o see ha he p oo educes
o ob aining a e sion o Lemma 5.1 wi h Mk+1 in place o M2. As Mk+1 ≈MΦk, he
decomposi ions o and wa e made wi h espec o his O licz maximal unc ion.
I we le D( ) = p0log(e+ )(k+1) p0−1+δ,E( ) = log(e+ )(k+2) −1+(e
D,e
E emain
he same), hen by means o Lemma 5.6 we ge ha he bumps o uand a e,
espec i ely, A( ) = plog(e+ )(k+2) p−1+δand B( ) = p0log(e+ )(k+1) p0−1+δ.
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 25
P oo o Lemma 5.6. We only need o p o e show he second inequali y. By homo-
genei y i su ices o assume ha kgkΦk,Q = 1. By he p ope ies o O licz no ms (see
[41]), his implies ha
(5.3) 1
|Q|ZQ
Φk(|g(x)|)dx ≤1.
The maximal ope a o MΦksa is ies he modula inequali y
(5.4) {x∈Rn:MΦkh(x)> λ}≤CZRn
Φk(|h(x)|/λ)dx.
The p oo is s anda d; see, o ins ance, [36]. Finally, we no e ha Φkis submul i-
plica i e: Φk(s )≤Φk(s)Φk( ). The e o e, since 0 < q < 1, i we w i e he Lq-no m
in e ms o he le el se s, hen by (5.3) and (5.4) we ha e ha
1
|Q|ZQ
MΦk(g χQ)qdx ≤Z1
0
q λqdλ
λ+CZ∞
1
λq1
|Q|ZRn
Φk(|g(x)χQ(x)|/λ)dx dλ
λ
≤C+CZ∞
1
λqΦk(1/λ)dλ
λ
1
|Q|ZQ
Φk(|g(x)|)dx ≤C.

Re e ences
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[4] M. Co la and C. Sadosky, On he Helson-Szeg¨o heo em and a ela ed class o modi ied Toepli z
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P oc. Symp. Pu e Ma h. 35, AMS, P o idence, 1979.
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Ha monic Analysis in Hono o An oni Zygmund, W. Beckne e . al. eds., ol. 1, 306-317,
Wadswo h In e na ional Ma hema ics Se ies, Wadswo h, Belmon , 1983.
[6] D. C uz-U ibe, SFO, The p oduc o unbounded Toepli z ope a o s, In eg al Equa ions Ope a o
Theo y 20 (1994), 231-237.
[7] D. C uz-U ibe, SFO, and A. Fio enza, The A∞p ope y o Young unc ions and weigh ed
no m inequali ies, Hous on J. Ma h., 28 (2002), 169-182.
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ions, J. Func . Anal. 213 (2004), 412-439.
[9] D. C uz-U ibe, SFO, and C. P´e ez, Sha p wo-weigh , weak- ype no m inequali ies o singula
in eg al ope a o s, Ma h. Res. Le . 6 (1999), 417-428.
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