T ans. Ame . Ma h. Soc. 352 (2000), 3265–3288.
Sha p weigh ed inequali ies o he ec o – alued maximal
unc ion
Ca los P´e ez
Depa men o de Ma em´a icas
Uni e sidad Au ´onoma de Mad id
28049 Mad id, Spain
e–mail: ca los.p[email p o ec ed]
wo k pa ially suppo ed by DGICYT g an PB940192, Spain
1
1 Mo i a ion and desc ip ion o he main esul s
The pu pose o his pape is o ob ain some sha p weigh ed inequali ies o he ec o – alued
maximal unc ion Mqwhich a e no wi hin he scope o he s anda d Ap heo y o ec o – alued
singula in eg als as can be ound in [RRT]. We s a wi h a e iew o some o he classical es ima es
and hen we shall s a e he main esul s.
1.1 Backg ound
Le Mbe he Ha dy–Li lewood maximal unc ion and le Mqbe he ec o – alued maximal
ope a o de ined by
Mq (x) = ∞
X
i=1
(M i(x))q!1/q
.
This nonlinea ope a o was in oduced by C. Fe e man and E. M. S ein in [FS] as a gene aliza ion
o bo h he (scala ) maximal unc ion Mand he classical in eg al o Ma cinkiewicz and since hen
i has played an impo an ole in he de elopmen o mode n Ha monic Analysis.
We ecall he wo basic es ima es ob ained in [FS] o 1 < q < ∞:
•Le 1 < p < ∞, hen he e exis s a cons an Csuch ha
ZRnMq (x)pdx ≤CZRn| (x)|p
qdx. (1)
•The ollowing weak ype (1,1) es ima e holds: he e exis s a cons an csuch ha
sup
λ>0
λ{x∈Rn:Mq (x)> λ}≤CZRn| (x)|qdx. (2)
We a e using he e he no a ion | (x)|q= (P∞
i=1 | i(x)|q)1/q =k (x)k`q.
Ano he undamen al gene aliza ion o he maximal heo em is due o B. Muckenhoup [M]
who ga e a cha ac e iza ion o ollowing “weigh ed no m inequali y”
ZRnM (x)pw(x)dx ≤cZRn| (x)|pw(x)dx, (3)
in e ms o he Apcondi ion o Muckenhoup : he e exis s a posi i e cons an csuch ha o all
cubes Q
Ap
1
|Q|ZQ
w(y)dy 1
|Q|ZQ
w(y)1−p0dyp−1
≤c. (4)
I is also well known ha he Apcondi ion (4) also cha ac e izes all he weigh s w o which
he weigh ed ec o – alued inequali y holds
ZRnMq (x)pw(x)dx ≤CZRn| (x)|p
qw(x)dx. (5)
This esul is due o K. F. Ande sen and R. T. John [AJ], and o V. Kokilash ili [K]. The e a e
by now h ee ways o p o ing (5):
2
•Re ining he a gumen o Fe e man and E. M. S ein in [FS] as done in [AJ] and [K].
•Looking a Mqas a ec o – alued singula in eg al wi h ope a o – alued ke nel sa is ying a
poin wise g adien condi ion as can be ound in [RRT].
•By applying he ex apola ion heo y o J. Ga cia-Cue a and J. L. Rubio de F ancia as
men ioned in [GCRdF] p. 521 which yields a simple p oo .
In his pape we in es iga e he wo weigh p oblem o he ec o - alued maximal unc ion Mq
ZRnMq (x)pw(x)dx ≤CZRn| (x)|p
q (x)dx, (6)
o which none o he abo e app oaches wo ks.
Recall ha he e is a cha ac e iza ion due o E. Sawye [S] o he wo weigh p oblem in he
scala si ua ion, he ollowing weigh ed inequali y:
ZRnM (x)pw(x)dx ≤CZRn| (x)|p (x)dx, (7)
holds i and only i he e exis s a cons an csuch ha o all cubes Q
SpZQ
M( 1−p0χQ)(y)pw(y)dy ≤cZQ
(y)1−p0dy. (8)
The ange p≤q o (6) is easy o handle since i coincides wi h he scala si ua ion. Indeed, i
p≤qwe claim ha he Spcondi ion is necessa y and su icien o (6). I is clea ha condi ion
(8) is necessa y. I we assume (8) we see ha (6) is immedia e o bo h q=pand q=∞. Then
he case 1 <p<q<∞ ollows by in e pola ion o “linea izable ope a o s” in he ec o – alued
con ex (c . he a gumen gi en in [GCRdF] p. 482).
Al hough we shall gi e a ull cha ac e iza ion o (6) in Theo em 2.3 we a e mo e in e es ed in
es ima es o he o m
ZRnMq (x)pw(x)dx ≤CZRn| (x)|p
qNw(x)dx, (9)
whe e Nis an app opia e (scala ) maximal ype ope a o . Needless o say ha he p o o ypical
es ima e ha we ha e in mind is he Fe e man–S ein weigh ed inequali y
ZRnM (x)pw(x)dx ≤CZRn| (x)|pMw(x)dx, (10)
which yields, as i is well known, he unweigh ed ec o – alued es ima e (1) when p>q.
One o he main obse a ions ha ollows om ou esul s is ha Mqdoes no e i y a simila
inequali y o (10) on he ange p > q (c . he i s ema k a e Theo em 1.1).
Inequali ies o he ype (10) e lec how singula is he ope a o unde s udy. This can be seen
o ins ance wi h he ollowing sha p inequali ies o singula in eg als ob ained in [P2] o p > 1
gene alizing some p e ious es ima es ob ained by M. Wilson in he ange 1 < p ≤2 [Wil2]:
Le Tbe any Calde ´on–Zygmund ope a o , and le 1 <p<∞. Then he e exis s a
cons an Csuch ha
3
ZRn|T (x)|pw(x)dx ≤CZRn| (x)|pM[p]+1w(x)dx, (11)
wi h Cindependen o wand . Fu he mo e, he es ima e is sha p since i does no
hold o M[p]. He e Mk=M◦(k)
. . . ◦M k = 1,2,· · ·, deno es he Ha dy–Li lewood
maximal ope a o Mi e a ed k imes.
Ano he example which s esses ou poin o iew is ela ed o he classical A ea unc ion. This
non linea ope a o is de ined by he in eg al
Sϕ( )(x) = ZB (x)
| ∗ϕ (y)|2d dy
n+1 !1/2
,
whe e ϕ∈C∞
0wi h Rϕ= 0 and ϕ (x) = −nϕ(x
), > 0. Then he A ea unc ion sa is ies he
ollowing inequali y:
Le 1 < p ≤2, hen he e exis s a cons an Csuch ha
ZRnSϕ( )(x)pw(x)dx ≤CZRn| (x)|pMw(x)dx. (12)
Fu he mo e, his inequali y is alse o p > 2.
The case p= 2 was i s ob ained by A. Chang, M. Wilson, and T. Wol in [CWW] and o
1<p<2 by S. Chanillo and R. Wheeden in [CW] as well as he coun e example o p > 2. See
also he wo k by M. Wilson [Wil1]–[Wil4].
1.2 Main esul s
Mo i a ed by he Theo ems men ioned abo e we s a e now he main esul o he pape .
THEOREM 1.1 Le 1< q < p < ∞.
a)The e exis s a cons an Csuch ha
ZRnMq (x)pw(x)dx ≤CZRn| (x)|p
qM[p
q]+1w(x)dx, (13)
o all locally in eg able unc ions ,w≥0.
b) Pa a) is sha p since he e exis s no cons an Csuch ha
ZRnMq (x)pw(x)dx ≤CZRn| (x)|p
qM[p
q]w(x)dx, (14)
o all locally in eg able unc ions ,w≥0. Likewise, he co esponding weak ype (p, p)es ima e
is alse.
We now make he ollowing ema ks.
(a) I ollows om pa b) o he Theo em ha he ec o - alued analogue o he Fe e man-S ein
inequali y (10)
4
ZRnMq (x)pw(x)dx ≤CZRn| (x)|p
qMw(x)dx. (15)
is alse in gene al in he ange p > q.
(b) I we look a he p oo o (13) we see ha we can e ine such an inequali y by eplacing
M[p
q]+1 by ML(log L)
p
q−1+, > 0, o by MAwhe e Asa is ies
Z∞
c
A( )(p
q)0−1d
<∞.
See Sec ion 3 o he app opia e de ini ion o he maximal ype unc ion MA.
(c) We emphasize on he ac ha he e is no assump ion on wo he han local in eg abil y.
In ac , i we assume ha w∈A∞ hen (15) holds being alse in gene al. Indeed, by he Lebesgue
di e en a ion Theo em we ha e
ZRnMq (x)pw(x)dx ≤ZRnMq (x)pMw(x)dx ≤CZRn| (x)|p
qMw(x)dx, (16)
whe e in he las inequali y we ha e used he Ap esul o Mq(5) since Mw ∈A1by s anda d
esul s (see he las pa o Sec ion 4).
Also, we may eplace M[p
q]+1wby he A1weigh M(w )(x)1/ , > 1, by applying again (5).
Howe e , he la e class o weigh s (essen ially he class A1) a e poin wise la ge han he non A∞
weigh s Mkwsince i may be shown using s anda d heo y ha
Fo each in ege k= 1,2,· · ·, each > 1 and each locally in eg able unc ion , we ha e
he ollowing poin wise inequali y o all x∈Rn:
w(x)≤Mkw(x)≤[M(w )1/ ]k−1
A1M(w )(x)1/ .
He e [w]A1deno es he “no m” o w∈A1, namely he smalles cons an Csuch ha
Mw ≤C w.
(d) Theo em 1.1 indica es ha Mqbeha es mo e as a singula in eg al ope a o a he han
as a maximal ope a o . Howe e , we wan o emphasize he ac ha inequali y (13) does no i
wi hin he scope o he heo y o ec o – alued singula in eg als as de eloped by J. L. Rubio de
F ancia, F. J. Ruiz and J. L. To ea in [RRT] whe e he pionee ing wo k [BCP] was upda ed. In
[RRT], he ope a o Mq, as well as many o he non–linea ope a o s such as he A ea unc ion
Sϕ, a e seen as singula in eg als aking alues in an app opia e Banach space. Using his poin
o iew, i is possible o ansla e o his mo e gene al con ex he one weigh scala Ap heo y a
leas o any ec o – alued singula in eg al wi h su icien ly smoo h ke nel. Howe e , his is no
he case o (13) (no o (12)) since he esul o he (scala ) Hilbe ans o m (11) is wo se han
(13) indica ing ha he ope a o Mqis less singula han H.
The p oo o he posi i e pa o Theo em 1.1 does no ollow he scheme used in [P2] o
ea singula in eg als since we canno dualize (13). We shall de i e (13) as a consequence o a
5

