Publicacions Ma em`a iques, Vol 39 (1995), 23–41.
IMPROVED MUCKENHOUPT-WHEEDEN
INEQUALITY AND
WEIGHTED INEQUALITIES
FOR POTENTIAL OPERATORS
Y. Rako ond a simba
Abs ac
By a a ian o he s anda d good λinequali y, we p o e he
Muckenhoup -Wheeden inequali y o measu es which a e no nec-
essa ily in he Muckenhoup class. Mo eo e we can deal wi h a
gene al po en ial ope a o , and consequen ly we ob ain a sui able
app oach o he wo weigh inequali y o such an ope a o when
one o he weigh unc ions sa isfies a e e se doubling condi ion.
0. In oduc ion
In his pape dµ,dω a e locally fini e posi i e Bo el measu es o Rn,
n≥1. Fo a nonnega i e locally-dµ in eg able unc ion K(x, y) (a.e.
con inuous in he fi s a iable) we define he po en ial ope a o
(T µ)(x)=y∈Rn
K(x, y) (y)dµ(y).
Fo each C1>0, we assume he exis ence o a C2>0 so ha
(H)K(x, y)≤C2K(z,y), o each x, y, z wi h 0 <|z−y|<C
1|x−y|.
The dual ope a o T∗is he ope a o defined by he ke nel K∗(x, y)=
K(y,z). The usual ac ional in eg al ope a o Is, wi h 0 <s<n,is
gi en by K(x, y)=|x−y|s−n. O he examples o ope a o s Ta e hose
in oduced by Chanillo-S ombe g-Wheeden [Ch-S -Wh] and gi en by
ke nels K(x, y)=a(y,|x−y|)
|x−y|n. He e ais conside ed as a unc ion defined
on balls o Rnand which sa isfies some g ow h condi ions we p ecise
below.
24 Y. Rako ond a simba
We a e in e es ed in finding a cons an C>0 o which
(PT)T µLq
ω≤C Lp
µ o all nonnega i e unc ions
wi h 1 <p,q<∞. The cons an Cdepends only on n,p,q,ω,µand
K; and when i is necessa y we deno e his dependance by w i ing C=
C(n, p, q, K, ω, µ). He e gL
ν=Rn
|g| dν1
. The inequali y (PT)
includes he usual wo weigh no m inequali y
T Lq
u≤C Lp
since i is sufficien o eplace by 1
p−1, and o ake dω =udx,dµ =
−1
p−1dx, whe e dx is he usual Lebesgue measu e on Rn. Inequali y (PT)
wi h T=Ishas been s udied ex ensi ely by many au ho s (see o
ins ance [Ke-Sa], [Sa-Wh] and [Pe] and he e e ence gi en by hem).
Ke man and Sawye [Ke-Sa] sol ed he p oblem (PIs) wi h dω =dx.
This pa icula case is fi s in e es ing since i is he usual o m which
appea s in many ma hema ic and physic a eas. I also appea s ha he
case dω =dx is na u ally sui able o be ea ed. In ac using a good
λ-inequali y, hey p o ed ha he le membe o (PT) is majo ized by
he Lqno m o he ac ional maximal unc ion. So (PT) is educed o a
weigh ed inequali y o maximal ope a o , whose s udy was done by he
fi s au ho [Sa]. P oblem (PT) wi h gene al measu es dµ and dω was
sol ed by Sawye and Wheeden [Sa-Wh].
Le us conside he ope a o T=Iswi h 0 <s<n. We ha e he
poin wise inequali y
Msg≤c(s, n)Isg
whe e Msis he ac ional maximal ope a o defined by
(Ms )(x) = sup{|Q|s
n−1 1I QL1(dy);Qa cube wi h Qx}.
