Endpoint estimates and weighted norm inequalities for commutators of fractional integrals

Author: Cruz-Uribe, David; Fiorenza, Alberto
Publisher: Dipòsit Digital de Documents de la UAB
Year: 2003
DOI: 10.5565/PUBLMAT_47103_05
Source: https://ddd.uab.cat/pub/pubmat/02141493v47n1/02141493v47n1p103.pdf
Publ. Ma . 47 (2003), 103–131
ENDPOINT ESTIMATES AND WEIGHTED NORM
INEQUALITIES FOR COMMUTATORS OF
FRACTIONAL INTEGRALS
D. C uz-U ibe, SFO and A. Fio enza
Abs ac
We p o e ha he commu a o [b, Iα], b∈BMO,Iα he ac-
ional in eg al ope a o , sa isﬁes he sha p, modula weak- ype
inequali y
|{x∈Rn:|[b, Iα] (x)|> }|≤CΨRn
BbBMO
| (x)|
dx,
whe e B( )= log(e+ ) and Ψ( )=[ log(e+ α/n)]n/(n−α).
These commu a o s we e ﬁ s conside ed by Chanillo, and ou
esul complemen s his. The hea o ou p oo consis s o he
poin wise inequali y,
M#([b, Iα] )(x)≤CbBMO [Iα (x)+Mα,B (x)],
whe e M#is he sha p maximal ope a o , and Mα,B is a gene -
aliza ion o he ac ional maximal ope a o in he scale o O licz
spaces. Using his inequali y we also p o e one-weigh inequali ies
o he commu a o ; o do so we p o e one and wo-weigh no m
inequali ies o Mα,B which a e o in e es in hei own igh .
1. In oduc ion
Gi en α,0<α<n, deﬁne he ac ional in eg al ope a o Iαby
Iα (x)=Rn
(y)
|x−y|n−αdy.
I b∈BMO we deﬁne he ﬁ s o de commu a o [b, Iα] obe he ope a o
[b, Iα] (x)=b(x)Iα (x)−Iα(b )(x)=Rn
(b(x)−b(y)) (y)
|x−y|n−αdy.
2000 Ma hema ics Subjec Classiﬁca ion. 42B20, 42B25.
Key wo ds. F ac ional in eg als, commu a o s, BMO, O licz spaces, maximal unc-
ions, no m inequali ies.
The au ho s would like o hank C. P´e ez o sugges ing his p oblem, and he e e ee
o a numbe o insigh ul commen s.
104 D. C uz-U ibe, SFO, A. Fio enza
Since b∈Lp(K) o any p>1 and Kcompac , his in eg al con e ges
o all ∈Cc(Rn).
The commu a o s [b, Iα]we e in oduced by Chanillo [3], who showed
ha o 1 <p<n/α,1/q =1/p −α/n,[b, Iα]: Lp(Rn)→Lq(Rn). This
co esponds o he no m inequali ies sa isﬁed by Iα.
The ac ional in eg al also sa isﬁes an endpoin inequali y:
|{x∈Rn:|Iα (x)|> }| ≤ C1
Rn| |dxn/(n−α)
.
Howe e , a s aigh o wa d compu a ion wi h (x)=χ[0,1] and b(x)=
log(1 + x)χ(1,∞)shows ha [b, Iα]isno weak (1,n/(n−α)). (Fo a
s onge coun e -example, see Sec ion 6 below.) Ou main esul is a
sha p endpoin inequali y o he commu a o .
Theo em 1.1. Gi en α,0<α<n, and a unc ion b∈BMO, le
B( )= log(e+ )and Ψ( )=[ log(e+ α/n)]n/(n−α). Then he e exis s
acons an Csuch ha
|{x∈Rn:|[b, Iα] (x)|> }|≤CΨRn
BbBMO | (x)|
dx.(1.1)
Fu he mo e, his esul is sha p: i (1.1) holds wi h Ψ eplaced by an
inc easing unc ion Ψ0, hen he e exis posi i e cons an s γand Ksuch
ha Ψ( /γ)≤KΨ0( ), >0.
Rema k 1.2.Since Band Ψ a e submul iplica i e, we could w i e he
igh hand side o (1.1) as
CΨ(B(bBMO )) Ψ Rn
B| (x)|
dx;
his appea s mo e na u al, bu (1.1) is s onge since i is homogeneous
in b:wecan mul iply bby a cons an wi hou changing he size o he
cons an C.
To p o e Theo em 1.1 we ﬁ s p o e a poin wise inequali y ela -
ing [b, Iα], Iα, and a ac ional maximal ope a o deﬁned using he scale
o O licz spaces. (Fo p ecise deﬁni ions, see Sec ion 2 below.) Gi en a
Young unc ion B( o example, B( )= log(e+ )), and α,0≤α<n,
deﬁne he ac ional O licz maximal ope a o Mα,B by
Mα,B (x)=sup
Qx|Q|α/n B,Q,
Commu a o s o F ac ional In eg als 105
whe e he sup emum is aken o e all cubes con aining x. When B( )=
his educes o he classical ac ional maximal ope a o ,
Mα (x)=sup
Qx
|Q|α/n
|Q|Q| |dy.
