Weighted inequalities and vector-valued Calderón-Zygmund operators on non-homogeneous spaces

Publicacions Ma em`a iques, Vol. 44 (2000), 613–640
WEIGHTED INEQUALITIES AND VECTOR-VALUED
CALDER´
ON-ZYGMUND OPERATORS ON
NON-HOMOGENEOUS SPACES
J. Ga c´
ıa-Cue a and J. M. Ma ell
Abs ac
Recen ly, F. Naza o , S. T eil and A. Volbe g (and independen ly
X. Tolsa) ha e ex ended he classical heo y o Calde ´on-Zygmund
ope a o s o he con ex o a “non-homogeneous” space (X,d,µ),
whe e, in pa icula , he measu e µmay be non-doubling. In he
p esen wo k we s udy weigh ed inequali ies o hese ope a o s.
Specifically, o 1 <p<∞, we iden i y sufficien condi ions o
he weigh on one side, which gua an ee he exis ence o ano he
weigh in he o he side, so ha he weigh ed Lpinequali y holds.
We deal wi h his p oblem by de eloping a ec o - alued he-
o y o Calde ´on-Zygmund ope a o s on non-homogeneous spaces
which is in e es ing in i s own igh . Fo he case o he Cauchy
in eg al ope a o , which is he mos impo an example, we e en
p o e ha he condi ions o he weigh s a e also necessa y.
1. In oduc ion
Le µbe a Bo el measu e in he complex plane. The Cauchy in eg al
ope a o is defined as
C (z)=Cµ (z)=C
(ξ)
z−ξdµ(ξ), o µ-a.e. z∈C supp .
I is na u al o wonde whe he his ope a o is bounded on L2(µ), on
Lp(µ)o e enbe weenL1(µ) and L1,∞(µ). Besides, since he p e i-
ous defini ion makes no sense o poin s in he suppo o he unc ion,
2000 Ma hema ics Subjec Classifica ion. 42B20, 30E20.
Key wo ds. Non-doubling measu es, Calde ´on-Zygmund ope a o s, ec o - alued in-
equali ies, weigh s, Cauchy in eg al.
Bo h au ho s a e pa ially suppo ed by DGES Spain, unde G an PB97-0030.
We would like o hank J. L. To ea o many commen s and sugges ions.
614 J. Ga c´
ıa-Cue a, J. M. Ma ell
ano he ques ion is o find condi ions on µin o de o ensu e he ex-
is ence o he p incipal alues on hese spaces. Fo example, when µ
is he one-dimensional Hausdo ff measu e o e a Lipschi z cu e, he
boundedness was p o ed in [Cal] o small Lipschi z cons an and he
ull esul was ob ained in [CMM]. Ano he app oach o his p oblem
is he T(b) heo em p o ed in [DJS], (see also [Da1]). Fo measu es
o e ec ifiable se s and he ela ion wi h analy ic capaci y see [Ch1],
[Ch2], [Mu ] and he e e ences gi en he e. See also he ecen su -
ey [Da2]. The answe o gene al measu es has been ob ained by Tolsa
in [To1], [To2]. In he fi s wo k, i is es ablished he equi alence o
he uni o m boundedness o he unca ed Cauchy in eg als in L2(µ)
and some geome ic condi ions on he measu e µ, namely: µhas lin-
ea g ow h —which means ha he measu e o each ball is con olled
by a cons a imes he adius— and i sa isfies ce ain local cu a u e
condi ion (see [To1], [Mel], [MV], [MMV]). In he second e e ence,
he au ho ob ains ha he boundedness in L2(µ) implies he exis ence
o p incipal alues. Besides, hose measu es, o which he exis ence o
p incipal alues holds, a e comple ely cha ac e ized.
In[NTV1]aT(1) heo em is p o ed o Calde ´on-Zygmund ope a o s
in Cwi h a measu e such ha µ(Q)≤(Q) o all squa es Q⊂C, whe e
(Q) s ands o he side leng h o Q. They p o e ha Tis con inuous
in L2(µ) i and only i Tand i s adjoin a e bounded o e cha ac e is ic
unc ions o squa es. (Ac ually, as i is poin ed ou in [NTV1], simila
esul s wo k o “n-dimensional” measu es in Rd,d≥nand Calde ´on-
Zygmund ope a o s wi h “n-dimensional” ke nels K.) In pa icula
o he Cauchy in eg al, his esul has been also ob ained by [Ve ].
In[NTV2] a gene aliza ion o his se ing is gi en. They deal wi h non-
homogeneous spaces which a e me ic spaces endowed wi h a posi i e
measu e in such a way ha he measu e o a ball is con olled by he a-
dius o he powe n, whe e n>0 is a fixed eal numbe . In hese spaces
(whe e he measu e is no assumed o sa is y any doubling condi ion)
om he L2(µ) boundedness, he au ho s manage o ob ain weak and
s ong ype es ima es o Calde ´on-Zygmund ope a o s and o he max-
imal ope a o s associa ed wi h hem. The main example is he Cauchy
in eg al whe e he me ic space is Cand n=1.
A non-homogeneous space (X,d) will be a sepa able me ic space en-
dowed wi h a non-nega i e “n-dimensional” Bo el measu e µ, ha is,
µ(B(x, )) ≤ n, o all x∈X, >0,
whe e B(x, )={y∈X:d(x, y)≤ }and nis a fixed posi i e numbe
(no necessa ily an in ege ).
Two-Weigh Theo y on Non-Homogeneous Spaces 615
Defini ion 1.1. A bounded linea ope a o Ton L2(µ) is said o be a
Calde ´on-Zygmund ope a o wi h “n-dimensional” ke nel Ki o e e y
∈L2(µ),
T (x)=X
K(x, y) (y)dµ(y), o µ-almos e e y x∈X supp ,
whe e, o some A>0, K:X×X−→ Csa isfies
(i) |K(x, y)|≤ A
d(x, y)n, o all x=y;
(ii) and he ollowing wo condi ions hold:
d(x,y)≥2d(x,x)|K(x, y)−K(x,y)|dµ(y)≤A,
