Publ. Ma . 49 (2005), 73–91
EXTRAPOLATION AND SHARP NORM ESTIMATES
FOR CLASSICAL OPERATORS ON WEIGHTED
LEBESGUE SPACES
Oli e D agiˇ
ce i´
c∗, Loukas G a akos†, Ma ´
ıa
C is ina Pe ey a‡and S e anie Pe e michl†
Abs ac
We ob ain sha p weigh ed Lpes ima es in he Rubio de F ancia
ex apola ion heo em in e ms o he Apcha ac e is ic cons an
o he weigh . P ecisely, i o a gi en 1 < <∞ he no m o
a sublinea ope a o on L (w) is bounded by a unc ion o he
A cha ac e is ic cons an o he weigh w, hen o p > i is
bounded on Lp( ) by he same inc easing unc ion o he Apcha -
ac e is ic cons an o , and o p < i is bounded on Lp( ) by
he same inc easing unc ion o he −1
p−1powe o he Apcha ac-
e is ic cons an o . Fo some ope a o s hese bounds a e sha p,
bu no always. In pa icula , we show ha hey a e sha p o he
Hilbe , Beu ling, and ma ingale ans o ms.
1. In oduc ion
1.1. Ex apola ion.
A posi i e locally in eg able unc ion on Rnis called a weigh . A
weigh wis said o be o class Ap, o 1 < p < ∞, i
sup
Q1
|Q|ZQ
w 1
|Q|ZQ
w−1
p−1p−1
<∞,
2000 Ma hema ics Subjec Classi ica ion. 42A50, 42B20, 42B25, 46M35 (44A15,
47B38).
Key wo ds. Ex apola ion, sha p weigh ed es ima es, dyadic squa e unc ion, dyadic
pa ap oduc , ma ingale ans o m, Hilbe ans o m, Beu ling ans o m.
∗Resea ch suppo ed by he Eu opean Commission (IHP ne wo k “Ha monic Analysis
and Rela ed P oblems” 2002–2006, con ac HPRN-CT-2001-00273-HARP).
†Wo k suppo ed by he NSF.
‡Resea ch pa ially done while isi ing he Cen e de Rece ca Ma em`a ica in
Ba celona, Spain.
74 O. D agiˇ
ce i´
c e al.
whe e he sup emum is aken o e all cubes Qin Rnwi h sides pa allel
o he axes (Qwill always deno e such cubes). The quan i y abo e is
called he Ap-cha ac e is ic cons an o he weigh wand will be deno ed
by kwkAp.
A weigh wis said o be o class A1i he e is a cons an C > 0 such
ha
Mw ≤Cw a.e.,
whe e Mis he (uncen e ed) Ha dy-Li lewood maximal unc ion, i.e.
M (x) = sup
x∈Q
1
|Q|ZQ| (y)|dy.
The smalles possible Cis deno ed by kwkA1.
Fo an ope a o Tbounded om a Banach space Xin o i sel (T∈
B(X)) we will deno e by kTkXi s ope a o no m. When 1 <q<∞,
q0shall s and o he dual exponen o q, i.e. 1
q+1
q0= 1. Gi en a weigh
on Rn,Lp( ) deno es he space o complex unc ions on Rnsuch ha
RRn| |p is ini e.
The ollowing esul is he celeb a ed ex apola ion heo em o Rubio
de F ancia.
Theo em (E). Assume we a e gi en a sublinea 1ope a o
T:[
w∈Aq
1≤q<∞
Lq(w)−→ {all measu able complex- alued unc ions}.
Suppose he e is 1≤ < ∞such ha T∈B(L (u)) o all weigh s u∈A ,
wi h bounds depending only on kukA . Then T∈ B(Lp(w)) o all 1<
p < ∞and all weigh s w∈Ap, wi h bounds depending only on kwkAp.
Mo e p ecisely, suppose o each B > 1 he e is a cons an N (B)>0
such ha we ha e
(1) kTkL (u)≤N (B) o all u∈A wi h kukA ≤B.
Then o any 1< p < ∞and B > 1 he e is Np(B)>0such ha o
all weigh s w∈Apwi h kwkAp≤B,
(2) kTkLp(w)≤Np(B).