The ela ionship be ween Mα,B and [b, Iα]in ol es he sha p maximal
unc ion o Feﬀe man and S ein [11]. Recall ha i is deﬁned by
M# (x)=sup
Qx
1
|Q|Q| (y)− Q|dy, whe e Q=1
|Q|Q
(x)dx.
Theo em 1.3. Le B( )= log(e+ ). Gi en α,0<α<n,b∈BMO
and a non-nega i e unc ion , he e exis s a cons an Csuch ha o
all x,
M#([b, Iα] )(x)≤CbBMO [Iα (x)+Mα,B (x)].(1.2)
Rema k 1.4.Theo em 1.3 emains ue i B( )= log(e+ )is eplaced
by any “la ge ” O licz unc ion. We will make his p ecise in Sec ion 2.
Rema k 1.5.Inequali ies simila o (1.1) and (1.2) o singula in eg al
ope a o s (which o mally co espond o he case α=0)a e ue. These
we e ﬁ s p o ed by P´e ez [22], and ou p oo s a e modeled on his.
Howe e , ou app oach o sha pness is diﬀe en om his.
We can also use Theo em 1.3 o p o e one-weigh no m inequali ies
o [b, Iα]. The ﬁ s is a s ong (p, q) inequali y due o Sego ia and
To ea [26] which gene alizes Chanillo’s o iginal esul .
Theo em 1.6. Gi en α,0<α<n, and p,1<p<n/α,ﬁxqso ha
1/q =1/p −α/n.Le wbeaweigh sa is ying he Apq condi ion: o all
cubes Q,
1
|Q|Q
wqdx1/q 1
|Q|Q
w−pdx1/p
≤K<∞.(1.3)
Then, gi en any unc ion b∈BMO, [b, Iα]sa isﬁes he s ong (p, q)
inequali y
Rn|[b, Iα] |qwqdx1/q
≤CbBMO Rn| |pwpdx1/p
.(1.4)
Rema k 1.7.We can also use Theo em 1.3, oge he wi h he ideas in [9]
and [6], o p o e wo-weigh no m inequali ies o commu a o s o ac-
ional in eg als. These will be ea ed in a sepa a e pape .
106 D. C uz-U ibe, SFO, A. Fio enza
The Apq condi ion go e ns he s ong (p, q) inequali ies o Iα; his
is due o Muckenhoup and Wheeden [18]. Gi en his ac , i seemed
na u al o conjec u e ha in he limi ing case p=1,q=n/(n−α),
he condi ion which go e ns he weak (1,n/(n−α)) inequali y o Iα,
wq∈A1, (also due o Muckenhoup and Wheeden) would go e n a
weigh ed e sion o (1.1). Howe e , his is no he case.
Example 1.8. The e exis s a unc ion wdeﬁned on Rsuch ha wq∈
A1,q=1/(1 −α), bu he e is no cons an Csuch ha
wq({x∈R:|[b, Iα] (x)|> })≤CΨR
BbBMO | (x)|
wdx
,(1.5)
whe e Band Ψ a e as in Theo em 1.1, holds o all .
This esul is e y su p ising, especially since he analogous weigh ed
inequali y wi h α=0holds o singula in eg al ope a o s. (See [22].)
We a e unsu e wha he co ec condi ion on he weigh wshould be
o (1.5) o hold.
The emainde o his pape is o ganized as ollows. In Sec ion 2 we
s a e some p elimina y deﬁni ions and esul s abou O licz spaces. In
Sec ions 3 and 4 we s a e and p o e an endpoin es ima e and weigh ed
no m inequali ies o he O licz ac ional maximal ope a o Mα,B.We
use hese, oge he wi h Theo em 1.3, o p o e Theo ems 1.1 and 1.6.
We ac ually p o e esul s which hold o a la ge class o Young unc-
ions B, since we can do so o essen ially no mo e wo k and hey a e o
independen in e es . In Sec ion 5 we p o e Theo em 1.3, in Sec ion 6
we p o e Theo em 1.1, and in Sec ion 7 we p o e Theo em 1.6, and
cons uc Example 1.8.
Th oughou his pape all no a ion is s anda d o will be deﬁned
as needed. All cubes a e assumed o ha e hei sides pa allel o he
coo dina e axes. Gi en a cube Qand >0, Q will deno e he cube
wi h he same cen e as Qand whose sides a e imes as long. By weigh s
we will always mean non-nega i e, locally in eg able unc ions which a e
posi i e on a se o posi i e measu e. Gi en a Lebesgue measu able
se Eand a weigh w,|E|will deno e he Lebesgue measu e o Eand
w(E)=Ewdx. Gi en 1 <p<∞,p=p/(p−1) will deno e he
conjuga e exponen o p.Cand cwill deno e posi i e cons an s whose
alue may change a each appea ance.