d(x,y)≥2d(x,x)|K(y,x)−K(y,x)|dµ(y)≤A.
No e ha his class o ope a o s is sligh ly la ge han hose conside ed
in [NTV2] whe e poin wise es ima es o he ke nel a e used a he
han in eg al es ima es. Obse e ha i we ake some measu e µin C
such ha he Cauchy in eg al is bounded in L2(µ), µwill ha e linea
g ow h (e.g. [To1]), ha is, µis “1-dimensional”. In his case, he
Cauchy in eg al is a Calde ´on-Zygmund ope a o wi h “1-dimensional”
ke nel K(z,ξ)= 1
z−ξ.
The aim o his pape is o ob ain some weigh ed inequali ies o hese
ope a o s. I 1 <p<∞, conside he ollowing wo-weigh inequali y
o T:X|T (x)|pu(x)dµ(x)≤X| (x)|p (x)dµ(x),(1)
o ∈Lp( )=Lp( dµ) and whe e u, a e µ-a.e. posi i e unc ions.
These inequali ies in Rdwi h he same weigh in bo h sides ha e
been ecen ly s udied by [OP]. They ha e ob ained some esul s abou
Muckenhoup weigh s and weigh ed inequali ies o Calde ´on-Zygmund
ope a o s. Howe e , we a e in e es ed in a diffe en ype o inequali ies,
namely, we shall be conce ned wi h he ollowing p oblem:
Find condi ions on 0≤ <∞µ-a.e. ( esp. u>0
µ-a.e.) such ha (1) is sa isfied by some u>0µ-a.e.
( esp. 0≤ <∞µ-a.e.).
616 J. Ga c´
ıa-Cue a, J. M. Ma ell
As we can see in Chap e VI o [GR] and in Chap e s II, V o [S e]
his ques ion is closely ela ed o ob aining ec o - alued inequali ies
o T. We shall use his connec ion o ge an answe o his p oblem,
ha is, we shall de elop a ec o - alued heo y o hese ope a o s o
ob ain he necessa y ec o - alued inequali ies. Some e e ences abou
classical ec o - alued heo y a e [BCP], [RRT] and [GR].
Fo hese ope a o s he ele an classes o weigh s will be, as usual,
Dpand Zp,1<p<∞, which a e defined as ollows:
Dp=0≤w<∞µ-a.e. : X
w(x)1−p(1 + d(x, x0))−np
dµ(x)<∞
Zp=w>0µ-a.e. : X
w(x)(1+d(x, x0))−npdµ(x)<∞,
o some x0∈X. No e ha hese classes o weigh s do no depend on
he poin x0and ha his defini ion becomes simple o spaces wi h
fini e diame e (see Sec ion 3). The conc e e esul is
Theo em. Take p,1<p<∞.I u∈Zp( esp. ∈Dp), hen he e
exis s some weigh 0< <∞µ-a.e. ( esp. 0<u<∞µ-a.e.) such
ha (1) holds. Mo eo e , ( esp. u) can be ound in such a way ha
α∈Zp( esp. uα∈Dp), p o ided ha 0<α<1.
Once we ha e ob ained sufficien condi ions on he weigh s in o de
o ensu e ha (1) holds, we shall s udy how sha p a e hese classes,
ha is, we shall p o e ha o a pa icula example hese condi ions a e
also necessa y. In [GR] his p oblem is ea ed o classical Calde ´on-
Zygmund ope a o s in Rn. The e he Riesz ans o ms a e used o show
ha hose classes o weigh s a e necessa y. In ou se ing his ˆole will be
played by he Cauchy in eg al. Take a measu e µ o which he Cauchy
in eg al is bounded in L2(µ), see [To1]. Now, he weigh ed inequali y is
C|C (z)|pu(z)dµ(z)≤C(u, )C| (z)|p (z)dµ(z),(2)
o any ∈Lp( dµ). We de o e Sec ion 4 o ge he ollowing heo em,
which is, essen ially, he con e se o he p e ious one.
Theo em. Take p,1<p<∞. Gi en 0<u<∞µ-a.e. ( esp. 0< <
∞µ-a.e.), i he e exis s some weigh 0< <∞µ-a.e. ( esp. 0<u<
∞µ-a.e.) such ha (2) holds, hen u∈Zp( esp. ∈Dp).
Two-Weigh Theo y on Non-Homogeneous Spaces 617
The plan o he pape is he ollowing. Sec ion 2 con ains a ec o -
alued e sion o he main heo em in [NTV2], which shall be p o ed
in h ee s eps. In Subsec ions 2.1 and 2.2 we shall ob ain he weak
ype (1,1) es ima e, whe eas he s ong inequali ies a e conside ed in
Subsec ion 2.3. An immedia e consequence is gi en in Subsec ion 2.4,
indeed he ec o - alued inequali ies ob ained he e will be he main ool
o sol ing he p oblem we a e conce ned wi h. Sec ions 3, 4 a e de o ed
o his p oblem: he fi s one o gene al ope a o s and he second one
o he pa icula case o he Cauchy in eg al, whe e he necessi y is
p o ed.
2. The ec o - alued heo em
Th oughou his sec ion we shall conside ec o - alued ope a o s,
ha is, ope a o s which ake hei alues in Banach spaces.
Le A,Bbe a couple o Banach spaces. L(A,B) will deno e he se o
bounded linea ope a o s om A o B. We shall say ha K:X×X−→
L(A,B) is a ( ec o - alued) “n-dimensional” Calde ´on-Zygmund ke nel
i , o some A>0, i e ifies
(i) K(x, y)L(A,B)≤A
d(x, y)n, o all x=y;
(ii) and he ollowing wo condi ions hold:
d(x,y)≥2d(x,x)K(x, y)−K(x,y)L(A,B)dµ(y)≤A,
d(x,y)≥2d(x,x)K(y,x)−K(y,x)L(A,B)dµ(y)≤A.
Defini ion 2.1. Le Tbe a linea ope a o mapping boundedly L2
A(µ)
in o L2
B(µ), such ha , o any ∈L2
A(µ),
T (x)=X
K(x, y) (y)dµ(y), o µ-a.e. x∈X supp ,
whe e Kis an “n-dimensional” Calde ´on-Zygmund ke nel. Then we shall
say ha Tis a ec o - alued Calde ´on-Zygmund ope a o .
Fo >0, he unca ed ope a o s a e defined as ollows
T (x)=X B(x, )
K(x, y) (y)dµ(y),
and we can conside he maximal ope a o associa ed wi h T,
T (x) = sup
>0T (x)B.