1I u ns ou ha Tdoes no need o be sublinea , jus well-de ined on i s domain,
see [G , Sec ion 9.5.b].
Ex apola ion and Sha p Weigh ed Es ima es 75
This esul i s appea ed in [R]. Di e en p oo s can be ound in he
books [GC-RF] and [G ].
Muckenhoup p o ed in [M] ha o 1 <p<∞ he maximal unc-
ion is bounded on Lp(w) i and only i he weigh wbelongs o he
class Ap. Hun , Muckenhoup and Wheeden p o ed in [HMW] ha he
Apcondi ion also cha ac e izes he boundedness o he Hilbe ans o m
H (x) = p. . 1
πZ (y)
x−ydy
in Lp(w). Coi man and Fe e man [CF] ex ended he heo y o gene al
Calde ´on-Zygmund ope a o s.
In 1993, Buckley [Buc] ob ained he ollowing esul conce ning he
Ha dy-Li lewood maximal unc ion2(1 < p < ∞):
(3) kMkLp(w)≤C(p)kwkp0/p
Ap,
whe e he cons an C(p) depends only on p(and he unde lying di-
mension n). These bounds a e sha p, i.e. kwkp0/p
Apcanno be eplaced
by ϕ(kwkAp) o any unc ion ϕ:R+→R+ ha g ows slowe han he
p0/p- h powe . This can be easily seen by using powe unc ions and
powe weigh s. Taking w≡1 we see ha he cons an s C(p) mus blow
up as p→1.
In his no e we use Buckley’s es ima e (3) o imp o e Theo em (E)
as ollows.
Theo em 1. Wi h he no a ion and hypo heses as in Theo em (E),
assume ha N (B)deno es he smalles cons an ha sa is ies inequal-
i y (1). Then o any 1<p<∞and all B > 1 he e is a cons an Np(B)
such ha (2) holds o all weigh s win Apsa is ying kwkAp≤B. Mo e-
o e ,
Np(B)≤


21
N (2C(p0)p−
p−1B)i p >
2 −1
N (2 −1(C(p)p− B) −1
p−1)i p < .
He e C(p)is he cons an appea ing in (3).
2Buckley ac ually ob ained his esul o he cen e ed maximal unc ion M0. How-
e e , he uncen e ed maximal unc ion M, he cen e ed one M0, and he dyadic
maximal unc ion Mda e compa able modulo dimensional cons an s, so (3) holds o
ei he one.
76 O. D agiˇ
ce i´
c e al.
This esul , applied o he Beu ling and ma ingale ans o ms
o = 2, N2(B) = CB and p > 2, was i s obse ed in [Pe V]. In
his case, a ca e ul ex apola ion o p > 2 yields Np(B)≤CpB. Tha
is, he linea dependence on he cons an when p= 2 is p ese ed also
o p > 2. Howe e , his is no he case when p < 2, which mo i a es a
mo e ca e ul examina ion o he p oblem.
1.2. Sha p bounds.
The linea bounds o he Beu ling ans o m in Lp(w) in e ms
o kwkAp o p≥2 ha e impo an consequences in he heo y o qua-
sicon o mal mappings. The connec ion is e y well explained in he
pape by As ala, Iwaniec and Saksman [AIS] who we e in e es ed in
inding he minimal q < 2 o which all solu ions o he Bel ami equa-
ion ¯
∂ =µ·∂ ha belong o he Sobole space W1,q
loc s ill sel -imp o e
o belonging o W1,2
loc , i.e. a e quasi egula . He e µis a bounded unc ion
wi h kµk∞=k < 1. A deep esul o As ala [A] says ha q > 1 + k
su ices. On he o he hand, Iwaniec and Ma in [IM] ound examples
showing ha he esul could in gene al no be ue o q < 1 + k.
In [AIS] he bo de line case q= 1 + kwas add essed; i was poin ed
ou by he au ho s ha quasi egula i y would be a consequence o he
linea dependence o he no m o he Beu ling ans o m on weigh ed
spaces Lp(w) o p≥2 in e ms o he Apcha ac e is ic o he weigh w.
This linea dependence was se led in [Pe V] and la e in [DV] o p≥2,
he only ange o which i is ue.
