a Xi :1304.5999 2 [ma h.CA] 9 Jul 2013
WEIGHTED ESTIMATES FOR THE MULTISUBLINEAR
MAXIMAL FUNCTION
WEI CHEN AND WENDOL´
IN DAMI´
AN
Abs ac . A o mula ion o he Ca leson embedding heo em in he mul ilinea
se ing is p o ed which allows o ob ain a mul ilinea analogue o Sawye ’s wo
weigh heo em o he mul isublinea maximal unc ion Min oduced in [8].
A mul ilinea e sion o he Bp heo em om [6] is also ob ained and a mixed
A~
P−W∞
~
Pbound o Mis p o ed as well.
1. In oduc ion
The beginning o he mode n heo y o weigh s was o igina ed in he wo ks o R.
Hun , B. Muckenhoup , R. Wheeden, R. Coi man and C. Fe e man in he decade
o he 70’s. In [12] B. Muckenhoup cha ac e ized he class o weigh s u, o which
he ollowing weak inequali y holds
(1.1) sup
λ>0
λpZ{M >λ}
u(x)dx ≤CZRn
| (x)|p (x)dx, ∈Lp( ),
whe e Mdeno es he Ha dy–Li lewood maximal ope a o and p≥1. This condi-
ion on he weigh s is known as Apcondi ion, namely
[u, ]Ap:= sup
Q1
|Q|ZQ
u(x)dx 1
|Q|ZQ
(x)−1
p−1p−1
<∞, p > 1
whe e he sup emum is aken o e all he cubes in Rn. When p= 1, he e m
(RQ
1
|Q| (x)−1
p−1)p−1mus be unde s ood as (in Q )−1. In he pa icula case when
u= , Muckenhoup also p o ed ha he ollowing s ong es ima e
ZRn
(M (x))p (x)dx ≤CZRn
| (x)|p (x)dx, ∈Lp( ),
The i s au ho is suppo ed by he Na ional Na u al Science Founda ion o China (G an
No. 11101353), he Na u al Science Founda ion o Jiangsu Educa ion Commi ee (G an No.
11KJB110018) and he Na u al Science Founda ion o Jiangsu P o ince (G an No. BK2012682).
The second au ho is suppo ed by Jun a de Andaluc´ıa (G an No. P09-FQM-4745)
2010 Ma hema ics Subjec Classi ica ion. 42B25.
Key wo ds and ph ases. Mul ilinea maximal ope a o , weigh ed bounds, e e se H¨olde ’s
inequali y.
1
2 W. CHEN AND W. DAMI ´
AN
holds i and only i sa is ies he Apcondi ion. Howe e , he p oblem o inding a
condi ion on he weigh s u, sa is ying he s ong es ima e abo e was mo e com-
plica ed. In [13] E. Sawye cha ac e ized he wo weigh inequali y, showing ha
M:Lp( )−→ Lp(u) i and only i he pai (u, ) sa is ies he ollowing es ing
condi ion known as Sawye ’s Spcondi ion
(1.2) [u, ]Sp= sup
Q RQM(χQσ)pudx
σ(Q)!1/p
<∞,
whe e σ= 1−p′and 1 < p < ∞. Mo i a ed by hese esul s he heo y o weigh ed
inequali ies de eloped apidly, no only o he Ha dy–Li lewood maximal ope a-
o bu also o some o he main ope a o s in Ha monic Analysis like Calde ´on–
Zygmund ope a o s. Much la e he in e es ocused in de e mining he sha p de-
pendence o he Lp(w) ope a o no m in e m o he ele an cons an in ol ing he
weigh s.
On his poin , he p oblem o he Ha dy–Li lewood maximal ope a o was sol ed
by S. Buckley [1] who p o ed
(1.3) ||M||Lp(w)≤C p′[w]
1
p−1
Ap,
whe e Cis a dimensional cons an . Mo i a ed by his esul and o he s, K. Moen
ound in [11] a quan i a i e o m o E. Sawye ’s esul men ioned abo e in e ms o
Sawye ’s condi ion (1.2), namely
(1.4) ||M||Lp( )−→Lp(u)≈[u, ]Sp.