Finally, we assume ha he eade is amilia wi h he deﬁni ion and
basic p ope ies o he Ha dy-Li lewood maximal ope a o M, i s dyadic
a ian Md, and he Muckenhoup Apweigh s, 1 ≤p≤∞.We e e he
eade o [10]o [13] o u he in o ma ion.
Commu a o s o F ac ional In eg als 107
2. Backg ound on O licz spaces
In he ollowing we a e going o use some no ions o O licz space
heo y. He e we summa ize some basic ac s; we e e he eade o [16],
[17], o [24] o u he de ails.
A unc ion B:[0,∞)→[0,∞)isaYoung unc ion i i is con inuous,
con ex and s ic ly inc easing, and i B(0) = 0, B( )→∞as →∞.
I A,Ba e Young unc ions, we w i e A( )≈B( )i he e a e con-
s an s 0,c
1,c
2>0 such ha c1A( )≤B( )≤c2A( ) o ≥ 0. Also, we
say ha Bdomina es A, and deno e his by AB,i he e exis s c>0
such ha o all >0, A( )≤B(c ). I his is ue o all ≥ 0>0, we
say ha ABnea inﬁni y.
AYoung unc ion Bis said o be doubling i he e exis s a posi i e
cons an Csuch ha B(2 )≤CB( ) o all >0; Bis called submul-
iplica i e i B(s )≤CB(s)B( ) o all s, > 0. Clea ly B( )= ,
≥1, is submul iplica i e. A s aigh o wa d compu a ion shows ha
B( )= a[log(e+ )]b,a≥1, b>0, is also submul iplica i e. (Fo sim-
plici y, he ea e we will omi he b acke s and w i e simply alog(e+ )b.)
Gi en a non-emp y open se Ein Rnand a Young unc ion B, he
O licz space LB(E)is he Banach space o Lebesgue measu able unc-
ions such ha B(| |/λ)is(Lebesgue) in eg able on E o some λ>0.
I is equipped wi h he Luxembu g no m
 LB(E)= in λ>0:E
B| |
λdx ≤1.
When Ehas ﬁni e measu e (e.g. i is a cube) we o en wan o no malize
by eplacing he measu e dx by dx/|E|.Inpa icula , gi en a cube Q,
we deﬁne he mean Luxembu g no m o on Qby
 B,Q = in λ>0: 1
|Q|Q
B| |
λdx ≤1.(2.1)
When B( )= ,1≤ <∞,
 B,Q =1
|Q|Q| | dx1/
,
so he Luxembu g no m coincides wi h he (no malized) L no m.
I ABnea inﬁni y hen he e exis s a cons an C, depending on A
and B, such ha o all cubes Qand unc ions , A,Q ≤C B,Q.
This ollows om he s anda d embedding heo em which shows ha
LB(Q)⊂LA(Q). Howe e , we s ess ha because his is he mean
Luxembu g no m, he cons an Cis independen o Q.

108 D. C uz-U ibe, SFO, A. Fio enza
I ollows om his ha i ABnea inﬁni y, hen Mα,A (x)≤
CMα,B (x). In pa icula , in Theo em 1.3 we can ake B o be any
Young unc ion such ha log(e+ )Bnea inﬁni y. (This makes
p ecise Rema k 1.4.)
Gi en a Young unc ion B, he complemen a y Young unc ion ¯
Bis
deﬁned by
¯
B( )=sup
s>0{s −B(s)}, >0.(2.2)
Band ¯
Bsa is y he ollowing inequali y: ≤B−1( )¯
B−1( )≤2 .
We will need a gene aliza ion o H¨olde ’s inequali y o O licz spaces
due o O’Neil [19]. (Also see [17]o [24].)
Lemma 2.1. Gi en a Young unc ion B, hen o all unc ions and g
and all cubes Q,
1
|Q|Q| g|dx ≤2 B,Qg¯
B,Q.(2.3)
Mo e gene ally, i A,Band Ca e Young unc ions such ha o all
>0,
B−1( )C−1( )≤A−1( ),
hen
 gA,Q ≤2 B,QgC,Q.(2.4)
I we se g≡1in(2.3), i immedia ely ollows ha o all Young
unc ions B,α,0≤α<n, and x∈Rn,
Mα (x)≤CMα,B (x).(2.5)
3. Endpoin inequali y o M
α,B
In his sec ion we p o e a modula endpoin inequali y o he O licz
ac ional maximal ope a o which we need o he p oo o Theo em 1.1.
To s a e i we need he ollowing deﬁni ion.
Deﬁni ion 3.1. Gi en a Young unc ion B, deﬁne he unc ion hBby
hB(s)=sup
>0
B(s )
B( ),0≤s<∞.