Fo he maximal unc ion, he bound o kMkL2(w)is also linea
in kwkA2, see (3). I 1 <p<2, ex apola ion yields sha p dependence
o kMkLp(w)on kwkAp. Howe e , o p > 2, ex apola ion only gi es
linea g ow h on kwkAp, when he sha p g ow h is kwkp0/p
Ap. In [Buc],
Buckley conside s wo mo e examples we e he same phenomena occu .
He shows ha a pa ame ic class o Ma cinkiewicz in eg al ope a o s is
uni o mly bounded on Lp(w) by kwkAp o all 1 < p < ∞and hese
linea es ima es a e sha p [Buc, Theo em 2.15]. In pa icula , ex ap-
ola ing om he sha p linea es ima e a p= 2 yields he igh sha p
linea es ima e o p > 2, bu o p < 2 i yields a wo se es ima e. Buck-
ley also shows ha a pa ame ic class o a e aging ope a o s is uni o mly
bounded on Lp(w)bykwk1/p
Ap o all 1 < p < ∞[Buc, Lemma 2.18].
In his case, s a ing om he es ima es on L (w) o any 1 < < ∞,
ex apola ion yields an es ima e ha is wo se han he sha p es ima e
o all p6= . The e o e he es ima es o Theo em 1 may no be sha p
o some ope a o s e en when he ini ial es ima e is sha p. Howe e , he
Ex apola ion and Sha p Weigh ed Es ima es 77
heo em i sel is sha p, as we will show ha o a a ie y o classical ope -
a o s ha ha e a sha p linea no m es ima e in L2(w), he ex apola ed
bounds a e also sha p o all 1 < p < ∞.
Buckley [Buc] also showed ha he Hilbe ans o m —and o ha
ma e con olu ion singula in eg al ope a o s wi h Calde ´on-Zygmund
ke nels— a e bounded on Lp(w) wi h an ope a o no m which is a
mos a mul iple o kwkα
Ap, whe e max{1, p0/p} ≤ α≤p0. In pa icula ,
o p= 2 he showed ha he dependence on kwkA2was a leas linea ,
and a mos quad a ic.
Recen ly he e has been enewed in e es in compu ing he exac de-
pendence o he ope a o no ms on he Apcha ac e is ic cons an o he
weigh . Sha p linea dependence on kwkA2was ob ained by Huko ic,
T eil, and Volbe g [Huk], [HukTV] o he dyadic squa e unc ion
on L2(w) and o he ma ingale ans o m, a dyadic model o sin-
gula in eg al ope a o s, by Wi we [W1], [W2]. As we al eady men-
ioned, analogous esul s we e ecen ly ob ained o he Beu ling ans-
o m by Pe e michl and Volbe g [Pe V], and la e by D agiˇce i´c and
Volbe g [DV]. Pe e michl and Po [Pe Po ] e y elegan ly showed
ha α≤3/2 o he Hilbe ans o m. Pe e michl [Pe ] imp o ed his
es ima e o α= 1 when p≥2. The di icul y in [Pe ] was o ob ain
linea dependence o kHkL2(w)on kwkA2; ex apola ion hen ga e he
same dependence o p > 2. Using Theo em 1 we ob ain ha he no ms
o hese ope a o s on Lp(w) a e bounded by a mos a mul iple o kwkα
Ap,
α= max{1, p0/p}, o all 1 < p < ∞.
As men ioned ea lie , using powe weigh s and powe unc ions, Buck-
ley [Buc] showed ha o con olu ion ope a o s wi h Calde ´on-Zygmund
ke nels he powe is a leas max{1, p0/p}. Hence in he cases o Hilbe
and Beu ling ans o m, Theo em 1 p o ides he sha p bounds. I we
could p o e linea bounds o all con olu ion ope a o s wi h CZ-ke nels,
hen by ex apola ion we will ob ain he same sha p bounds in Lp(w) as
o he Hilbe and he Beu ling ans o m. Ob aining he linea bounds
in L2(w) can be e y di icul . Fo ins ance, i is no ye known o he
au ho s whe he he e is a bound o he i s -o de Riesz ans o ms
on L2(w) depending linea ly on kwkA2.
