Michigan Ma h. J. 49 (2001)
Sha p Weigh ed Endpoin Es ima es o
Commu a o s o Singula In eg als
Ca los Pé ez & Gladis P adolini
1. In oduc ion and S a emen s o he Main Resul
The main pu pose o his pape is o imp o e he main esul in [P2] by means
o a di ec p oo ha a oids he classical good-λ echnique conside ed he e. The
good-λme hod, in oduced by Bu kholde and Gundy in [BG], is a powe ul ool
bu has hedisad an age ha i isessen iallyadap ed omeasu essa is ying heA∞
condi ion, such as he Lebesgue measu e. The app oach we conside he e is mo e
ela ed o he classical a gumen o Calde ón and Zygmund o p o ing ha sin-
gula in eg al ope a o s sa is y he weak- ype (1,1)-p ope y, an app oach whose
ad an ageis ha i allows us o conside mo e gene al measu e. The me hod, how-
e e , mus be di e en because commu a o s o singula in eg al ope a o s wi h
BMO unc ions a e no o weak- ype (1,1), as shown in [P2].
Le bbe a locally in eg able unc ion on Rn,usually called he symbol, and le
Tbe a Calde ón–Zygmund singula in eg al ope a o (see [C] o [J]). Conside
he commu a o ope a o [b, T ] de ined o , say, smoo h unc ions by
[b, T ] =bT ( ) −T(b ). (1)
A now classical esul o Coi man, Rochbe g, and Weiss [CRW] s a es ha [b, T ]
is a bounded ope a o on Lp(Rn), 1<p<∞,when bis a BMO unc ion. In
ac , BMO is also a necessa y condi ion o he commu a o [b, R] o be bounded
on Lp(Rn), whe e R=(R1,...,Rn)is he ec o - alued Riesz ans o m. We
will always assume ha b∈BMO(Rn)unless o he wise no ed.
None o he di e en p oo s o his esul ollows he usual scheme o he clas-
sical Calde ón–Zygmund heo y o singula in eg al ope a o s T. Indeed, he key
esul in his heo y is ha any o hese ope a o s sa is ies he weak- ype (1,1)-
p ope y, which is de i ed om he assump ion ha Tis bounded on L2(Rn)com-
bined wi h a mild egula i y o he ke nel. Once he weak- ype (1,1)-inequali y
is ob ained, in e pola ion and duali y yield he boundedness o he ope a o on
Lp(Rn) o all 1 <p<∞.Howe e , simple examples show ha commu a o s
(wi h BMO symbols) ail o be o weak- ype (1,1), as ound in [P2]. To emedy
Recei ed May 3, 2000. Re ision ecei ed Oc obe 11, 2000.
This esea ch was ca ied ou du ing a s ay a he Uni e sidad Au ónoma de Mad id. Resea ch o he
i s au ho was pa ially suppo ed by DGESIC g an PB98-0106, Spain. Resea ch o he second
au ho was suppo ed by Uni e sidad Nacional del Li o al (FOMEC) and CONICET, República
A gen ina.
23
24 Ca los Pé ez & Gladis P adolini
he si ua ion, i is shown he e ha commu a o s sa is y a “L(logL)”- ype es i-
ma e. To be p ecise, we ha e he ollowing esul .
Theo em 1.1 [P2]. Le Tbe any Calde ón–Zygmund singula in eg al ope a-
o . Then he e exis s a posi i e cons an Cdepending upon he BMO no m o b
such ha , o all unc ions and all λ>0,
|{y∈Rn:|[b, T ] (y)|>λ}| ≤ CZRn
| (y)|
λ1+log+| (y)|
λdy. (2)
Thep oo o his es ima eisbased on showing ha he e is anin ima e ela ionship
be ween commu a o s and i e a ions o he Ha dy–Li lewood maximal unc ion
(in his case, M2=MBM) ia he good-λ echnique o Bu kholde and Gundy
[BG]. Mo e p ecisely, i we le 8( ) = (1+log+ ) and le wbe a weigh sa is-
ying he A∞condi ion, hen he e exis s a posi i e cons an Cdepending on he
BMO cons an o bsuch ha , o any smoo h unc ion wi h compac suppo ,
sup
>0
1
8(1/ )w({y∈Rn:|[b, T ] (y)|> })
≤C[w]A∞sup
>0
1
8(1/ )w({y∈Rn:M2 (y)> }). (3)
Using his es ima e wi h w=1 and analyzing he beha io o M2,we ob ain he
desi ed es ima e (2)—whe e, in ac , he Lebesgue measu e can be eplaced by
any weigh unc ion sa is ying he A1condi ion. The Lp e sions o hese es i-
ma es and hei consequences a e u he exploi ed in [P3].