We gene ally use he le e Q o deno e a cube o Rn, and by which we
mean a p oduc o nin e als [ai,a
i+ ](0< <∞). The Muckenhoup -
Wheeden inequali y [Mu-Wh] yields a so o con e se (in no m) o he
abo e inequali y, and asse s ha o 0 <q<∞:
Is Lq
ω≤c(s, n, q, ω)Ms Lq
ω o all unc ions
whene e he measu e dω sa isfies he Muckenhoup condi ion A∞, i.e.
he e a e c=c(ω),δ >0 such as
|E|ω
|Q|ω
≤c|E|
|Q|δ
o all cubes Qand all measu ables se s E⊂Q.
Muckenhoup -Wheeden inequali y 25
In his hesis Pe ez [Pe] ga e a weake condi ion han he A∞’s. He
p o ed he abo e Muckenhoup -Wheeden inequali y o measu es dω sa -
is ying D∞and Bρcondi ions wi h ρ>1−s
n, and which can be no ed
as dω ∈D∞∩Bρ. These condi ions espec i ely mean:
|2Q|ω=2Q
ω≤C(ω)|Q|ω o all cubes Q,
(he e 2Qis he cube ha ing he same cen e as Qand he leng h expanded
wice)
|Q|ω
|Q|ω
≤C(ω)|Q|
|Q|ρ
o all cubes Q, Qwi h Q⊂Q.
Con a y o he Muckenhoup -Wheeden echnique [Mu-Wh], he
Pe ez’s analysis [Pe] is no based on he s anda d good-λinequali ies.
This las au ho used some es ima es ob ained by F azie and Jawe h
[F -Ja] o local maximal ope a o , and mo eo e he was able o ea
he p oblem wi h a gene al con olu ion ope a o .
In his pape we also p o e he Muckenhoup -Wheeden inequali y o
measu es which a e no necessa ily in he Muckenhoup class (see Co ol-
la y 4), and wi h he gene al po en ial ope a o Tdesc ibed abo e. We
do his, wi h a so o a a ian o he s anda d good λinequali y and
by in oducing a sui able maximal ope a o MT,ω (see Theo em 1). The
addi ional condi ions on he measu e dω a ise only in o de o ela e his
“exo ic” maximal ope a o o a mo e s anda d one like Ms(see Theo-
em 3). Consequen ly we ob ain a sui able app oach o he wo weigh
inequali y o such an ope a o when one o he weigh unc ions sa isfies
a e e se doubling condi ion (see Theo em 5).
1. S a emen s o esul s
Le us define he dyadic maximal ope a o
(Md
T,ω,µ )(x) = sup{|Q|ω−1 (T∗1I Qω)1I3QL1(dµ)
;
Qa dyadic cube wi h Qx}.
A dyadic cube Qis a p oduc o nin e als o he o m [2kai,2k(ai+1)],
whe e kand aia e in ege s. Fix q≥1. By using he Holde inequali y we
can obse e ha (Md
T,ω,µ )(x) is a.e. fini e o all bounded unc ions wi h
compac suppo s whene e measu es dω and dµ sa is y he condi ion
(ST)T1I |x|<RµLq
ω≤c(R)<∞ o all R>0.
Ou fi s esul is as ollow:
26 Y. Rako ond a simba
Theo em 1.
Le 0<q<∞and Kbe a nonnega i e ke nel sa is ying he hypo hesis
H. Assume he measu es dω and dµ sa is y he condi ion (ST). Then
he e is C=C(n, q, K)>0so ha
T µLq
ω≤CMd
T,ω,µ Lq
ω.
I Md
ωis he dyadic maximal ope a o defined by
(Md
ω )(x) = sup{|Q|ω−1 1I QL1(dω)
;Qa dyadic cube wi h Qx},
hen (see Lemma 1 below)
(Md
T,ω,µg)(.)≤(Md
ω(Tgµ))(.)
and consequen ly we ge
P oposi ion 2.
Le K,dω,dµ be as abo e Then o q>1we ha e
T µLq
ω≈Md
T,ω,µ µLq
ω.
Mo eo e his equi alence also holds o he ange o q∈]0,1] whene e
Md
ω(Tg)(.)≤c(n, K, ω)(Tg)(.)
o all nonnega i e unc ions g.