We will show ha he bounds ob ained by ex apola ion o he ma -
ingale ans o m a e also sha p o all 1 < p < ∞. We can show ha
he ex apola ed bounds o he squa e unc ion a e sha p o p < 2. I
is no clea ye ha he linea bound ob ained by ex apola ion o p > 2
is sha p, so a we can show ha i mus be a leas o he o de kwkp0/p
Ap.
We can summa ize all hese esul s in he ollowing heo em.

78 O. D agiˇ
ce i´
c e al.
Theo em 2. Le Tbe any o he Hilbe ans o m, he Beu ling ans-
o m, he ma ingale ans o m, o he dyadic squa e unc ion. Then
o any 1< p < ∞ he e exis posi i e cons an s Cpsuch ha o all
weigh s win Apwe ha e
(4) kTkLp(w)≤Cpkwkα
Ap,
whe e α= max{1, p0/p}. The exponen αin his es ima e is sha p o
he Hilbe , Beu ling and ma ingale ans o ms o all 1<p<∞. Fo
he dyadic squa e unc ion he exponen is sha p o 1< p ≤2.
All esul s es ablishing he linea bounds o he abo e ope a o s
on L2(w) ha e been ob ained using he echnique o Bellman unc ions
in oduced by Naza o , T eil and Volbe g [NTV] in he ha monic anal-
ysis con ex ; see [NT] o an ex ensi e in oduc ion o his echnique.
The linea uppe bound in Theo em 2 o p > 2 was p e iously known
o he ma ingale, Hilbe and Beu ling ans o ms.
Un o una ely, ex apola ion does no p ese e he na u e o he ini-
ial es ima e on L (w) o all 1 < p < ∞, only o p > . The e o e sha p-
ness a he gi en does no au oma ically ans e o all o he p∈(1,∞).
One has o check sha pness by o he means o each p6= . In all he
examples discussed we sea ch o a unc ion and a weigh (o a amily o
unc ions and weigh s) ha will p o ide a lowe bound es ima e o he
same o de o he uppe bound, he e o e showing ha he es ima e is
indeed sha p.
Acknowledgemen . The au ho s would like o hank he e e ee o
some e y use ul sugges ions ha imp o ed he p esen a ion.
2. Some Lemma a
The i s wo lemma a below co espond o IV.5.16 and IV.5.17
in [GC-RF]. The case = 2 and p > 2 was ca e ully calcula ed
in [Pe V].
Lemma 1. Take p, s > 1,w∈Apand u∈Ls(w). Le
(5) S(u) = w−1M(|u|s/p0w)p0/s .
(a) Then Sis bounded in Ls(w), mo eo e ,
kSkLs(w)≤C(p0)p0/skwkp0/s
Ap.
(b) Le p,sbe such ha := p/s0∈[1,∞). Take a nonnega i e
unc ion u∈Ls(w).
Ex apola ion and Sha p Weigh ed Es ima es 79
I > 1, hen he pai (uw, S(u)w)belongs o he class A .
Fu he mo e,
sup
Q1
|Q|ZQ
uw 1
|Q|ZQ
(S(u)w)−1
−1 −1
≤ kwk1−p0
s
Ap.
I = 1, he A1condi ion on he pai (uw, S(u)w)also holds and
ansla es in o
M(uw)≤S(u)w.
P oo : (a) Es ima ing di ec ly he no m we ob ain
kSukLs(w)=Zhw−1M(|u|s/p0w)ip0
w1
s
=ZhM(|u|s/p0w)ip0
w1−p01
s
≤ kMkp0/s
Lp0(w1−p0)k|u|s/p0
wkp0/s
Lp0(w1−p0)=kMkp0/s
Lp0(w1−p0)
kukLs(w).
I only emains o inse Buckley’s sha p (3) es ima e and o ecall ha
w∈Apimplies w1−p0∈Ap0, mo eo e
(6) kw1−p0kAp0=kwk
1
p−1
Ap=kwkp0/p
Ap.
All oge he , hese ac s imply
kMkLp0(w1−p0)≤C(p0)kw1−p0k(p0)0/p0
Ap0=C(p0)(kwkp0/p
Ap)p/p0=C(p0)kwkAp.
Thus, kSkLs(w)≤C(p0)p0/skwkp0/s
Ap, as claimed.
(b) I s=p0, we ha e = 1, hen S(u)w=M(uw). Au oma ically
he wo-weigh A1condi ion, M(uw)≤S(u)w, holds.