cha ac e iza ion o he wo weigh p oblem gi en in Theo em 2.3. The condi ion we ob ain is a
blend o Sawye ’s condi ion Sp oge he wi h Rubio de F ancia’s cha ac e iza ion o ec o – alued
inequali ies o sublinea ope a o s as can be ound in [GCRdF] Chap e VI.
I should be men ioned ha Y. Rako ond a simba has ob ained in [R] a di e en cha ac e iza-
ion o he wo weigh p oblem which is much close in spi i o Sawye ’s condi ion Sp.
1.3 Sha p su icien condi ions close o Ap
In his sec ion we ake up he wo weigh p oblem o Mq ha we w i e in he ollowing mo e
con enien o m
ZRn(w(x)Mq (x))pdx ≤CZRn( (x)| (x)|q)pdx. (17)
The ask is o p o ide sha p su icien condi ions on he weigh s “close” in s uc u e o he Ap
condi ion.
Le us b ie ly e iew some esul s ela ed o he scala si ua ion
ZRn(w(x)M (x))pdx ≤cZRn( (x)| (x)|)pdx. (18)
I is well known ha he necessa y Apcondi ion o his p oblem
1
|Q|ZQ
w(x)pdx1/p 1
|Q|ZQ
(x)−p0dx1/p0
≤c(19)
is no su icien , and ha he co ec necessa y and su icien condi ion is, as we men ioned abo e,
Sawye ’s condi ion which wi h ou no maliza ion on he weigh s has he ollowing o m
ZQ
(w(y)M( −p0χQ)(y))pdy ≤CZQ
(y)−p0dy.
The d awback o his condi ion is ha i in ol es he ope a o Mi sel , and i would be in e es ing
o ob ain su icien condi ions close in o m o he Apcondi ion (19). Pe haps, he i s esul in
ha di ec ion was ob ained by C. Neugebaue in [N]. He no iced ha i (w, ) is a couple o weigh s
such ha o some > 1
1
|Q|ZQ
w(y)p dy1/p 1
|Q|ZQ
(y)−p0 dy1/p0
≤c(20)
o all cubes Q, hen
ZRn(w(y)M (y))pdy ≤cZRn( (y)| (y)|)pdy. (21)
In ac Neugebaue p o es ha (20) is equi alen o showing ha he e is an Apweigh inse ed
(poin wise) be ween wpand pand he esul ollows i ially. This p oblem has been conside ed in
[P1] whe e i is shown ha such a s ong condi ion is no needed. In pa icula i is no necessa y
o “bump” he le weigh wand ha much less han a powe “bump” is equi ed on he igh
weigh o ge he esul . We ex ac he ollowing esul om [P1]. Recall ha o a gi en Young
unc ion Aand a cube Qon Rnwe de ined he A-a e age o a unc ion o e Qby
k kA,Q = in {λ > 0 : 1
|Q|ZQ
A| (y)|
λdy ≤1}.
6
THEOREM 1.2 [P1] Le 1<p<∞, and le Bbe a doubling Young unc ion such ha
Z∞
c p0
B( )!p−1d
<∞,(22)
o some posi i e cons an c. Le (w, )be a couple o weigh s such ha he e is a posi i e cons an
K o which
1
|Q|ZQ
w(y)pdy1/p