The abo e equi alence means
C1(n, q, K)T µLq
ω≤Md
T,ω,µ µLq
ω≤C2(n, q, ω)T µLq
ω.
The ex a assump ion in his esul is sa isfied o ins ance o he ke nel
K(x, y)=|x−y|s−n, wi h dω =dx he Lebesgue measu e, and mo e
gene ally o measu es dω ∈D∞∩Bρwi h 1 −s
n<ρ.
Thus in iew o Theo em 1, he inequali y (PT) is educed o he
ollowing one, ela ed o Md
T,ω,µ
(˜
PT)Md
T,ω,µ µLq
ω≤C Lp
µ.
In o de o ge his las one, we impose mo e hypo hesis on he ke nel
K. So as o simpli y, we only deal wi h ke nels
K(x, y)=Ka(x, y)=a(y,|x−y|)
|x−y|n=a(B(y,|x−y|))
|x−y|n
Muckenhoup -Wheeden inequali y 27
whe e ais a unc ion defined on balls sa is ying he ollowing hypo heses
H:
(i) a(B1)≤c1(n, a)a(B2) o all balls B1,B
2wi h B1⊂B2;
he e a e λ, σ > 0 so ha
(ii)
c
1(n, a) nλ a(B)≤a( B)≤c
2(n, a) nσ a(B) o all balls Band ≥1.
We also define he unc ion aon cubes by a(Q)=a(B), whe e Bis he
smalles ball which con ains he cube Q. A sui able dyadic maximal
ope a o ela ed o he po en ial ope a o T=Ta(wi h ke nel K=Ka)
is
(Md
Φ )(x) = sup{a(Q)|Q|−1 1I QL1(dy);Qa dyadic cube wi h Qx}.
The nondyadic e sion o Md
Φis me ely deno ed by MΦ. The measu e
dω sa isfies he condi ion RDρwi h ρ>0 (and we w i e as dω ∈RDω)
when he e c=c(ω,n)>0 o which
nρ |B|ω≤c| B|ω o all balls Band >1.
Ou second esul ensu es he link be ween he wo maximal ope a o s
we ha e defined abo e.
Theo em 3.
Le K=Kabe a ke nel sa is ying Hi)-ii) wi h 0<λ,σ≤1. Suppose
dω ∈RDρwi h 1−λ<ρ. Then
C1(Md
Φ µ)(.)≤(Md
Ta,ω,µ )(.)≤C2(MΦ µ)(.) o all unc ions
he e C1=C1(n, a)>0and C2=C2(n, a, ω).
In ac C2does no depend on he indi idual measu e dω bu only on
he RDρcons an o dω. The claim we announced in he in oduc ion
can be s a ed as
Co olla y 4.
Le 0<s<nand 0<q<∞. Suppose dω ∈RDωwi h (1 −s
n)<ρ.
Then
IsgLq
ω≈MsgLq
ω o all nonnega i e unc ions g
whene e R<|x|
|x|(s−n)qdω(x)<c(R)<∞ o all R>0.

28 Y. Rako ond a simba
This is an immedia e consequence o Theo ems 1 and 2. Indeed since o
all R>0:
(Is1I B(0,R))(.)1IB(0,2R)(.)≈Rs1I B(0,2R)(.) and (Is1I B(0,R))(x)1I|x|>2R(x)≈
|x|s−n1I |x|>2R(x) so he condi ion Is1I B(0,R)Lq
ω<∞is educed
o he one w i en in his co olla y. No e also in s udying he
wo weigh inequali y Is Lq
ω≤c Lq
νi is necessa y ha
Is1I B(0,R)Lq
ω≤c1I B(0,R)Lp
ν<∞.
By Theo ems 1 and 3, he p oblem (PT) is hen educed o he ollow-
ing maximal inequali y
MΦ µLq
ω≤c Lp
µ.
By he s udy o his las case (see [Ra1], o adap he p oo gi en in
[Sa]) hen we ge
Theo em 5.