I s > p0>1, hen p > s0>1 and > 1. No e ha ( −1) =
(p−1) 1−p0
sand, by de ini ion o he maximal unc ion,
Dus/p0wEQ≤sup
x∈Q
M(us/p0w)(x).
80 O. D agiˇ
ce i´
c e al.
He e h iQdeno es he mean o he unc ion o e he cube Q. Con-
sequen ly,
huwiQD[S(u)w]
−1
−1E −1
Q=huwiQD[(w−1M(us/p0w))p0/sw]
−1
−1E −1
Q
=Du wp0/sw1−p0/sEQ
D[M(us/p0w)]p0
s
−1
−1w
−1
p−1E −1
Q
≤Dus/p0
wEp0/s
Qhwi1−p0
s
QDus/p0
wE−p0
s
QDw
−1
p−1E(p−1)
“1−p0
s”
Q
=hwiQDw
−1
p−1Ep−1
Q1−p0
s
≤kwk1−p0
s
Ap.
Taking sup emum on he le -hand-side, o e all cubes Qwi h sides
pa allel o he axis, we ob ain he desi ed inequali y.
Lemma 2. Le p,s, and wbe as in he p e ious lemma. Then o
each u≥0,u∈Ls(w), he e exis s ∈Ls(w)such ha
(a) u(x)≤ (x)a.e. and k kLs(w)≤2kukLs(w).
(b) w ∈A , mo eo e , k wkA ≤2C(p0)p0/skwkAp.
P oo : De ine ia he ollowing con e gen Neumann se ies:
=
∞
X
n=0
Sn(u)
2nkSkn=u+S(u)
2kSk+···,
whe e kSk=kSkLs(w). Then (a) is clea ly sa is ied.
(b) I ollows om he de ini ion o and he sublinea i y o S ha
S ≤2kSk( −u)≤2kSk .
Suppose > 1. By he p e ious lemma, he pai ( w, S( )w) lies in A
wi h i s A -cons an bounded by kwk1−p0
s
Ap. Also ecall ha kSk ≤
C(p0)p0/skwkp0/s
Ap. We can now es ima e k wkA :
h wiQD( w)
−1
−1E −1
Q≤ h wiQD(S( )w)
−1
−1E −1
Q2kSk
≤ kwk1−p0
s
Ap2C(p0)p0/skwkp0/s
Ap= 2C(p0)p0/skwkAp.
Ex apola ion and Sha p Weigh ed Es ima es 81
Taking sup emum on he le hand side, o e all cubes Qwi h sides
pa allel o he axis, we ob ain he desi ed es ima e o k wkA , > 1.
When = 1, hen s=p0and kSk ≤ C(p0)kwkAp, u he mo e
M( w)≤S( )w≤2kSk w ≤2C(p0)kwkAp w.
We conclude ha k wkA1≤2C(p0)kwkAp, as claimed.
The nex lemma appea s as IV.5.18 in [GC-RF]; see also Lemma 9.5.4
in [G ] o a sligh ly di e en me hod o pa (b) which yields he
same bounds as he e. A en ion was paid o he cons an s in [G ] bu
Buckley’s sha p es ima e o kMkLp(w)was missing; wi h his addi ional
in o ma ion, he cons an s in [G ] would be o he same o de as he
ones ob ained he e.
Lemma 3. Fix sa is ying 1≤ < ∞.
(a) Le 1≤ < p < ∞and s= (p/ )0. Le w∈Ap, hen o e e y
u≥0,u∈Ls(w), he e exis s ≥0, ∈Ls(w), such ha
u(x)≤ (x)and k kLs(w)≤2kukLs(w).
Mo eo e , w ∈A and k wkA ≤2C(p0)p−
p−1kwkAp.
(b) Le 1< p < and s=p
−p. Le w∈Ap, hen o e e y u≥0,
u∈Ls(w), he e exis s ≥0, ∈Ls(w)such ha , u(x)≤ (x),
and k kLs(w)≤2 −1kukLs(w).
Mo eo e , −1w∈A and k −1wkA ≤2 −1(C(p) −pkwkAp) −1
p−1.
He e C(p)deno es he cons an in (3).