Recen ly, T. Hy ¨onen and C. P´e ez in [6] (see also [7] o a be e esul and
a simpli ied p oo ) imp o ed Buckley’s bound (1.3) eplacing a po ion o he Ap
cons an by he weake A∞cons an as de ined by Fujii in [4] and la e used in he
wo k o J.M. Wilson [14]. The A∞cons an is de ined as ollows
(1.5) [w]A∞:= sup
Q
1
w(Q)ZQ
M(wχQ),
whe e he sup emum is aken o e all he cubes Qin Rn. In [6] he au ho s show in
a wo-weigh se ing and o p > 1 ha
(1.6) ||M( σ)||Lp(w)≤Cp′(Bp[w, σ])1/p|| ||Lp(σ),
and
(1.7) ||M( σ)||Lp(w)≤Cp′([w]Ap[σ]A∞)1/p|| ||Lp(σ),
whe e Cin bo h inequali ies is a dimensional cons an and
WEIGHTED ESTIMATES FOR THE MULTISUBLINEAR MAXIMAL FUNCTION 3
(1.8) Bp[w, σ] := sup
Q1
|Q|ZQ
w 1
|Q|ZQ
σp
exp 1
|Q|ZQ
log σ−1,
is known as he Bpcons an o he weigh s wand σ. This cons an clea ly sa is ies
[w]Ap≤Bp[w, σ]≤[w]Ap[σ]′
A∞,
whe e [σ]′
A∞deno es he A∞cons an in oduced by S. H usˇcˇe in [5] de ined as
ollows
[w]′
A∞= sup
Q1
|Q|ZQ
wexp 1
|Q|ZQ
log w−1.
In he one weigh se ing and by a s anda d change-o -weigh a gumen , (1.7) implies
(1.9) ||M||Lp(w)≤Cp′([w]Ap[σ]A∞)1/p,
whe e σ=w1−p′.
The aim o his a icle is o gi e some mul ilinea analogues o some o he abo e
men ioned esul s ollowing he spi i o he heo y o mul iple weigh s de eloped in
[8].
The pape is o ganized as ollows. Some p elimina y de ini ions and esul s a e
summa ized in Sec . 2. In Sec . 3we gi e he s a emen s o he main esul s on his
pape and some ema ks on hem. Finally, in Sec . 4we gi e all he p oo s o ou
esul s.
Th oughou his pape , we will use he no a ion A.B o indica e ha he e is
a cons an c, independen o he weigh cons an , such ha A≤cB.
2. P elimina ies
Be o e s a ing and p o ing ou main esul s, we i s ecall some basics ela ed o
he heo y o mul ilinea weigh ed inequali ies as well as we in oduce he de ini ion
o some cons an s in ol ed in he mul iple heo y o weigh s and dyadic g ids.
2.1. Some basics on mul ilinea weigh ed inequali ies. One o he main ob-
jec s o he heo y o mul iple weigh s is he ollowing ex ension o he classical
Ha dy–Li lewood maximal unc ion. Gi en −→
= ( 1,..., m), we de ine ollowing
[8] he mul i(sub)linea maximal ope a o Mby
M(−→
)(x) = sup
Q∋x
m
Y
i=1
1
|Q|ZQ
| i(yi)|dyi,
whe e he sup emum is aken o e all cubes Qcon aining x. The impo ance o
his ope a o s ems om he ac ha i con ols he class o mul ilinea Calde ´on–
Zygmund ope a o s as i is shown in [8]. A pa icula example o his ela ionship is
he class o weigh s cha ac e izing he weigh ed Lpspaces o which bo h ope a o s
4 W. CHEN AND W. DAMI ´
AN
a e bounded. To de ine his class o weigh s we le −→
w= (w1,...,wm) and −→
P=
(p1,...,pm) such ha 1 < p1,...,pm<∞. Se 1
p=1
p1+···+1
pmand ν−→
w=
Qm
i=1 wp/pi
i. We say ha −→
wsa is ies he A−→
Pcondi ion i
[−→
w]A−→
P= sup
Q1
|Q|ZQ
ν−→
wm
Y
i=1 1
|Q|ZQ
w1−p′
i
ip/p′
i<∞.