Le 1<p,q<∞, and K=Kabe a ke nel sa is ying Hi)-ii) wi h
0<λ,σ<1. Suppose dω ∈RDωwi h (1 −λ)<ρ. Then he inequali y
(PT)T µLq
ω≤c Lp
µ
holds i and only i
(1) |x|>Ra(x, |x|)
|x|nq
dω(x)<c(R)<∞ o all R>0,
and
(2) (T
k
εk1I Qkµ)1IQkLq
ω
≤C
k
εk1I QkLp
µ
,
whe e C>0is a cons an which does no depend o each sequence (Qk)k
o cubes and (εk)ko nonnega i e eals εk.
Mo eo e in he case 1<p≤q he condi ion (2) can be eplaced by
(2’) (T1I Qµ)1IQLq
ω≤C1I QLp
µ o all cubes Q
Sawye and Wheeden [Sa-Wh] p o ed ha o 1 <p≤qand o all
gene al measu es dω and dµ, hen (PT) is equi alen o (2) and
(2”) (T∗1I Qω)1IQLp
µ≤c1I QLq
ωp=p
p−1,q=q
q−1.
Wi h an addi ional hypo hesis on he measu e dµ we can simpli y he
condi ions in Theo em 5. We fi s conside he case p≤q.
Muckenhoup -Wheeden inequali y 29
P oposi ion 6.
Le 1<p≤q<∞, and K=Kabe a ke nel sa is ying Hi)-ii) wi h
0<λ,σ<1. Suppose dω ∈RDρ,dµ ∈RDρwi h (1 −λ)<ρand
ρ>0. Then he inequali y (PT)holds i and only i
(T1I Qµ)1IQLq
ω≤C1I QLp
µ.
I mo eo e dµ ∈RDρwi h (1 −λ)<ρ
o dµ ∈A∞ hen, an easy
necessa y and sufficien condi ion o (PT)is
a(Q)
|Q||Q|1−1
p
µ|Q|
1
q
ω≤C o all cubes Q.
This second pa is al eady known [Sa-Wh], and he e we deduce i by
using esul s on maximal unc ions (see [Ra2] and [Pe]). To deal wi h he
ange o q<p, we in oduce he wo condi ions dµ ∈
RD(p), dω ∈Dε,q
wi h ε∈[1,∞[ (see [Ch-S -Wh]) and which mean espec i ely

j≥0
k
εk|Qk|µ
|2jQk|µ1I 2jQkLp
µ
≤c(µ)
k
εk1I QkLp
µ

k
εk1I QkLq
ω
≤c(ω) nε 1
q
k
εk1I QkLq
ω
o all ≥1, εk>0 and all cubes Qand Qk. Thus we can s a e
P oposi ion 7.
Le 1<q<p<∞, and K=Kabe a ke nel sa is ying Hi)-ii) wi h
0<λ,σ<1, and dω ∈RDρ, wi h (1 −λ)<ρ.
Suppose dµ ∈
RD(p). Then he inequali y (PT)holds i and only i
o some m≥4and C>0

k
εk(T1I Qk)1I(mQk)Lq
ω
≤C
k
εk1I QkLp
µ
o all cubes Qkand all εk>0.
Fo dµ ∈RDρ∩Dε,p wi h max(1 −λ, 1
pε)<ρ
, a necessa y and
sufficien condi ion o (PT)is

k
εka(Qk)
|Qk||Qk|µ1I QkLq
ω
≤C
k
εk1I QkLp
µ
.
30 Y. Rako ond a simba
This equi alence is also ue when dµ ∈RDρ∩D∞,dω ∈Dε,q wi h
1−λ<ρ
and ε<q(1 −σ).
Rema k. Now we show ha he use o he sha p maximal M#(see
[Ya] o a defini ion) is no well adap ed o weaken he weigh condi ion
in he Muckenhoup -Wheeden inequali y
(1) Is Lq
ω≤cMs Lq
ω.
Indeed such a pu pose is based on he wo inequali ies:
(2) (Is )#≤c(Ms );
(3) gLq
ω≤cg#Lq
ω.