As men ioned be o e, we p o ide a di e en p oo o (2) whose ad an age is
ha i allows us o de i e a sha p wo-weigh inequali y ha is simila in spi i o
he ollowing one o Calde ón–Zygmund singula in eg al ope a o s.
Theo em 1.2 [P1]. Le Tbe any Calde ón–Zygmund ope a o and le ε>0.
Then, o any weigh w, unc ion , and >0, he e is a cons an Cεsuch ha
w({x∈Rn:|T (x)|> })≤Cε
ZRn
| (x)|ML(logL)ε(w)(x) dx. (4)
The poin he e is ha no assump ion on he weigh is assumed. He e MA=MA(L)
deno es a maximal- ype unc ion de ined by he exp ession
MA(L) (x) =sup
Q3x
k kA,Q,
whe e Ais anyYoung unc ion and k kA,Qdeno es he A-a e age o e Qde ined
by means o he Luxembou g no m
k kA,Q=in λ>0: 1
|Q|ZQ
A| |
λdx ≤1.(5)
Fo ou applica ions, he main examples a e gi en by A( ) = (1+log+ )α,
α≥0.
We will conside a mo e gene al e sion o (1) deno ed by Tm
b(m =0,1,2,...)
and usually called highe -o de commu a o s. They a e de ined by he o mula
Sha p Weigh ed Endpoin Es ima es o Commu a o s o Singula In eg als 25
Tm
b=[b,...,[b, T ]]
|{z }
(m imes)
;
in he pa icula case o Calde ón–Zygmund ope a o s, hey can be exp essed by
means o i s ke nel K:
Tm
b (x) =ZRn
(b(x) −b(y))mK(x,y) (y)dy,
whe e is an app op ia e es unc ion. As usual, we assume ha he ke nel K
sa is ies he so-called s anda d es ima es (c . [C] o [J]).
Ou esul is he ollowing.
Theo em 1.3. Le Tbe a Calde ón–Zygmund singula in eg al ope a o , and le
b∈BMO and ε>0.Then he e exis s a posi i e cons an Csuch ha
w({x∈Rn:|Tm
b (x)|>λ})
≤CZRn
8mkbkm
BMO
| (x)|
λML(logL)m+ε(w)(x) dx, (6)
whe e 8m( ) = (1+log+ )m.The cons an Cis independen o he weigh w,
he unc ion , and λ>0.
Obse e ha he eisno es ic ionon heclasso weigh sconside ed. Obse ealso
ha , since 8mis submul iplica i e (i.e., 8m(ab) ≤C8m(a)8m(b) wi h a, b ≥
0), we ha e
w({x∈Rn:|Tm
b (x)|>λ})
≤C8m(kbkm
BMO)ZRn
8m| (x)|
λML(logL)m+ε(w)(x) dx.
Inequali ies simila o (6) ha e u ned ou o be e y use ul in he s udy o he
wo-weigh p oblem o singula in eg alope a o s (see [CP1; CP2]). On he o he
hand, i would be in e es ing o know whe he o no his inequali y holds when
ε=0.
2. Some P elimina ies and No a ion
In his sec ion we summa ize a ew ac s abou O licz spaces. (Fo mo e in-
o ma ion, see Benne and Sha pley [BS] o Rao and Ren [RR].) A unc ion
B:[0,∞)→[0,∞)isadoublingYoung unc ion i :
(a) i is con inuous, con ex, and inc easing;
(b) B(0)=0 and B( ) →∞as →∞;and
(c) i sa is ies B(2 ) ≤CB( ) o all >0.
Fo O licz no ms we a e usually conce ned abou he beha io o Young unc ions
o la ge. Gi en wo unc ions Band C, we w i e B( ) ∼
=C( ) i B( )/C( ) is
bounded and bounded below o ≥c>0.