 −1
B,Q ≤K, (23)
o all cubes Q. Then ZRn(w(y)M (y))pdy ≤cZRn( (y) (y))pdy (24)
o all nonnega i e unc ions .
As we may expec we need o conside s onge condi ions on he weigh s o ge co esponding
esul s o Mqin he ange p > q. In pa icula we need o “bump” he le weigh was well since
o he wise he esul is alse as he coun e example (w, Mw) in (15) shows. Indeed, obse e ha
his pai o weigh s sa s ies (23) o any cube Qand any Young unc ion B:
1
|Q|ZQ
w1/p 

(Mw)−1/p

B,Q
≤1
|Q|ZQ
w1/p 



1
|Q|RQw−1/p



B,Q
=1
|Q|ZQ
w1/p 1
|Q|ZQ
w−1/p
k1kB,Q = 1
since 1
|Q|RQw≤Mw(x) o x∈Q.
THEOREM 1.3 Le 1<q<p<∞, and le =p
q. Le A, B be doubling Young unc ions such
ha bo h
Z∞
c
A( ) 0−1d
and Z∞
c q0
B( )!q−1d
,(25)
a e ini e o some posi i e cons an c, ha is ¯
A∈B 0and ¯
B∈Bq. Le (w, )be a couple o weigh s
such ha he e is a posi i e cons an K o which
kwqk1/q
A,Q 
 −1
B,Q ≤K, (26)
o all cubes Q. Then he wo weigh ed ec o – alued inequali y