I is easy o see ha in he linea case ( ha is, i m= 1) [−→
w]A−→
P= [w]Apis he
usual Apcons an .
In [8] he ollowing mul ilinea ex ension o he Muckenhoup Ap heo em o he
maximal unc ion was ob ained: he inequali y
(2.1) kM(−→
)kLp(ν−→
w)≤C
m
Y
i=1
k ikLpi(wi)
holds o e e y −→
i and only i −→
wsa is ies he A−→
Pcondi ion.
Ve y ecen ly in [3] A. Le ne , C. P´e ez and he second au ho p o ed a mul ilinea
e sion o Buckley’s esul as well as a ull analogue o (1.7). In his wo k he au ho s
ound ha he mul ilinea e sion o (1.7) is sha p when m≥1, al hough is much
mo e complica ed o do he same o Buckley’s esul . In his case, se e al pa ial
esul s we e ob ained in [3] which ha e been imp o ed in [9].
2.2. Some cons an s on mul iple weigh heo y. Nex we s a e he no a ion
ha we will ollow in he sequel ela ed o some cons an s in ol ed in he mul iple
heo y o weigh s. To de ine hese cons an s, le w1,...,wmand be weigh s and
le us deno e −→
w= (w1,...,wm). Also le 1 < p1,...,pm<∞and pbe numbe s
such ha 1
p=1
p1+···+1
pmand deno e −→
P= (p1,...,pm).
We say ha ( , −→
w) sa is ies he A−→
Pcondi ion i
(2.2) [ , −→
w]A−→
P:= sup
Q1
|Q|ZQ
m
Y
i=1 1
|Q|ZQ
w1−p′
i
ip/p′
i<∞.
In pa icula when =ν−→
w:= Qm
i=1 w
p
pi
i, we will w i e [ν−→
w,−→
w]A−→
Pas [−→
w]A−→
P.
Nex we de ine he mul ilinea analogues o he A∞cons an de ined by Fujii in
[4], he Bpcons an de ined by Hy ¨onen and P´e ez in [6] and he Spcons an de ined
by Sawye in [13], espec i ely. We say ha
(1) −→
wsa is ies he W∞
−→
Pcondi ion i
[−→
w]W∞
−→
P= sup
QZQ
m
Y
i=1
M(wiχQ)p
pidxZQ
m
Y
i=1
w
p
pi
idx−1
<∞.
WEIGHTED ESTIMATES FOR THE MULTISUBLINEAR MAXIMAL FUNCTION 5
(2) ( , −→
w) sa is ies he B−→
Pcondi ion i
[ , −→
w]B−→
P:= sup
Q
(Q)
|Q|m
Y
i=1
wi(Q)
|Q|pexp 1
|Q|ZQ
log
m
Y
i=1
w−p
pi
idx<∞.
(3) ( , −→
w) sa is ies he S−→
Pcondi ion i
[ , −→
w]S−→
P= sup
QZQ
M(−−→
σχQ)p dx1
pm
Y
i=1
σi(Q)1
pi−1
<∞,
whe e −−→
σχQ= (σ1χQ,...,σmχQ) and σi=w1−p′
i
i o all i= 1,...,m and all
he sup ema in he abo e de ini ions a e aken o e all cubes Qin Rn.