Recall ha we de ined he localized Luxembou g no m by equa ion (5); an
equi alen no m ha is o en use ul in calcula ions is due o K asnosel’ski˘ı and
Ru icki˘ı [KR, p. 92] (also see [RR, p. 69]):
26 Ca los Pé ez & Gladis P adolini
k kA,Q≤in
µ>0µ+µ
|Q|ZQ
A| |
µdx≤2k kA,Q.(7)
Gi en aYoung unc ion A, we use ¯
A o deno e he complemen a yYoung unc-
ion associa ed o A;i has he p ope y ha , o all >0,
≤A−1( ) ¯
A−1( ) ≤2 .
The basic p ope y ha we will use is he ollowing gene alized Hölde inequali y:
1
|Q|ZQ
| g|≤2k kA,Qkgk¯
A,Q.(8)
In pa icula , we shall wo k wi h A( ) = (1+log+ )m,m=1,2,..., wi h
maximal unc ion deno ed by ML(logL)m.The complemen e y Young unc ion is
gi en by ¯
A( ) ≈exp( 1/m), wi h he co esponding maximal unc ion deno ed by
Mexp L1/m .
The i s gene alizedYoung inequali y s a es ha A−1( ) ·B−1( ) ≤C−1( ) o
>0;i ollows ha
C(s ) ≤A(s) +B( ) (9)
holds o all s, > 0.
3. P oo o he Theo em
In his sec ion we p o e Theo em 1.3 by induc ion om he case m=1.We will
use he ollowing s ong- ype e sion o ou es ima e de i ed in [P3].
Theo em 3.1 [P3]. Le Tbe any Calde ón–Zygmund singula in eg al ope a-
o , and le 1<p<∞and b∈BMO.Then o each δ>0 he e exis s a posi i e
cons an C=Cδsuch ha , o all unc ions g,
ZRn
|Tm
bg(x)|pw(x) dx
≤Cδkbkmp
BMO ZRn
|g(y)|pML(logL)(m+1)p−1+δ(w)(y) dy. (10)
3.1. The Case m=1
A simple homogenei y shows ha we may assume kbkBMO =1.Gi en ha as-
sump ion, we need only show ha
w({x∈Rn:|[b, T ] (x)|>λ})≤CZRn
8| (x)|
λML(logL)1+ε(w)(x) dx,
whe e 8( ) =81( ) = (1+log+ ).
We conside he s anda dCalde ón–Zygmund decomposi iono a le elλand
ob ain a collec ion o dyadic non-o e lapping cubes Qj=Qj(xQj,
j) ha sa is y
λ< 1
|Qj|ZQj
| |≤2nλ. (11)
Sha p Weigh ed Endpoin Es ima es o Commu a o s o Singula In eg als 27
We se =λ=SjQj; hen | (x)|≤λa.e. x∈Rn . We w i e =
g+h, whe e gis de ined by
g(x) = (x) i x∈Rn ,
Qji x∈Qj.
As usual, we use he no a ion Q=1
|Q|RQ o a locally in eg able unc ion
and a cube Q. Obse e ha |g(x)|≤2nλa.e.
We spli he “bad pa ” has h=Pjhj,whe e hj(x) =( (x) − Qj)χQj(x).
We will use he no a ion w∗(x) =w(x)χRn ˜
(x) and wj(x) =w(x)χRn 3Qj,
whe e ˜
Qj=3Qjand ˜
=Sj˜
Qj.Then
w({x∈Rn:|[b, T ] (x)|>λ})
≤w({x∈Rn ˜
:|[b, T ]g(x)|>λ/2})+w( ˜
)
+w({x∈Rn ˜
:|[b, T ]h(x)|>λ/2})
=I+II+III.
As we will see om he p oo , pa I (p ecisely he piece associa ed o he “good
pa ”) is he one ha ca ies a highe deg ee o singula i y. Now we use Theo em
3.1, wi h m=1and wi h p, δ such ha 1 <p<1+ε/2 and δ=ε−2(p −1)>
0.Then
I≤C
λpZRn
|[b, T ]g(x)|pw∗(x)dx
≤C
λpZRn
|g(x)|pML(logL)1+ε(w∗)(x) dx
≤C
λZRn
|g(x)|ML(logL)1+ε(w∗)(x) dx
=C
λZRn 
| (x)|ML(logL)1+ε(w)(x) dx +Z
|g(y)|ML(logL)1+ε(w∗)(y) dy.