(P∞
i=0(w M i)q)1/q

Lp(Rn)≤C

(P∞
i=0 | i|q)1/q

Lp(Rn)(27)
holds o all i.
Some in e es ing examples a e gi en by A( )≈ (log ) −1+δand B( )≈ q0(log )q0−1+δwi h
δ > 0.
7
1.4 Endpoin es ima es
Al hough he ope a o Mqis, o some ex en , mo e closely ela ed o a singula in eg al his is no
he case when we look a endpoin es ima es such as he ollowing.
THEOREM 1.4 The e exis s a cons an Csuch ha o each weigh wand o all λ > 0
w({x∈Rn:Mq (x)> λ})≤C
λZRn| (x)|qMw(x)dx. (28)
This esul e lec s once again ha sha p esul s o Mqa e independen om he heo y o
ec o – alued singula in eg als since we do no know whe he he (scala ) Hilbe ans o m H
sa is ies
w({x∈Rn:|H (x)|> λ})≤C
λZRn| (x)|Mw(x)dx. (29)
See [P2] o sha p esul s.
The e exis s an in e es ing ela ionship be ween (28) and a possible ec o – alued e sion o
he classical Besico i ch lemma. We shall o mula e his as a conjec u e. Mc
wdeno es he weigh ed
cen e ed maximal unc ion.
CONJECTURE 1.5
w({x∈Rn: ∞
X
i=1
(Mc
w i(x))q!1/q
> λ})≤C
λZRn| (x)|qw(x)dx. (30)
One can show (c . Sec ion 6) ha i he conjec u e we e ue hen he inequali y (28) ollows
immedia ely.
Acknowledgemen s. The au ho is e y g a e ul o A. Va gas o se e al con e sa ions con-
ce ning he p oblems conside ed in his pape .
2 A cha ac e iza ion o he wo weigh p oblem
The pu pose o his sec ion is o gi e a cha ac e iza ion o he wo weigh p oblem o he ec o –
alued maximal unc ion Mq. We ecall ha he case 1 < p ≤qis cha ac e ized by means o
Sawye ’s condi ion Sp. The main esul is Theo em 2.3. Fo he p oo o his Theo em i will be
mo e e icien o wo k wi hin a mo e gene al con ex . Le Bbe a basis in Rn, and by his we
mean a collec ion o open se s in Rn. We say ha wis a weigh associa ed o he basis Bi wis a
non-nega i e measu able unc ion in Rnsuch ha w(B) = RBw(y)dy < ∞ o each Bin B.MB,w
is he co esponding maximal ope a o de ined by
MB,w (x) = sup
x∈B
1
w(B)ZB
| (y)|w(y)dy
i x∈ ∪B∈B and MB,w (x) = 0 o he wise. I w≡1, we jus w i e MB (x).
8
PROPOSITION 2.1 Le 1< q < p < ∞, and le =p
q. Suppose ha
MB,σ :Lp
`q(σ)→Lp
`q(σ)whe e σ= 1−p0.
Then he wo weigh ec o alued inequali y


(P∞
i=0(MB i)q)1/q

Lp(w)≤C

(P∞
i=0 | i|q)1/q

Lp( )(31)
holds i and only i he e exis s a cons an csuch ha o each g∈L 0(Rn)we can ind G∈L 0(Rn)
wi h kGkL 0(Rn)≤ kgkL 0(Rn)such ha
ZΩ
MB(σχΩ)(x)qw(x)1/ g(x)dx ≤cZΩ
σ(x)1/ G(x)dx, (32)
o e e y se Ωwhich is a union o se s in B.
P oo : We i s show ha condi ion (32) is necessa y. Fi s obse e ha inequali y (31) is
equi alen o



P∞
i=0(w1/p MB( i
1/p ))q1/q


Lp(Rn)
≤c

(P∞
i=0 | i|q)1/q

Lp(Rn).
Now, by Rubio de F ancia’s heo em (c . [GCRdF] p. 555) his es ima e is equi alen o showing
ha o each g∈L 0(Rn) he e exis s G∈L 0(Rn) wi h kGkL 0(Rn)≤ kgkL 0(Rn)and
ZRn(w(y)1/pMB(
1/p )(y))qg(y)dy ≤CZRn| (y)|qG(y)dy,
o all , o wha is he same
ZRnMB( )(y)qw(y)1/ g(y)dy ≤CZRn| (y)|q (y)1/ G(y)dy,
o all . Tes ing his inequali y wi h =σχΩ= 1−p0χΩgi es he necessa y condi ion (32).
To p o e he su iciency o (32) we use ha L and L 0a e dual spaces. We adap he basic
ideas om [GCRdF]. I we de ine Ias
I=


P∞
i=0(w1/p MB i)q1/q



q
Lp(Rn)
,
hen
I=
∞
X
i=0 ZRnMB( i)(y)qw(y)1/ g(y)dy
o some g∈L 0(Rn) wi h uni no m. Fix i, and o each in ege kconside he se Ei
k={y∈
Rn: 2k< MB i(y)≤2k+1}. F om he de ini ion o MB,Ei
k⊂ ∪jBi
k,j, whe e Bi
k,j ∈ B sa is ies
2k<1
Bi
k,jZBi
k,j
i(y)dy.
De ine now Ei
k,1=Bi
k,1∩Ei
k,and o j > 1Ei
k,j =Bi
k,j ∪s<j Bi
k,s∩Ei
k. Fo any ixed k, each
o he se s Ei
kis he disjoin union o he se s Ei
k,j. We now can w i e
9
We also pos pone he p oo o his lemma un il he end o he p oo o he heo em.
Now, using (49) and (51) we can es ima e he le side o (43) as ollows
ZRnMB (y)pw(y)dy =X
kZΩk−Ωk+1
MB (y)pw(y)dy (52)
≤apX
k
akpw(Ωk)≤CX
k,j
akpw(3Qk,j)≤
≤CX
k,j
k kp
B,Qk,j w(3Qk,j) = CX
k,j
k kp
B,Qk,j
w(3Qk,j)
|3Qk,j||Qk,j|
≤CX
k,j 