Inequali y (2) is alid o all unc ions wi h (Is )∈L1
loc and was p o ed
in [Ad]. Al hough (3) is well known o be ue o w∈A∞, Yabu a [Ya]
had ob ained such an inequali y wi h a weak condi ion he deno ed as
w∈C (wi h >q). Thus we hink ge (1) wi h his las condi ion.
Bu since Ms ≤cIs =h hen
(4) h#Lq
ω≤chLq
ω
I was p o ed in [Ya] ha condi ion like (4) implies necessa ily w∈A∞.
2. Some Lemmas
We fi s s a e wo Lemmas we need and hen we gi e hei p oo s.
Lemma 1. Le be a nonnega i e (dµ-locally in eg able) unc ion.
Then
(Md
T,ω,µ )(.)≤(Md
ω(T µ))(.).
Lemma 2. Le T=Tabe an ope a o wi h he ke nel K=Kasa is-
ying Hi)-ii), and le dν be a posi i e Bo el measu e.
A) I 0<σ≤1 hen he e is C=C(a, n)>0so ha o all cubes Q
a(Q)
|Q||Q|ν1I Q(.)≤C(T1I Qν)(.)1IQ(.).
B) Le m≥1. The e is C=C(a, n, m)>0so ha
(T1I Qν)(.)1ImQ(.)≤C[S1(.)+S2(.)]
Muckenhoup -Wheeden inequali y 31
whe e
S1(.)=a(Q)
|Q||Q|ν1I mQ(.)
and
S2(.)=a(Q)
|Q|
j≥0
2−jn[λ−1] Q∩{|y−.|∼2−j|Q|
1
n}
ν1I mQ(.)
C) Le m≥4. The e is C=C(a, n, m)>0so ha
(T1I Qν)(.)1I(mQ)c(.)≤C|Q|ν
j≥0
a(2jQ)
|2jQ|1I 2jQ.
P oo o Lemma 1:
Le Qbe a dyadic cube. Then we ha e
Q
(T µ)dω =Rn
[T∗1I Qω] dµ ≥3Q
[T∗1I Qω] dµ.
Di ing by |Q|ω his inequali y and aking he sup emum, we ob ain he
conclusion.
P oo o Lemma 2:
A) Le Qbe a cube wi h cen e x0and leng h 2R>0. Then |x−y|≤
c2R o all x, y ∈Qwi h c=c(n). We ob ain
a(Q)
|Q||Q|ν1I Q(y)≤c1(a, n)x∈Q
a(y,c2R)
Rndν1I Q(y)
≤c2(a, n)x∈Q|x−y|
c2Rn1
|x−y|na
y, c2R
|x−y||x−y|dν(x)1I Q(y)
≤c3(a, n)x∈Q|x−y|
c2Rn[1−σ]a(y,|x−y|)
|x−y|ndν(x)
1I Q(y).
Since 0 <σ≤1 and |x−y|<2cR we ge
a(Q)
|Q||Q|ν1I Q(y)≤c(a, n)(Ta1I Qν)(y)1IQ(y).
38 Y. Rako ond a simba
≤c2|Q|µ
j≥0
|2jQ|−1
µ(T1I 2jQµ)1I(2jQ)Lq
ω
≤c2A|Q|
1
p
µ
j≥0|Q|µ
|2jQ|µ1−1
p
≤c3A|Q|
1
p
µ.
Fo he second pa o his p oposi ion, he poin is o no e ha MΦµ:
Lp
µ→Lq
ωis equi alen o
a(Q)
|Q||Q|1−1
p
µ|Q|
1
q
ω<A<∞
whene e dµ ∈A∞(see [Pe]) o dµ ∈RD∞wi h 1 −λ<ρ
(see
[Ra2]).