I is clea ha we need only es ima e he second e m in he las exp ession; o do
so, we use he ollowing ac :
Fo a bi a yYoung unc ionA, nonnega i emeasu ewwi hMAw(x) <
∞a.e., cube Q, and R>1,we ha e
MA(χRn RQw)(y) ≈MA(χRn RQw)(z)
o each y,z ∈Q;hence
MA(χRn RQw)(y) ≈in
y∈QMA(χRn RQw)(y) (12)
o each y∈Q.
This is an obse a ion whose p oo ollows exac ly as o he case o he Ha dy–
Li lewood maximal ope a o M, which co esponds o he case A( ) = (see e.g.
[GR, p. 159]).

28 Ca los Pé ez & Gladis P adolini
Hence we can con inue es ima ing he second e m wi h
Z
|g(x)|ML(logL)1+ε(w∗)(x) dx
≤X
jZQj
| Qj|ML(logL)1+ε(wj)(x) dx
=X
jZQj
| (x)|dx1
|Qj|ZQj
ML(logL)1+ε(wj)(x) dx
≤CX
jZQj
| (x)|dxin
Qj
ML(logL)1+ε(wj)
=CX
jZQj
| (x)|ML(logL)1+ε(w)(x) dx
≤CZRn
| (x)|ML(logL)1+ε(w)(x) dx.
Fo IIweha e
II =w( ˜
) ≤CX
j
w( ˜
Qj)
|˜
Qj||Qj|≤C
λX
j
w( ˜
Qj)
|˜
Qj|ZQj
| (x)|dx
≤C
λX
jZQj
| (x)|Mw(x) dx
≤C
λZRn
| (x)|Mw(x) dx.
Obse e ha his pa is smoo he han I since we ob ain a smalle ope a o Mon
he igh -hand side. Simila ly, pa III is smoo he han I bu oughe han II, as
we now show. Indeed, i s no e ha
[b, T ]h(x) =X
j
[b, T ]hj(x)
=X
j
(b(x) −bQj)T hj(x) −X
j
T ((b −bQj)h
j)(x),
whe e (as be o e) bQ=1
|Q|RQb. Then
III ≤wx∈Rn ˜
:X
j
(b(x) −bQj)T hj(x)>λ
4
+wx∈Rn ˜
:X
j
T ((b −bQj)h
j)(x)>λ
4
=A+B.
Sha p Weigh ed Endpoin Es ima es o Commu a o s o Singula In eg als 29
Using he s anda d es ima es o he ke nel K, we ha e
A≤C
λZRn ˜
X
j
|b(x) −bQj||Th
j(x)|w(x) dx
≤C
λX
jZRn 3Qj
|b(x) −bQj|w(x) ZQj
|hj(y)||K(x −y) −K(x −xQj)|dy dx
≤C
λX
jZQj
|hj(y)|ZRn 3Qj
|K(x −y) −K(x −xQj)||b(x) −bQj|wj(x)dx dy
≤C
λX
jZQj
|hj(y)|
∞
X
k=1
Z2k j≤|x−xQj|<2k+1 j
|y−xQj|
|x−xQj|n+1|b(x) −bQj|wj(x)dx dy
≤C
λX
jZQj
|hj(y)|dy∞
X
k=1
2−k
(2k+1 j)nZ|x−xQj|<2k+1 j
|b(x) −bQj|wj(x)dx.
To con ol he sum on k, we use s anda d es ima es oge he wi h he gene alized
Hölde inequali y and he John–Ni enbe g heo em. Indeed, i y∈Qj hen we
ha e
∞
X
k=1
2−k
(2k+1 j)nZ|x−xQj|<2k+1 j
|b(x) −bQj|wj(x)dx
≤C
∞
X
k=1
2−k
(2k+1 j)nZ2k+1Qj
|b(x) −b2k+1Qj|wj(x)dx
+
∞
X
k=1
2−k
(2k+1 j)nZ2k+1Qj
|b2k+1Qj−bQj|wj(x)dx
≤C
∞
X
k=1
2−kkb−b2k+1QjkexpL,2k+1QjkwjkLlogL,2k+1Qj
+
∞
X
k=1
2−k(k +1)M(wj)(y)
≤CML(logL)(wj)(y)
∞
X
k=1
2−k+M(w
j)(y)
∞
X
k=1
2−k(k +1)

≤CMLlogL(wj)(y).