w(3Qk,j )
|3Qk,j|1/p




p
B,Qk,j
|Ek,j|
≤CX
k,j ZEk,j
MB( (Mw)1/p)(y)pdy ≤CZRnMB( (Mw)1/p)(y)pdy
≤CZRn (y)pMw(y)dy,
since we a e assuming ii). This p o es iii).
Le us assume ha iii) holds. Obse e ha (44) is equi alen wi h
ZRnM( g)(y)pw(y)
[M¯
B(g)(y)]pdy ≤cZRn (y)pMw(y)dy,
o all nonnega i e unc ions ,g, and w. Then i ) ollows immedia ely om (43) a e an appli-
ca ion o he inequali y
M( g)(y)≤MB (y)M¯
Bg(y)y∈Rn
which is a consequence o he gene alized H¨olde ’s inequali y (41).
To p o e ha i ) implies i) we le w= 1 in (44) ob aining
ZRnM (y)p1
[M¯
B(u1/p)(y)]pdy ≤cZRn (y)p1
u(y)dy,
o all nonnega i e unc ions , and u. Tes ing his inequali y wi h =u=χQ(0,1) , whe e Q(x, )
deno es he cube cen e ed a x∈Rnand wi h sideleng h equal o , we ha e
ZRnM (y)p1
[M¯
B( )(y)]pdy ≤C. (53)
On he o he hand we ha e o la ge x ha
M¯
B( )(x)≈1
¯
B−1(1
|x|n).
The e o e we ge
16

ZRnM (y)p1
[M¯
B( )(y)]pdy ≥CZ|y|>c
1
|y|np
1
¯
B−1(1
|y|n)pdy
=CZ∞
c
1
np
1
¯
B−1(1
n)p nd
≈Z∞
c
B( )
p
d
.
This es ima e combined wi h (53) shows ha i ) ⇒i).
To conclude he p oo o he Theo em, apa om he p oo s o Lemmas 3.3 and 3.4, we need
o show ha i) ⇔ ). Tha i) is necessa y is i ial since ) implies he scala case, namely ii). To
show ha i) is su icien obse e ha he case p<s ollows by in e pola ion om he cases s=p,
s=∞. Now he case p > s ollows om he weigh ed inequali y iii) by s anda d a gumen s. 2
P oo o Lemma 3.3: The p oo is a simple adap a ion o a gumen s in [GCRdF] Ch. 2.
Since is bounded wi h compac suppo , say supp ⊂K,
k kB,Q ≤ k kL∞kχKkB,Q
=k kL∞
1
B−1|Q|
|Q∩K|,
and i ollows ha
k kB,Q →0
as Q↑Rn. Hence, i he e a e any dyadic cubes Qwi h k kB,Q > , hey a e con ained in cubes
o his ype which a e maximal wi h espec o inclusion. We le C ={Pj}be he amily o he
dyadic maximal nono e lapping cubes sa is ying
< k kB,Pj.
Le P0
jbe he only dyadic cube con aining Pjwi h sideleng h wice ha o Pj. Then
< k kB,Pj≤2nk kB,P 0
j
.
The las inequali y can easily be deduced om he de ini ion o he Luxembu g no m using he
ac ha →B( )
is non dec easing. Hence by he maximali y o he cubes {Pj}we ge
< k kB,Pj≤2n . (54)
Obse e ha om his discussion i is clea ha
{y∈Rn:Md
B (y)> }=∪jPj.(55)
Le x∈Ω . By de ini ion, he e is a cube Rcon aining xsuch ha
< k kB,R .(56)
17
Le kbe he unique in ege such ha 2−(k+1)n<|R| ≤ 2−kn. The e is some dyadic cube wi h side
leng h 2−k, and a mos 2no hem, {Ji:i= 1, . . . , n}, mee he in e io o R. I is easy o see
ha o one o hese cubes, say J1,
2n<