P oo : (a) Clea ly ≥1 implies s0≤p, and we can now use Lemma 2
a e obse ing ha p0
s=p−
p−1.
(b) Take p, and sas in he o mula ion o he lemma. (No ice
ha e e y hing ha is being said s ill holds i 0 <s<1.) Now he
dual exponen s sa is y he opposi e inequali y, 0< p0, and i we de ine
s∗:= (p0/ 0)0>1, hen s∗=s( −1).
We apply he p e ious case wi h p0, 0and w1−p0∈Ap0ins ead o p,
and w∈Ap, espec i ely. I u≥0, u∈Ls(w), hen u0=us/s∗wp0/s∗∈
Ls∗(w1−p0) and by (a) he e exis s 0∈Ls∗(w1−p0) such ha
u0≤ 0a.e.,k 0kLs∗(w1−p0)≤2ku0kLs∗(w1−p0),and
0w1−p0∈A 0,k 0w1−p0kA 0≤2C(p)p0− 0
p0−1kw1−p0kAp0
= 2C(p) −p
p−1kwk
1
p−1
Ap.
88 O. D agiˇ
ce i´
c e al.
So a his was ue o any σ. Now choose σIk= (−1)k. We ge
kTσ kp
Lp(w)≥1
δp
∞
X
n=1 
(−1)n+1
2n+1 −X
k>n+1
(−1)k
2k
p
2n(p+δ−pδ)
=2p
(3δ)pX
n≥1
2−n(p−1)δ=2p
(3δ)p(2(p−1)δ−1)
∼1
δp+1(p−1) ∼C0(p)kwkp0
Apk kp
Lp(w).
Taking p- h oo s we conclude ha
sup
σkTσkLp(w)≥C00(p)kwkp0/p
Ap.
Thus ϕ(x) = xp0/p is sha p o p < 2, and by duali y ϕ(x) = xis sha p
o p > 2.
4.3. The dyadic pa ap oduc .
A locally in eg able unc ion bis said o be in dyadic BMOd, i he
a e age oscilla ion o bis uni o mly bounded on dyadic in e als. Mo e
p ecisely, i
kbkBMOd= sup
J∈D
1
|J|ZJ|b(x)−hbiI|dx < ∞.
Fo each unc ion b∈BMOd he dyadic pa ap oduc πbis de ined by
πb (x) = X
I∈D h iIhb, hIihI(x).
I is known ha he dyadic pa ap oduc is bounded in Lp(w) whene e
w∈Ap; see [KPe ]. The ollowing quad a ic es ima e can be shown o
hold [Pe Pe ]
kπb kL2(w)≤K(kbkBMO )kwk2
A2k kL2(w).
(We do no hink his is sha p, we belie e he sha p es ima e should be
linea as o all o he ope a o s s udied in his pape .) Theo em 1 hen
gi es he uppe bound
kπb kLp(w)≤Kp(kbkBMO )kwk2α
Apk kLp(w),
whe e α= max{1, p0/p}.

Ex apola ion and Sha p Weigh ed Es ima es 89
Re e ences
[A] K. As ala, A ea dis o ion o quasicon o mal mappings, Ac a
Ma h. 173(1) (1994), 37–60.
[AIS] K. As ala, T. Iwaniec and E. Saksman, Bel ami ope a-
o s in he plane, Duke Ma h. J. 107(1) (2001), 27–56.
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Ex apola ion and Sha p Weigh ed Es ima es 91
Oli e D agiˇce i´c:
Scuola No male Supe io e
Piazza dei Ca alie i 7
56126 Pisa
I aly
E-mail add ess:[email p o ec ed]
Loukas G a akos:
Depa men o Ma hema ics
Uni e si y o Missou i
Columbia, MO 65211
USA
E-mail add ess:[email p o ec ed]
Ma ´ıa C is ina Pe ey a:
Depa men o Ma hema ics and S a isi ics
Uni e si y o New Mexico
Albuque que, NM 87131
USA
E-mail add ess:[email p o ec ed]
S e anie Pe e michl:
Depa men o Ma hema ics
B own Uni e si y
Box 1917
P o idence, RI 02912
USA
E-mail add ess:[email p o ec ed]
P ime a e si´o ebuda el 16 de desemb e de 2003,
da e a e si´o ebuda el 26 d’ab il de 2004.