Addi ionally we de ine a mul iple Re e se H¨olde condi ion ha we will use in he
ollowing. We say ha −→
wsa is ies he RH−→
Pcondi ion i he e exis s a posi i e
cons an Csuch ha
(2.3)
m
Y
i=1 ZQ
σidxp
pi≤CZQ
m
Y
i=1
σ
p
pi
idx,
whe e σi=w1−p′
i
i o i= 1,...,m. We deno e by [−→
w]RH−→
P he smalles cons an C
in (2.3).
2.3. Dyadic g ids. Recall ha he s anda d dyadic g id Din Rnconsis s o he
cubes
2−k([0,1)n+j), k ∈Z, j ∈Zn.
By a gene al dyadic g id Dwe mean a collec ion o cubes wi h he ollowing
p ope ies:
(1) Fo any Q∈Di s sideleng h ℓQis 2k, k ∈Z
(2) Q∩R∈ {Q, R, ∅} o any Q, R ∈D.
(3) The cubes o a ixed sideleng h 2k o m a pa i ion o Rn.
We say ha {Qk
j}is a spa se amily o cubes i :
(1) The cubes Qk
ja e disjoin in j, wi h k ixed.
(2) I Ωk=∪jQk
j, hen Ωk+1 ⊂Ωk.
(3) |Ωk+1 ∩Qk
j| ≤ 1
2|Qk
j|.
Wi h each spa se amily {Qk
j}we associa e he se s Ek
j=Qk
j Ωk+1. Obse e ha
he se s Ek
ja e pai wise disjoin and |Qk
j| ≤ 2|Ek
j|.
In he sequel we will use he ollowing lemmas ha could be ound in [6] and [3],
espec i ely.
Lemma 2.1. The e a e 2ndyadic g ids Dαsuch ha o any cube Q⊂Rn he e
exis s a cube Qα∈Dαsuch ha Q⊂Qαand ℓQα≤6ℓQ.

6 W. CHEN AND W. DAMI ´
AN
Lemma 2.2. Fo any non-nega i e in eg able i, i = 1,...,m, he e exis spa se
amilies Sα∈Dαsuch ha o all x∈Rn,
M(−→
)(x)≤(2 ·12n)m
2n
X
α=1
ADα,Sα(−→
)(x),
whe e −→
= ( 1,..., m)and gi en a spa se amily S={Qk
j}o cubes om a dyadic
g id D, he ope a o AD,Sis gi en by
AD,S(−→
) = X
j,k m
Y
i=1
( i)Qk
j!χQk
j.
3. Main esul s
In his sec ion we summa ize he main esul s on his wo k. Fi s ly we s a e he
main ool o his pape . This lemma ex ends o he mul ilinea se ing a nons anda d
o mula ion o he (dyadic) Ca leson embedding heo em p o ed in [6] and i will
allow us o p o e ou main esul s.
Lemma 3.1. Suppose ha he nonnega i e numbe s {aQ}Qsa is y
(3.1) X
Q⊂R
aQ≤AZR
m
Y
i=1
σ
p
pi
idx, ∀R∈D
whe e σia e weigh s o i= 1,...,m. Then o all 1< pi<∞and p∈(1,∞)
sa is ying 1
p=1
p1+···+1
pmand o all i∈Lpi(σi),
X
Q∈D
aQm
Y
i=1
1
σi(Q)ZQ
i(yi)σi(yi)dyip!1/p
≤A||Md
−→
σ(−→
)||Lp(ν−→
σ)
≤A
m
Y
i=1
p′
i|| i||Lpi(σi),
(3.2)
whe e Md
−→
σ(−→
) = sup
Q∋x
Q∈D
m
Y
i=1
1
σi(Q)ZQ
| i(yi)|σi(yi)dyi.
Nex we es ablish a gene aliza ion o Sawye ’s heo em o he mul ilinea se ing.