χJ1 

B,R
.(57)
This can be seen as ollows. I o each i= 1, . . . , 2nwe had


χJi 

B,R
≤
2n,
we would ge since R⊂ ∪2n
i=1Ji ha
k kB,R =



χ∪2n
i=1Ji 


B,R
≤
2n
X
i=1 

χJi 

B,R
≤2n
2n= ,
con adic ing (56). Since |R| ≤ |J1|<2n|R|one can also show
4n<k kB,J1.(58)
By le ing C /(4)n={Qj}, we ha e by (54) ha
4n<k kB,Qj≤
2n,(59)
o each j, yielding (46). Equa ion 48) also ollows since {y∈Rn:Md
B (y)>
4n}=∪jQj. Also,
we see om (58) ha J1⊂Qk, o some k, and hen R⊂3J1⊂3Qk. This gi es
Ω ⊂ ∪j3Qj,
which is (45). Now, by he le side o he inequali y (59), and he de ini ion o k kB,Q we ge
|Ω | ≤ CX
j
|Qj|
≤CX
jZQj
B4n (y)
dy ≤CZRnB (y)
dy. (60)
To ob ain (47) we jus use he s anda d idea o w i ing as = 1+ 2, whe e 1(x) = (x) i
(x)>
2, and 1(x) = 0 o he wise. Then MB (x)≤MB 1(x) + MB 2(x)≤MB 1(x) +
2. Finally,
since (60) holds o each ≥0, > 0 we ha e
|Ω | ≤ {y∈Rn:MB 1(y)>
2}≤CZRnB 1(y)
dy
=CZ{y∈Rn: (y)> /2}
B (y)
dy,
concluding he p oo o Lemma 3.3. 2
18
We now conclude he p oo o he Theo em by p o ing Lemma 3.4.
P oo o Lemma 3.4: The amily Ek,j is clea ly disjoin . We no e ha (49) and he de ini ion
o he Luxembu g no m implies ha
1<1
|Qk,j|ZQk,j
B4n
ak (y)dy,
and 1
|Qk,j|ZQk,j
B2n
ak (y)dy ≤1.
Hence by s anda d p ope ies o he dyadic cubes we can es ima e wha po ion o Qk,j is co e ed
by Dk+1 as in [GCRdF] p. 398
|Qk,j ∩Dk+1|
|Qk,j|=X
i
|Qk,j ∩Qk+1,i|
|Qk,j|=X
i:Qk+1,i⊂Qk,j
|Qk+1,i|
|Qk,j|
<X
i:Qk+1,i⊂Qk,j
1
|Qk,j|ZQk+1,i
B4n
ak+1 (y)dy
≤2n
a
1
|Qk,j|ZQk,j ∩∪iQk+1,i
B2n
ak (y)dy ≤2n
a.
He e we ha e used ha B(2n
a )≤2n
aB( ), > 0, since 2n
a<1, and because →B( )
is inc easing.
This gi es (50). Finally
|Ek,j|
|Qk,j|>1−2n
a>0,
comple ing he p oo o he Lemma and hence ha o Theo em 3.2. 2
4 P oo o he main Theo em
In his sec ion we gi e he p oo o Theo em 1.1. We s a wi h he p oo o pa a), he posi i e
pa . We apply Theo em 2.3 by e i ying condi ion (37). The weigh wis ixed and will be chosen
ina a momen . Recall ha =p
qand ha σ= 1−p0. We need o show ha he e exis s a cons an
csuch ha o a bi a y g∈L 0(Rn) he e is G∈L 0(Rn) wi h kGkL 0(Rn)≤ kgkL 0(Rn)and such
ha o all cubes Q
ZQ
M(σχQ)(x)qw(x)1/ g(x)dx ≤cZQ
σ(x)1
G(x)dx. (61)
We use he Fe e man-S ein inequali y (10) oge he wi h he gene alized H¨olde ’s inequali y (41)
o es ima e he le hand side o (61) by a mul iple o
ZQ
σ(x)qM(w1/ g)(x)dx ≤ZQ
σ(x)qMB(w1/ )(x)M¯
B(g)(x)dx
=ZQ
σ(x)q[(MB(w1/ )(x)) ]1/ M¯
B(g)(x)dx.
19
I we le = (MB(w1/ )) we ha e ha
ZQ
M(σχQ)(x)qw(x)1/ g(x)dx ≤CZQ
σ(x)1/ M¯
B(g)(x)dx.
To conlude he p oo all we ha e o do know is o choose Bsuch ha ¯
B∈B 0, namely ha
Z∞
c
B( ) 0−1d
<∞
since by he cha ac e iza ion in Theo em 3.2 M¯
B:L 0(Rn)→L 0(Rn). I we le 
M¯
B
be he
no m o his ope a ion we can ake G=M¯
B(g)

M¯
B

such ha kGkL 0(Rn)≤ kgkL 0(Rn)and we ha e
ZQ
M(σχQ)(x)qw(x)1/ g(x)dx ≤C
M¯
B(g)
ZQ
σ(x)1/ G(x)dx.
Finally we a e le wi h showing ha we can pick Bsuch ha = (MB(w1/ )) ≤M[ ]+1w.
Indeed, le B( )≈ (log(1 + ))[ ], hen B( 1/ )≈ (log(1 + )[ ]and (MB(w1/ )) =ML(log L)[ ](w).
Now, we make he ollowing obse a ion.
Le k= 1,2,3,· · ·, hen he e exis s a cons an C=Cnsuch ha o all bounded
unc ions wi h suppo con ained in Q
k kL(log L)k,Q ≤C
|Q|ZQ
Mk (y)dy.
Indeed, by homogenei y we can assume ha he igh hand side is one. Then by he de ini ion
o he Luxembu g no m i is enough o p o e
1
|Q|ZQ
(y)(1 + log+( (y)))kdy ≤C,
which is a consequence o i e a ing he ollowing well known inequali y o E.M. S ein:
ZQ
(y)(1 + log+( (y)))kdy ≤CZQ
M (y)(1 + log+(M (y)))k−1dy, (62)
wi h k= 1,2,3,· · ·.
The e o e we inally ha e ha =ML(log L)[ ](w)≤M[ ]+1wconcluding he p oo o he Theo-
em.
This concludes he p oo o he i s pa , o he coun e example we ake n= 1 and we le N
be a la ge posi i e in ege and =p
q>1. Se w=χ(0,1) and de ine
i(x) = (log x)−1/qχ(ei,ei+1)(x)
o each i= 1,· · · , N −1, and i= 0 o i≥N. Then,


(P∞
i=1 | i|q)1/q


p
Lp(M[ ]w)=ZR N−1
X
i=1
(log x)−1χ(ei,ei+1)(x)!
M[ ]w(x)dx
20
≈ZeN
e
(log x)− (log x)[ ]−1dx
x.
When is an in ege his is compa able o log N, and when is no an in ege he in eg al is a
cons an independen o Nsince [ ]− + 1 >0. In any case, i is less han a cons an imes log N.
On he o he hand