Rema k 3.2.The unc ion hBcould be inﬁni e i s>1, bu i Bis
doubling hen i is ﬁni e o all 0 <s<∞. (See [17, Theo em 11.7].)
I Bis submul iplica i e hen hB≈B. Mo e gene ally, gi en any B, o
all s, ≥0, B(s )≤hB(s)B( ).
Commu a o s o F ac ional In eg als 109
Theo em 3.3. Gi en α,0≤α<n, le Bbe aYoung unc ion such
ha B( )/ n/α is dec easing o all >0. Then he e exis s a cons an
depending only on Bsuch ha o all >0,Mα,B sa isﬁes he modula
weak- ype inequali y
Φ(|{x∈Rn:Mα,B (x)> }|)≤CRn
B (x)
dx,(3.1)
o all non-nega i e ∈LB(Rn), whe e Φis any unc ion such ha
Φ(s)≤C1Φ1(s)=


0i s=0
s
hB(sα/n)i s>0.
Be o e p o ing Theo em 3.3 we make a numbe o obse a ions abou
i s s a emen .
Rema k 3.4.When α=0we in e p e he g ow h condi ion o mean
ha Bcan be any Young unc ion.
Rema k 3.5.The unc ion Φ1is well-deﬁned: by Lemma 3.12 below, i
ollows om he ac ha B( )/ n/α is dec easing ha 0 <h
B(sα/n)<∞
o all s>0. Also no e ha i Bis submul iplica i e, B≈hB,so
Φ1(s)≈s/B(sα/n).
Rema k 3.6.Suppose Φ is con inuous and in e ible. I we le Ψ = Φ−1,
and i Ψ is doubling, hen inequali y (3.1) can be w i en as
|{x∈Rn:Mα,B (x)> }| ≤ CΨRn
B (x)
dx.(3.2)
By Lemma 3.12, o any Bsa is ying he assump ions o Theo em 3.3
he e exis unc ions Φ, in e ible, sa is ying Φ(s)≤CΦ1(s). In he
p oo o Theo em 1.1 we use (3.2) wi h B( )= log(e+ ).
Rema k 3.7.We can weaken he g ow h condi ion on Bas ollows: i
I(B)<n/α, whe e I(B) deno es he uppe Boyd index o B, hen
he e exis s a Young unc ion B1such ha B1≈Band B1( )/ n/α is
dec easing. See [12, Theo em 1.1] o u he de ails.
Rema k 3.8.When B( )= ,Φ( )= 1−α/n (o Ψ( )= n/(n−α)in (3.2)),
and Theo em 3.3 educes o he weak (1,n/(n−α)) inequali y o he
classical ac ional maximal ope a o (c . [10,p.89]). When α=0,
Φ( )=Ψ( )= , and (3.1) becomes he modula endpoin inequali y o
M0,B due o P´e ez [21].
110 D. C uz-U ibe, SFO, A. Fio enza
Rema k 3.9.In he p oo o Theo em 1.3 we need Theo em 3.3 o he
case B( )= log(e+ ). Since Bis submul iplica i e, we can ake
Φ( )= 1−α/n
log(e+ α/n),
o equi alen ly,
Ψ( )=[ log(e+ α/n)]n/(n−α).
We could eplace α/n by in he deﬁni ion o Φ and Ψ; we chose no o
since wi h he gi en deﬁni ions we ecap u e he case α=0.
Rema k 3.10.In he limi ing case B( )= n/α, Theo em 3.3 is i ial:
o all x∈Rn,
Mα,B (x)=sup
Qx|Q|α/n 1
|Q|Q| |n/α dxα/n
=Rn| |n/α dxα/n
,
and he e o e Mα,B is cons an . Then (3.1) is i ially ue wi h Φ
deﬁned by Φ(0) = 0, Φ(s)=1,s>0.
The p oo o Theo em 3.3 equi es ou lemmas.
Lemma 3.11. I Bis a Young unc ion hen hBis nonnega i e, submul-
iplica i e, inc easing in [0,∞), s ic ly inc easing in [0,1] and hB(1)=1.
Fo he (easy) p oo see [6, Lemma 3.1] o [17,p.84].
Lemma 3.12. Gi en α,0≤α<n, le Bbe aYoung unc ion such
ha B( )/ n/α is dec easing o all >0. Then he unc ion Φ1in
Theo em 3.3 is inc easing, and Φ1(s)/s is dec easing. Mo eo e , he e
exis s Φsuch ha Φ(s)≤C1Φ1(s)and Φis in e ible.
P oo o Lemma 3.12: I α=0, he asse ion is i ial. I 0 <α<n,
o 0 <s<σand >0, sα/n <σ
α/n . Since he unc ion B( )/ n/α is
dec easing,
B(σα/n )
σ n/α ≤B(sα/n )
s n/α ;
he e o e,
B(σα/n )
σB( )≤B(sα/n )
sB( ).
I we ake he sup emum o e all , hen
hB(σα/n)
σ≤hB(sα/n)
s.