Then we can con inue he es ima e o Ausing (12) as ollows:
30 Ca los Pé ez & Gladis P adolini
A≤C
λX
jZQj
|hj(y)|MLlogL(wj)(y) dy
≤C
λX
jZQj
| (y)|MLlogL(w)(y) dy +X
jZQj
| Qj|MLlogL(wj)(y) dy
≤C
λZRn
| (y)|MLlogL(w)(y) dy
+X
jZQj
| (x)|dx 1
|Qj|ZQj
MLlogL(wj)(y) dy
≤C
λZRn
| (y)|MLlogL(w)(y) dy +X
jZQj
| (x)|MLlogL(w)(x) dx
≤C
λZRn
| (y)|MLlogL(w)(y) dy.
To es ima e B, we combine inequali y (4) o singula in eg als oge he wi h
(again) obse a ion (12):
B=w∗x∈Rn:TX
j
(b −bQj)h
j(x)>λ
4
≤C
λZRnX
j
(b(x) −bQj)h
j)(x)ML(logL)ε(w∗)(x) dx
≤C
λX
jZQj
|b(x) −bQj|| (x)− Qj|ML(logL)ε(wj)(x) dx
≤C
λX
j
in
Qj
ML(logL)ε(wj)(x)ZQj
|b(x) −bQj|| (x)|dx
+ZQj
|b(x) −bQj|| Qj|dx
=B1+B2.
The es ima e o B2is simple since, by (12),
B2=C
λX
j
in
Qj
ML(logL)ε(wj)(x) ZQj
|b(x) −bQj|| Qj|dx
≤C
λX
j
1
|Qj|ZQj
|b(x) −bQj|ZQj
| (x)|ML(logL)ε(wj)(x) dx
≤CZRn
| (x)|ML(logL)ε(w)(x) dx.
Sha p Weigh ed Endpoin Es ima es o Commu a o s o Singula In eg als 31
Fo B1we ha e, by he gene alized Hölde inequali y (8),
B1=C
λX
j
in
Qj
ML(logL)ε(wj)(x) ZQj
|b(x) −bQj|| (x)|dx
≤C
λX
j
in
Qj
ML(logL)ε(wj)(x)|Qj|k kLlogL,Qj.
Now, combining o mula (7) wi h (11) and ecalling ha 8( ) = (1+log+ ),
we ha e
1
λ|Qj|k kLlogL,Qj≤1
λ|Qj|in
µ>0µ+µ
|Qj|ZQj
8| (x)|
µdx
≤|Qj|+ZQj
8| (x)|
λdx
≤1
λZQj
| (x)|dx +ZQj
8| (x)|
λdx
≤2ZQj
8| (x)|
λdx.
Then
B1≤CZQj
8| (x)|
λML(logL)ε(wj)(x) dx
≤CZRn
8| (x)|
λML(logL)ε(w)(x) dx.
This concludes he p oo o he case m=1.
3.2. The Gene al Case
We will use an induc ion a gumen and will omi some echnical a gumen s ha
a e simila o he case m=1.Again, a simply homogenei y a gumen using ha
Tm
b( /kbkm
BMO)=Tm
b/kbkBMO( ) shows ha we may assume kbkBMO =1.We con-
side again he Calde ón–Zygmund decomposi ion o a le el λ. Then, wi h he
same no a ion as in he p oo o he case m=1,we ha e
w({y∈Rn:|Tm
b (y)|>λ})≤w({y∈Rn ˜
:|Tm
bg(y)|>λ/2})+w( ˜
)
+w({y∈Rn ˜
:|Tm
bh(y)|>λ/2})
=I+II+III.
F om (10) wi h pand δsuch ha
1<p<1+ε/(m +1)and δ=ε−(m +1)(p −1)>0,