Ve y ecen ly i was shown in [10] a mul ilinea e sion o Sawye ’s heo em using
a kind o mono one p ope y on he weigh s. We es ablish he e ano he condi ion
ha is a so o e e se H¨olde inequali y in he mul ilinea se ing (see Sec . 2 o
de ini ion) and ha was used by he i s au ho in [2] in he se ing o ma ingale
spaces. When m= 1 his e e se H¨olde condi ion is supe luous and we eco e he
linea esul o Moen (1.4).
WEIGHTED ESTIMATES FOR THE MULTISUBLINEAR MAXIMAL FUNCTION 7
Theo em 3.2. Le 1< pi<∞,i= 1,...,m and 1
p=1
p1+...+1
pm. Le and
wibe weigh s. I we suppose ha −→
w∈RH−→
P hen he e exis s a posi i e cons an C
such ha
(3.3) ||M(−→
σ)||Lp( )≤C
m
Y
i=1
|| i||Lpi(σi), i∈Lpi(σi),
whe e σi=w1−p′
i
i, i and only i ( , −→
w)∈S−→
P. Mo eo e , i we deno e he smalles
cons an Cin (3.3)by ||M||, we ob ain
(3.4) [ , −→
w]S−→
P.||M|| .[ , −→
w]S−→
P[−→
w]1/p
RH−→
P
.
He e we make some ema ks ela ed o he p e ious heo em.
Rema k 3.3. In he pa icula case when =ν−→
w, he ollowing s a emen s a e
equi alen :
(1) −→
w∈A−→
P.
(2) σi=w1−p′
i
i∈Amp′
i, o i= 1,...,m and ν−→
w∈Amp.
(3) (ν−→
w,−→
w)∈S−→
P.
(4) The e exis s a posi i e cons an Csuch ha
(3.5) ||M(−→
)||Lp(ν−→
w)≤C
m
Y
i=1
|| i||Lpi(wi), i∈Lpi(wi).
Indeed, he equi alence be ween 1., 2.and 4.was p o ed in [8, Th. 3.6, Th. 3.7].
I can be easily seen ha in his pa icula case [ν−→
w,−→
w]S−→
P.||M|| whe e ||M||
deno es he smalles cons an in (3.5) and [−→
w]A−→
P.[ν−→
w,−→
w]p
S−→
P. The e o e we ha e
ha 4.implies 3.and 3.implies 1.. So we ha e ob ained ha all he s a emen s a e
equi alen .
Addi ionally, ollowing [3, Th. 1.1], we also ha e ha ||M|| .[−→
w]1/p
A−→
PQm
i=1[σi]
1
pi
∞.
So, we ha e ob ained
(3.6) [−→
w]1/p
A−→
P.[ −→
w,−→
w]S−→
P.||M|| .[−→
w]1/p
A−→
P
m
Y
i=1
[σi]
1
pi
∞.
Rema k 3.4. As we ha e obse ed in he p e ious ema k, RH−→
Pcondi ion is no
necessa y when =ν−→
win Theo em 3.2. We a e no su e i his condi ion can be
emo ed in he gene al case.
Making use o he analogue o he Bpcons an wi hin he mul ilinea se ing
al eady de ined in Sec . 2, we ob ain an ex ension o (1.6).
8 W. CHEN AND W. DAMI ´
AN
Theo em 3.5. Le 1< pi<∞,i= 1,...,m and 1
p=1
p1+...+1
pm. Le and wi
be weigh s. Then
(3.7) ||M(−→
σ)||Lp( ).[ , −→
σ]1/p
B−→
P
m
Y
i=1
|| i||Lpi(σi), i∈Lpi(σi),
whe e σi=w1−p′
i
i,−→
σ= (σ1,...,σm)and −→
σ = ( 1σ1,..., mσm).
And inally, using he gene aliza ion o he Fujii–Wilson A∞cons an [−→
w]W∞
−→
Pand
he wo-weigh cons an [ , −→
w]A−→
Pde ined in Sec . 2, we ge a mixed A−→
P−W∞
−→
P
bound o M ha ex ends (1.7) o he mul ilinea se ing.