I ollows immedia ely ha Φ1(s)≤Φ1(σ). Fu he mo e, by Lemma 3.11,
Φ1(s)/s is dec easing.
Commu a o s o F ac ional In eg als 111
Finally, i C(s)isany con inuous and s ic ly inc easing unc ion such
ha C(0)=1,C(s)→2ass→+∞, hen i ially he unc ion Φ
deﬁned by Φ(s)=C(s)Φ1(s)isin e ible and sa isﬁes Φ(s)≤2Φ1(s)
since Φ1(s)>0i s>0.
Lemma 3.13. I Φ( )/ is dec easing, hen o any posi i e sequence {xj},
Φ

j
xj
≤
j
Φ(xj).
Fo a p oo , see [14,p.83, n. 103].
Lemma 3.14. Gi en a non-nega i e, locally in eg able unc ion and
α,0≤α<n, suppose ha o some Young unc ion B, cube Qand
>0,
|Q|α/n B,Q > .
Then he e exis s a dyadic cube Psuch ha Q⊂3Pand a posi i e
cons an βα,n, depending only on αand n, such ha
|P|α/n B,P >β
α,n .
When B( )= his esul is p o ed in [5], and when α=0 his is
implici in [21, Lemma 4.1]. Ei he p oo eadily adap s and we omi
he de ails.
P oo o Theo em 3.3: Fix a non-nega i e unc ion in LB(Rn). Fix
>0 and deﬁne
E ={x∈Rn:Mα,B (x)> }.
I is such ha he se E is emp y, we ha e no hing o p o e. O he wise,
o each x∈E he e exis s a cube Qxcon aining xsuch ha
|Qx|α/n B,Qx> .
By Lemma 3.14, he e is a cons an βsuch ha o each x he e exis s
adyadic cube Pxwi h Qx⊂3Pxand
|Px|α/n B,Px>β .(3.3)
Since ∈LB(Rn), we can eplace he collec ion {Px}wi h a maximal
disjoin subcollec ion {Pj}. Each Pjsa isﬁes (3.3), and by ou choice o
he Qx’s, E ⊂j3Pj.
118 D. C uz-U ibe, SFO, A. Fio enza
As we no ed abo e, since Csa isﬁes he Bpcondi ion, M0,C is bounded
on Lp(see [21]); hence
≤CRn| |p pdxq/p
.
This comple es ou p oo .
P oo o Co olla y 4.4: Fix α,pand qas in he hypo heses. I will suﬃce
o show ha i w∈Apq hen he pai (w,w) sa isﬁes he hypo heses o
Theo em 4.2.
Le =1+p/q; hen we can e-w i e he Apq condi ion as
1
|Q|Q
w−pdx1
|Q|Q
(w−p)1− dx −1
≤K<∞.
Hence w−pis in A , and so sa isﬁes he e e se H¨olde inequali y wi h
exponen s>1. Le A( )= sp; hen A−1( )= 1/sp. The e o e, i we
le
C−1( )= 1/(sp)
log(e+ ),
we ha e ha
A−1( )C−1( )=
log( )≈B−1( ).
Fu he mo e,
C( )≈( log(e+ ))(sp);
since sp>p
,(sp)<p, and so Csa isﬁes he Bpcondi ion.
Inequali y (4.1) now ollows a once:
|Q|α/n+1/q−1/p 1
|Q|Q
wqdx1/q
 −1A,Q
=1
|Q|Q
w−pdx1/q 1
|Q|Q
w−psdx1/sp
≤C1
|Q|Q
w−pdxq1
|Q|Q
w−pdx1/p
≤K.

Commu a o s o F ac ional In eg als 119
5. P oo o Theo em 1.3
Ou p oo equi es se e al ac s abou unc ions in BMO and abou
he ac ional in eg al ope a o . We ga he hese in wo lemmas.
Lemma 5.1. The ollowing a e ue:
(1) A unc ion bis in BMO i o each cube Q he e exis s a con-
s an cQsuch ha
sup
Q
1
|Q|Q|b(x)−cQ|dx < ∞.
Fu he , his sup emum is compa able o bBMO .
(2) Fo each p,1<p<∞, he e exis s a cons an Cpsuch ha
sup
Q1
|Q|Q|b(x)−bQ|pdx1/p
≤CpbBMO .
(3) I b∈BMO hen he e exis s a cons an Csuch ha o e e y
cube Q,
1
|Q|Q
exp |b(x)−bQ|
CbBMO dx < ∞.
(4) I b∈BMO hen o any cube Qand k≥0,
|b2k+1Q−bQ|≤2(k+1)bBMO .
Fo ap oo o (1)–(3), see [10, Chap e 6]. Fo a p oo o (4), see [27,
p. 206].