(P∞
i=1(M i)q)1/q


p
Lp(w)=Z1
0 N−1
X
i=1
(M i(x))q!
dx
≥Z1
0 N−1
X
i=1
1
i!
dx ≈(log N) ,
since o 0 < x < 1 and i= 1,· · · , N −1,
M i(x)≥1
ei+1 Zei+1
0
(log y)−1/qχ(ei,ei+1)(y)dy ≈1
i1/q .
To conclude, obse e ha (log N) ≤Clog Ndoes no make sense o la ge Nwi h Cinde-
penden o N.2
We conclude he sec ion by gi ing he ollowing simple a gumen showing ha Mw ∈A1
assuming ha w∈A∞which was used o p o e inequali y (16). Indeed since ws ais ies o some
> 1 he e e se H¨olde inequali y
1
|Q|ZQ
w 1/
≤C
|Q|ZQ
w
wi h Cindependen o he cube Q. Now o ixed Qand x∈Qwe ha e
1
|Q|ZQ
Mw ≤1
|Q|ZQ
M(wχ2Q) + 1
|Q|ZQ
M(wχRn 2Q)
≤1
|Q|ZQ
M(wχ2Q) 1/
+Cin
QM(w)≤C1
|2Q|Z2Q
w 1/
+M(w)(x)
≤C
|2Q|Z2Q
w+M(w)(x)≤C M(w)(x).
He e we ha e used ha M(χRn 2Qw)(y)≈M(χRn 2Qw)(z) o each y, z ∈Q, [GCRdF] p. 159 and
he L boundedness o M. This means ha Mw ∈A1.
5 P oo o he sha p su icien condi ions
We wan o poin ou ha we do no know how o p o e his heo em di ec ly om he cha ac e i-
za ion gi en in Theo em 2.3. We a e going o modi y and combine he p oo o his Theo em wi h
he esul s in Theo em 3.2 which in ac con ains he key es ima e.
P oo o Theo em 1.3:
21



(P∞
i=0(wM i)q)1/q


q
Lp(Rn)=
∞
X
i=0 ZRnM i(y)qw(y)qg(y)dy (63)
o some g∈L 0(Rn) wi h uni no m.
Le ibe ixed. Fo each in ege k, and o any a bi a y cons an a > 2nwe le Ωi
k, and Di
kbe
he se s
Ωi
k={x∈Rn:ak< M i(x)},
Di
k={x∈Rn:Md i(x)>ak
4n}.
By he classical Calde ´on–Zygmund decomposi ion (c . [GCRdF] p. 137) he e exis s a amily o
maximal nono e lapping dyadic cubes {Qi
k,j} o which Ωi
k⊂ ∪j3Qi
k,j,Di
k=∪jQi
k,j, and
ak
4n<1
Qi
k,jZQi
k,j
i(y)dy ≤ak
2n.(64)
We can now es ima e he in eg al in (63) as ollows
ZRnM i(y)qw(y)qg(y)dy =X
kZΩi
k−Ωi
k+1
M i(y)qw(y)qg(y)dy (65)
≤aqX
k
akq(wqg)(Ωi
k)≤CX
k,j
akq(wqg)(3Qi
k,j)
≤CX
k,j 

1
Qi
k,jZQi
k,j
i(y)dy

q
(wqg)(3Qi
k,j)
=CX
k,j 

1
Qi
k,jZQi
k,j
i(y) (y) (y)−1dy

q
(wqg)(3Qi
k,j)
≤CX
k,j 

1
3Qi
k,jZ3Qi
k,j
i(y) (y) (y)−1dy

q1
3Qi
k,jZ3Qi
k,j
w(y)qg(y)dy Qi
k,j.
Fo each in ege k, j we se Ei
k,j =Qi
k,j −Qi
k,j ∩Di
k+1. Then {Ei
k,j}is a disjoin amily o se s, and
by Lemma 3.3 wi h B( ) = , he e is a posi i e cons an βsuch ha o each k, j Qi
k,j< β Ei
k,j.
This oge he wi h he gene alized H¨olde ’s inequali y (41) allows o domina e he las sum by a
mul iple o
X
k,j
k ikq
¯
B,3Qi
k,j 
 −1

q
B,3Qi
k,j
kwqkA,3Qi
k,j
kgk¯
A,3Qi
k,j Ei
k,j
≤KqX
k,j ZEi
k,j
M¯
B( i)(y)qM¯
A(g)(y)dy ≤CZRnM¯
B( i)(y)qM¯
A(g)(y)dy.
since he se s {Ei
k,j}a e pai wise disjoin when iis ixed. Hence, by H¨olde ’s inequali y wi h
exponen s and 0we can es ima e (63) by
22