Theo em 3.6. Le 1< pi<∞,i= 1,...,m and 1
p=1
p1+...+1
pm. Le and wi
be weigh s. Then
(3.8) ||M(−→
σ)||Lp( ).([ , −→
w]A−→
P[−→
σ]W∞
−→
P)1/p
m
Y
i=1
|| i||Lpi(σi), i∈Lpi(σi),
whe e σi=w1−p′
i
i,−→
σ= (σ1,...,σm)and −→
σ = ( 1σ1,..., mσm).
4. P oo s
We s a p o ing Lemma 3.1, ha is, he mul ilinea e sion o he dyadic Ca leson
embedding heo em. I ollows a scheme o p oo simila o he one used by Hy ¨onen
and P´e ez in [6].
Lemma 3.1.Le us see he sum
X
Q∈D
aQ m
Y
i=1
1
σi(Q)ZQ
i(yi)σi(yi)dyi!p
as an in eg al on a measu e space (D,2D, µ) buil o e he se o dyadic cubes D,
assigning o each Q∈D he measu e aQ. Thus
X
Q∈D
aQ m
Y
i=1
1
σi(Q)ZQ
i(yi)σi(yi)dyi!p
=
=Z∞
0
pλp−1µ(Q∈D:
m
Y
i=1
1
σi(Q)ZQ
i(yi)σi(yi)dyi> λ)
=: Z∞
0
pλp−1µ(Dλ)dλ.
WEIGHTED ESTIMATES FOR THE MULTISUBLINEAR MAXIMAL FUNCTION 9
Le us deno e by D∗
λ he se o maximal dyadic cubes Rwi h he p ope y ha
Qm
i=1
1
σi(Q)RR i(yi)σi(yi)dyi> λ. Then he cubes R∈D∗
λa e disjoin and hei
union is equal o he se {Md
−→
σ(−→
)> λ}. Thus
µ(Dλ) = X
Q∈Dλ
aQ≤X
R∈D∗
λX
Q⊂R
aQ
≤AX
R∈D∗
λZR
m
Y
i=1
σ
p
pi
idx
=AZ{Md
−→
σ(−→
)>λ}
m
Y
i=1
σ
p
pi
idx.
Then we ob ain
X
Q∈D
aQ m
Y
i=1
1
σi(Q)ZQ
i(yi)σi(yi)dyi!p
≤AZ∞
0
pλp−1Z{Md
−→
σ(−→
)>λ}
m
Y
i=1
σ
p
pi
idxdλ
=AZRn
Md
−→
σ(−→
)p
m
Y
i=1
σ
p
pi
idx
≤AZRn
m
Y
i=1
((Md
σi( i))piσi)p
pidx
≤A
m
Y
i=1 ZRn
(Md
σi( i))piσidxp
pi
≤A
m
Y
i=1
(p′
i)pZRn
| i|piσidxp
pi,
whe e we ha e used ha Md
−→
σ(−→
)≤Qm
i=1 Md
σi( i), H¨olde ’s inequali y and he
boundedness p ope ies o Md
σi( i) in Lpi(σi). 
Nex we p o e Theo em 3.2 making use o Lemma 3.1.
Theo em 3.2.I is clea ha (3.3) implies he S−→
Pcondi ion wi hou using ha
( , −→
w)∈RH−→
P. Thus, i emains o p o e ha ( , −→
w)∈S−→
Pimplies (3.3) o comple e
he p oo o he heo em.
By Lemma 2.1, i su ices o p o e he heo em o he dyadic maximal ope a o s
MDα. Since he p oo is independen o he pa icula dyadic g id, wi hou loss o
gene ali y we conside Md aken wi h espec o he s anda d dyadic g id D. Nex
we p oceed as in he p oo o Lemma 2.2. Le a= 2m(n+1) and o k∈Zconside
he ollowing se s