Lemma 5.2. Gi en α,0<α<n, and a non-nega i e unc ion , he
ollowing a e ue:
(1) The e exis s a cons an Csuch ha o any cube Q,
Q
Iα dx≤C|Q|α/n Rn
dx;(5.1)
(2) Iα ∈A1, ha is, he e exis s a cons an Csuch ha M(Iα )(x)≤
CIα (x) o almos e e y x. Hence, i sa isﬁes he e e se H¨olde
inequali y o some exponen s>1.
(3) Iαis weak (1,n/(n−α)): o all λ>0,
|{x∈Rn:|Iα(x)|>λ}| ≤ C1
λRn| (x)|dxn/(n−α)
.(5.2)
120 D. C uz-U ibe, SFO, A. Fio enza
P oo : Inequali y (5.1) ollows easily om Fubini’s heo em:
Q
Iα dx=QRn
(y)dy
|x−y|n−αdx
=Rn
(y)Q|x−y|α−ndxdy
≤C|Q|α/n Rn
(x)dx.
To see ha Iα ∈A1,i suﬃces o no e ha |x|α−n∈A1, and he
con olu ion o a non-nega i e unc ion wi h an A1weigh is again in A1.
(C . [25].) Fo he weak (1,n/(n−α)) inequali y, see [10,p.89].
P oo o Theo em 1.3: By homogenei y, i will suﬃce o p o e (1.2) o
x=0.Fu he , by Lemma 5.2(2), and Lemma 5.1(1), i will suﬃce o
show ha , gi en a cube Qcen e ed a he o igin, he e exis s a con-
s an cQsuch ha
1
|Q|Q|[b, Iα] (y)−cQ|dy ≤CbBMO [M(Iα )(0) + Mα,B (0)].(5.3)
Decompose as 1+ 2, whe e 1= χQ∗, and Q∗is he cube cen e ed a
he o igin whose sides a e 2√n imes la ge . Le cQ=(Iα((b−bQ∗) 2))Q.
Then, since [b, Iα] =[b−bQ∗,I
α] ,
1
|Q|Q|[b, Iα] (y)−cQ|dy
≤1
|Q|Q|(b(y)−bQ∗)Iα (y)|dy
+1
|Q|Q|Iα((b−bQ∗) 1)(y)|dy
+1
|Q|Q|Iα((b−bQ∗) 2)(y)−(Iα((b−bQ∗) 2))Q|dy
=I1+I2+I3.
We es ima e each in eg al in u n. Fo I1,byLemma 5.2(2), Iα
sa isﬁes he e e se H¨olde inequali y wi h exponen s>1. The e o e,
Commu a o s o F ac ional In eg als 121
i we apply H¨olde ’s inequali y wi h exponen s,byLemma 5.1(2),
I1≤1
|Q|Q|(b(y)−bQ∗)|sdy1/s
1
|Q|Q
Iα (y)sdy1/s
≤CbBMO 1
|Q|Q
Iα (y)dy
≤CbBMO M(Iα )(0).
To es ima e he second in eg al, no e ha by Lemma 5.1(3),
1
|Q∗|Q∗
exp |b(y)−bQ∗|
CbBMO dy < ∞.
Since B( )= log(e+ ), ¯
B( )e −1; he e o e, by (2.1),
b−bQ∗¯
B,Q∗≤CbBMO .
Hence, by Lemma 5.2(1), and by he gene alized H¨olde inequali y (2.3),
I2≤C|Q|α/n
|Q|Rn|b(y)−bQ∗| 1(y)dy
=C|Q∗|α/n
|Q∗|Q∗|b(y)−bQ∗| (y)dy
≤C|Q∗|α/nb−bQ∗¯
B,Q∗ B,Q∗
≤CbBMOMα,B (0).
Finally, we es ima e he hi d in eg al. By he mean- alue heo em,
i |x|>2|h| hen he e exis s θ,0≤θ≤1, such ha

1
|x|n−α−1
|x+h|n−α≤C|h|
|x+θh|n−α+1 ≤C|h|
|x|n−α+1 .
122 D. C uz-U ibe, SFO, A. Fio enza
I x, y ∈Qand z∈Rn 2kQ∗,k≥0, hen |x−z|>2k+1|y−x|.
The e o e,
I3≤1
|Q|Q
1
|Q|QRn
Q∗
|(b(z)−bQ∗) (z)|
1
|y−z|n−α−1
|x−z|n−α
dz dx dy
≤1
|Q|2QQ
∞

k=0 2k+1Q∗ 2kQ∗|(b(z)−bQ∗) (z)||y−x|
|x−z|n−α+1 dz dx dy
≤C
|Q|2QQ
∞

k=0
2−k
|2k+1Q∗|1−α/n 2k+1Q∗|(b(z)−bQ∗) (z)|dz dx dy
≤C
∞

k=0
2−k
|2k+1Q∗|1−α/n 2k+1Q∗|(b(z)−b2k+1Q∗) (z)|dz
+C
∞

k=0
2−k
|2k+1Q∗|1−α/n 2k+1Q∗|(b2k+1Q∗−bQ∗) (z)|dz
≤
∞

k=0
2−k|2k+1Q∗|α/nb−b2k+1Q∗¯
B,2k+1Q∗ B,2k+1 Q∗
+CbBMO
∞

k=0
(k+ 1)2−k
|2k+1Q∗|1−α/n 2k+1Q∗| (z)|dz
≤CbBMO Mα,B (0) + CbBMO Mα (0)
≤CbBMO Mα,B (0).