(P∞
i=0(wM i)q)1/q


q
Lp(Rn)≤C

P∞
i=0(M¯
B( i ))q1/q


q
Lp(Rn)
M¯
A(g)
L 0(Rn)
≤C

(P∞
i=0( i)q)1/q


q
Lp(Rn)kgkL 0(Rn)=

(P∞
i=0( i)q)1/q


q
Lp(Rn)
since M¯
B:Lp
`q(Rn)→Lp
`q(Rn) and M¯
A:L 0(Rn)→L 0(Rn) by Theo em 3.2.
2
6 Endpoin es ima es and he Besico i ch lemma
P oo o Theo em 1.4: I is enough o conside ≥0 in he sense ha i≥0 o all i. Le
Ω = {x∈Rn:Md(| |q)(x)> λ}=∪Q, whe e he dyadic cubes Qa e maximal nono e lapping
sa is ying
λ < 1
|Q|ZQ
| (x)|qdx ≤2nλ, (66)
and | (x)|q≤λ, a.e. x∈Rn Ω. W i e = ·χRn−Ω+ ·χΩ=g+b. Since Mq ≤Mqg+Mqb,
i is su icien o es ima e he dis ibu ion se o Mqgand Mqbsepa e ly.
Obse ing ha |g(x)|q≤λwe ge by (10)
w({x∈Rn:Mqg(x)> λ/2})≤C
λqZRnMqg(x)qw(x)dx
≤C
λqX
iZRn(Mgi)(x)qw(x)dx ≤C
λqX
iZRngi(x)qMw(x)dx
=C
λqZRn|g(x)|q
qMw(x)dx ≤C
λZRn| (x)|qMw(x)dx.
Fo bwe spli he dis ibu ion se o Mqbse as ollows. Le ˜
Ω = ∪˜
Q,˜
Q= 3Q. Then
w({x∈Rn:Mqb(x)> λ/2})≤w({x∈Rn ˜
Ω : Mqb(x)> λ/2}) + w(˜
Ω).
The second e m is es ima ed by he le hand side o (66):
w(˜
Ω) ≤C
λX
Q
w(˜
Q)1
|Q|ZQ
| (x)|qdx
≤C
λX
QZQ
| (x)|qMw(x)dx
≤C
λZRn| (x)|qMw(x)dx.
Fo he i s e m we use he a gumen in [FS] p. 110 which shows ha Mbi(x)≤M(¯
bi)(x),
x∈Rn ˜
Ω whe e ¯
biis he unc ion
23
¯
bi(x) = n0x∈Rn Ω1
|Q|RQ i(y)dy x ∈Q
w({x∈Rn ˜
Ω : Mqb(x)> λ/2} ≤ 1
λqX
iZRn ˜
Ω
(Mbi(x))qw(x)dx
≤1
λqX
iZRn ˜
Ω
(M(¯
bi)(x))qw(x)dx ≤C
λqX
iZRn
¯
bi(x)qM(wχRn ˜
Ω)(x)dx
=C
λqX
iX
Q1
|Q|ZQ
i(y)dyqZQ
M(wχRn ˜
Q)(x)dx
≤C
λqX
Q"X
i1
|Q|ZQ
i(y)dyq#1
qqZQ
M(wχRn ˜
Q)(x)dx
≤C
λqX
Q1
|Q|ZQ
| (y)|qdyq
in
QM(w)|Q| ≤ C
λX
Q
1
|Q|ZQ
| (y)|qdy in
QM(w)|Q|
≤C
λX
QZQ
| (x)|qMw(x)dx ≤C
λZRn| (x)|qMw(x)dx.
2
As we men ioned in he in oduc ion he e exis s a close connec ion be ween he weigh ed scala
inequali y
w({x∈Rn:M (x)> λ})≤C
λZRn| (x)|Mw(x)dx (67)
and he he Besico i ch co e ing lemma [dG]. Indeed, he i s obse a ion is ha (67) is equi alen
o
w({x∈Rn:M( w
Mw)(x)> λ})≤C
λZRn| (x)|w(x)dx.
The second is ha we i ially ha e he poin wise inequali y
M( w
Mw)(x)≤cnMc
w (x),
whe e Mc
wis he weigh ed cen e ed maximal unc ion
Mc
w (x) = sup
>0
1
w(B (x)) ZB (x)
| (y)|w(y)dy. (68)
The e o e (67) ollows om
w({x∈Rn:Mc
w (x)> λ})≤C
λZRn| (x)|w(x)dx
which is a consequence o he Besico i ch co e ing lemma.
24
We can epea his a gumen wi h M eplaced by he ec o – alued maximal ope a o Mq
excep o he ac ha he e is no ec o – alued analogue o he Besico i ch lemma, namely
w({x∈Rn: ∞
X
i=1
(Mc
w i(x))q!1/q
> λ})≤C
λZRn| (x)|qw(x)dx.
Combining his es ima e oge he wi h (68) would yield a di e en p oo o Theo em 1.4.
Re e ences
[AJ] K. F. Ande sen and R. T. John, Weigh ed inequali ies o ec o – alued maximal unc ions
and singula in eg als, S udia Ma h. 69 (1980), 97–101.
[BCP] A. Benedek, A. P. Calde ´on, R. Panzone, Con olu ion ope a o s on Banach space alued
unc ions, P oc. Na . Acad. Aci. USA 48, (1962), 356–365.
[CWW] S. Y. A. Chang, J. M. Wilson, and T. H. Wol , Some weigh ed no m inequali ies conce ning
he Sch ¨odinge ope a o s, Commen . Ma h. Hel e ici 60 (1985), 217–286.
[CW] S. Chanillo and R. Wheeden, Some weigh ed no m inequali ies o he A ea in eg al, Indiana
Uni . Ma h. J. 36 (1987).
[FS] C. Fe e man and E. M. S ein, Some maximal inequali ies, Ame . J. Ma h. 93 (1971),
107-115.
[GCRdF] J. Ga cia-Cue a and J. L. Rubio de F ancia, Weigh ed no m inequali ies and ela ed
opics, No h Holland Ma h. S udies 116, No h Holland, Ams e dam, (1985).
[dG] de Guzm´an M., Di e en ia ion o In eg als in Rn, Lec . No es in Ma h. 481, Spinge –
Ve lag, (1975).
[K] V. Kokilash ili, Maximal inequali ies and mul iplie s in weigh ed Lizo kin-T iebel spaces .
So ie Ma h. Dokl. 19 (2), 272–276.
[M] B. Muckenhoup , Weigh ed no m inequali ies o he Ha dy–Li lewood maximal unc ion,
T ans. Ame . Ma h. Soc. 165 (1972), 207–226.
[N] C. J. Neugebaue , Inse ing Ap–weigh s, P oc. Ame . Ma h. Soc. 87 (1983), 644–648.
[P1] C. P´e ez, On su icien condi ions o he boundedness o he Ha dy–Li lewood maximal
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