P oo o P oposi ion 7:
I is clea ha a necessa y condi ion o (PT)is
(*) 
k
εk(T1I Qkµ)1I(mQk)Lq
ω
≤A
k
εk1I QkLp
µ
wi h m≥4 and o all cubes Q,Qkand all εk>0. Con e sely we
suppose his condi ion be sa isfied and dµ ∈
RD(p). Once we ha e
(**) 
k
εk(T1I Qkµ)1I(mQk)cLq
ω
≤cA
k
εk1I QkLp
µ
hen (i) and (ii) hold as in p oo o Theo em 5, and consequen ly he
inequali y (PT) is sa isfied. Now using Pa C) o Lemma 2, he abo e
condi ion (∗) and he hypo hesis dω ∈
RD(p)weha e
S=
k
εk(T1I Qkµ)1I(mQk)cLq
ω
≤c1
k
εk
j≥0
a(2jQk)
|2jQk||Qk|µ1I (2jQk)Lq
ω
by pa C o Lemma 2
≤c2
k
εk
j≥0|Qk|µ
|2jQk|µ(T1I 2jQkµ)1I(2jQk)Lq
ω

Muckenhoup -Wheeden inequali y 39
≤c2A
j≥0
k
εk|Qk|µ
|2jQk|µ1I (2jQk)Lp
µ
by he condi ion (∗)
≤c3A
k
εk1I QkLp
µ
since dµ ∈
RD(p).
I is also clea ha a necessa y condi ion o (PT)is
(**’) 
k
εka(Qk)
|Qk||Qk|µ1I QkLq
ω
≤A
k
εk1I QkLp
µ
.
Con e sely we assume his condi ion be sa ified and dµ ∈Dε,p ∩RDρ
wi h 1 −λ<ρ
and ε<pρ
. I is sufficien o ge he condi ions in he
fi s pa o he p esen P oposi ion. As in he p oo o Theo em 3 by
using pa A) o Lemma 2) and since dµ ∈D∞ hen
(T1I Qkµ)1I(mQk)≤ca(Qk)
|Qk||Qk|µ1I (mQk)
and consequen ly

k
εk(T1I Qkµ)1I(mQk)Lq
ω
≤c
k
εka(Qk)
|Qk||Qk|µ1I (mQk)Lq
ω
≤c
k
εk1I (mQk)Lp
µ
.
Now using dµ ∈Dε,p ∩RDρwi h ε<pρ
we can ge he condi ion
dµ ∈
RD(p) as ollow:
S=
k
εk
j≥0|Qk|µ
|2jQk|µ
|Qk|µ1I (2jQk)Lp
µ
≤c1
j≥0
2−jnρ
k
εk1I (2jQk)Lp
µ
≤c2
j≥0
2−jn[ρ−1
pε]
k
εk1I QkLp
µ
=c3
k
εk1I QkLp
µ
.
Finally we suppose dµ ∈D∞∩RDρand dω ∈Dε,q ∩RDρwi h
1−λ<ρ
and ε<q(1 −σ). I emains o ge he abo e condi ion (∗∗).
Thus we ha e
S=
k
εk(T1I Qkµ)1I(mQk)cLq
ω
40 Y. Rako ond a simba
≤c1
k
εk
j≥0a(2jQk)
|2jQk||Qk|µ1I (2jQk)Lq
ω
≤c2
j≥0
2−jn[1−σ]
k
εka(Qk)
|Qk||Qk|µ1I (2jQk)Lq
ω
≤c3
j≥0
2−jn[1−σ−1
qε]
k
εka(Qk)
|Qk||Qk|µ1I QkLq
ω
≤c3A
j≥0
2−jn[1−σ−1
qε]
k
εk1I QkLp
µ
≤c4A
k
εk1I QkLp
µ
.
Acknowledgemen . I would like o hank he e e ee o his help ul
commen s and sugges ions.
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Uni e si ´e d’O l´eans
D´epa emen de Ma h´ema iques
U.F.R. Facul ´e des Sciences
B.P. 6759
45067 O l´eans Cedex 2
FRANCE
P ime a e si´o ebuda el 16 de Ma ¸c de 1993,
da e a e si´o ebuda el 15 de Gene de 1995