The ﬁ h inequali y ollows om Lemma 5.1(4), and he las inequali y
ollows om (2.5). This comple es he p oo .
6. P oo o Theo em 1.1
In ou p oo we need a a ian o he good-λinequali y o Feﬀe man
and S ein [11] ela ing he dyadic maximal ope a o and he sha p max-
imal ope a o . (Also see [10,p.121].) This exac esul is gi en by
P´e ez [22]; he p oo is a s aigh o wa d modiﬁca ion o he p oo o
he s anda d esul and so is omi ed.
Commu a o s o F ac ional In eg als 123
Lemma 6.1. Le ϕbe aposi i e unc ion on (0,∞)such ha ϕ(2 )≤
ϕ( ) o all >0. Then he e exis s a posi i e cons an Csuch ha
sup
λ>0
ϕ(λ)|{y∈Rn:Md (y)>λ}|≤Csup
λ>0
ϕ(λ)|{y∈Rn:M# (y)>λ}|
o all unc ions such ha he le hand side is ﬁni e.
P oo o Theo em 1.1: Fi s , ﬁx a unc ion in BMO.I bBMO =0
hen bis cons an and he esul is i ial. We may he e o e assume
ha bBMO =0.
Now ﬁx a unc ion . Since we can decompose an a bi a y unc ion
in o he sum o i s posi i e and nega i e pa s, wi hou loss o gene ali y
we may assume ha is non-nega i e. By homogenei y, i will suﬃce
o p o e (1.1) when =1, ha is,
|{x∈Rn:|[b, Iα] (x)|>1}| ≤ CΨRn
B(bBMO | (x)|)dx.(6.1)
Bu in his case,
|{x∈Rn:|[b, Iα] (x)|>1}|
≤Ψ(B(1)) sup
>0
1
Ψ(B(1/ ))|{x∈Rn:|[b, Iα] (x)|> }|
≤Ψ(B(1)) sup
>0
1
Ψ(B(1/ ))|{x∈Rn:Md([b, Iα] )(x)> }|.
Le ϕ( )=1/Ψ(B(1/ )); hen a s aigh o wa d bu somewha edious
calcula ion shows ha
lim
→0
ϕ(2 )
ϕ( )= lim
→∞
ϕ(2 )
ϕ( )=2
n/(n−α),

124 D. C uz-U ibe, SFO, A. Fio enza
and so ϕ(2 )≤Cϕ( ) o all >0. The e o e, by Lemma 6.1 and
Theo em 1.3,
|{x∈Rn:|[b, Iα] (x)|>1}|
≤Csup
>0
1
Ψ(B(1/ ))|{x∈Rn:M#([b, Iα] )(x)> }|
≤Csup
>0
1
Ψ(B(1/ )) x∈Rn:Iα (x)+Mα,B (x)>
CbBMO 
≤Csup
>0
1
Ψ(B(1/ )) x∈Rn:Iα (x)>
CbBMO 
+Csup
>0
1
Ψ(B(1/ )) x∈Rn:Mα,B (x)>
CbBMO 
.
By (5.2) and (3.2), and since Ψ and Ba e submul iplica i e,
≤Csup
>0
1
Ψ(B(1/ )) bBMO
Rn| (x)|dxn/(n−α)
+Csup
>0
1
Ψ(B(1/ ))ΨRn
BbBMO | (x)|
dx
≤Csup
>0
1
Ψ(B(1/ )) ·1
n/(n−α)RnbBMO | (x)|dxn/(n−α)
+Csup
>0
1
Ψ(B(1/ )) ·Ψ(B(1/ ))ΨRn
B(bBMO | (x)|)dx
=J1+J2.
No e ha
sup
>0
1
Ψ(B(1/ )) ·1
n/(n−α)≤C,(6.2)
since
1
Ψ(B(1/ )) ·1
n/(n−α)=1
log(e+1/ ) log e+ [(1/ ) log(e+1/ )]α/n n
n−α
is con inuous and has ﬁni e limi s as →0, →∞(0 and 1 espec i ely).
Commu a o s o F ac ional In eg als 125
Fu he mo e, since ≤B( ) and n/(n−α)≤Ψ( ), we ha e ha
RnbBMO | (x)|dxn/(n−α)
≤ΨRn
B(bBMO | (x)|)dx.(6.3)
F om (6.2) and (6.3) we ge ha
J1+J2≤CΨRn
B(bBMO | (x)|)dx,
which p o es (6.1).
We will now show ha (1.1) is sha p, in he sense ha i we can
eplace Ψ by Ψ0, hen o all >0, Ψ( /γ)≤KΨ0( ). To do so, we
will adap he a gumen in Rema k 3.15, which explo ed sha pness in
Theo em 3.3. Le n=1and 0 <α<1, ﬁx x>0 and le Nbe such ha
0<N<x; he exac alue o Nwill be chosen below. Le =χ[0,N].
Fi s , we show ha he e is a cons an K, depending only on αsuch
ha o all y>N,
Mα,B (y)≤Kyα
B−1(y/N).(6.4)
(As we showed in Rema k 3.15, he opposi e inequali y holds wi h con-
s an 1.) To see his, no e ha since is a non-inc easing unc ion
on [0,∞),
Mα,B (y)=sup
z≥y
zα B,[0,z]= sup
z≥y
zα
B−1(z/N).
The e o e, i will suﬃce o show ha he e exis s K>0, such ha i
z≥y,
zα
B−1(z/N)≤Kyα
B−1(y/N).
Le H( )= α/B−1( /N); since o all >0, B−1( )≈ / log(e+ ), we
ha e ha
H(z)
H(y)≤C(z/N)α−1log(e+z/N)
(y/N)α−1log(e+y/N).
The unc ion log(e+ )/ 1−αis ei he dec easing o has a unique local
maximum on [1,∞); i ollows, he e o e, ha he igh hand e m is dom-
ina ed by a cons an which depends only on α. This es ablishes (6.4).
Now le b(y)=log(e+y/N)χ(N,∞)(y). Since he BMO no m is
dila ion in a ian , he BMO no m o bdoes no depend on N.Fo all z,
126 D. C uz-U ibe, SFO, A. Fio enza
Iα(b )(z)=0,so o all ysuch ha N<y<x,
|[b, Iα] (y)|=b(y)Iα (y)
= log(e+y/N)N
0
dz
|z−y|1−α
≥Nlog(e+y/N)
y1−α
≥cyα
B−1(y/N);
by inequali y (6.4),
≥cMα,B (y)
>cxα
B−1(x/N).
The cons an cdepends only on α.
We now conside wo cases, depending on whe he xis la ge o
small. Suppose ﬁ s ha xis such ha cxα>B
−1(2), and le N=
x/B(cxα)<x/2. Then he abo e inequali y shows ha o all y∈
[x/2,x], [b, Iα] (y)>1. Hence, i inequali y (1.1) holds o some in-
c easing unc ion Ψ0 hen we ha e ha
x/2≤|{y∈R:|[b, Iα] (y)|>1}|
≤CΨ0R
B(bBMO (y)) dy
≤CΨ0(NB(bBMO )).
On he o he hand, he e is a cons an γ, depending only on α, such
ha
B(bBMO )N=B(bBMO )x
B(cxα)≤γΦ(c1/αx),
whe e Φ = Ψ−1. The e o e, i se he igh hand side equal o and sol e
o x,wege ha
c1/αx=Ψ( /γ).
I we combine his wi h he inequali y abo e we ge ha o all suﬃ-
cien ly la ge, Ψ( /γ)≤KΨ0( ).
We will now show ha he same inequali y holds o all suﬃcien ly
small. Fix x>0 small and le 0 <N<x/2; he exac alue o Nwill be
Commu a o s o F ac ional In eg als 127
ﬁxed below. Then, by he abo e a gumen we ha e ha i N<y<x,
[b, Iα] (y)≥cxα
B−1(x/N).
The e o e, i we ﬁx usuch ha
2u=cxα
B−1(x/N),
hen a guing as abo e and as in Rema k 3.15, we ha e ha
x/2≤CΨ0R
BbBMO
(y)
udy
=CΨ0NBbBMO
2B−1(x/N)
cxα.
Again a guing as in Rema k 3.15 we can choose Nsuﬃcien ly small such
ha
≤CΨ0(CB(bBMO )x1−α)
≤CΨ0(CB(bBMO )Φ(x)).
We can now a gue as we did abo e o ge ha o all suﬃcien ly small,
Ψ( /γ)≤KΨ0( ). This comple es ou p oo .
7. P oo o Theo em 1.6 and Cons uc ion o
Example 1.8
Ou p oo o Theo em 1.6 equi es wo ac s which we gi e as a lemma.
Lemma 7.1. The ollowing a e ue:
(1) I w∈Ap o some p>1, hen o all q>0, he e exis s a
cons an Cqsuch ha
Rn
(Md )qwdx≤CqRn
(M# )qw dx.
(2) Gi en α,0<α<n,p,1<p<n/α, le 1/q =1/p −α/n. Then
i w∈Apq,
Rn|Iα |qwqdx1/q
≤CRn| |pwpdx1/p
.
The ﬁ s inequali y is due o Jou n´e[15]; also see [10,p.144]. The
second is due o Muckenhoup and Wheeden [18].