618 J. Ga c´
ıa-Cue a, J. M. Ma ell
Fo 1 ≤p≤∞, i is well known ha Lp
A∗(µ)⊂(Lp
A(µ))∗.I
he Banach space Ais eflexi e equali y holds, howe e i ails in gen-
e al. When we deal wi h a eflexi e Banach space A, we can define
T∗, he adjoin o T, which u ns ou o be a ec o - alued Calde ´on-
Zygmund ope a o ha maps boundedly L2
B∗(µ)in oL2
A∗(µ). The ke nel
is 
K(x, y)=K(y,x)∗∈L(B∗,A∗) ( he adjoin ope a o o K(y,x)). Be-
sides, T∗L2
B∗(µ)→L2
A∗(µ)≤TL2
A(µ)→L2
B(µ). Since K(y,x)∗L(B∗,A∗)≤
K(y,x)L(A,B)and
K(y1,x
1)∗−K(y2,x
2)∗L(B∗,A∗)≤K(y1,x
1)−K(y2,x
2)L(A,B),

Kwill be an “n-dimensional Calde ´on-Zygmund ke nel wi h he same
cons an A.
Le M(X) be he space o all complex- alued Bo el measu es on X.
The space A⊗M(X) will consis o all fini e linea combina ions o
elemen s o he o m aη wi h a∈Aand η∈M(X). Fo one o hese
elemen s we define by con enience
T(aη)(x)=X
K(x, y)adη(y),x∈X supp η.
As in [NTV2], we conside he ollowing e sion o he Ha dy-Li le-
wood maximal unc ion:

M (x) = sup
>0
1
µ(B(x, 3 )) B(x, )| |dµ.
This maximal unc ion is bounded in Lp(µ), 1 <p≤∞, and ac s
con inuously om L1(µ) oL1,∞(µ).
We shall need he ollowing esul , which is a kind o boundedness o
an “a om” away om i s suppo .
Lemma 2.2. Fo η=J
i=1 aiηi∈A⊗M(X)wi h supp η⊂B(x, ρ)
and
η(X)=X
dη =
J

i=1
aiX
dηi=
J

i=1
aiηi(X)=0,
we ha e
X B(x,2ρ)Tη(y)Bdµ(y)≤A
J

i=1 aiAηi,
whe e Ais he cons an in he defini ion o he ke nel.
Two-Weigh Theo y on Non-Homogeneous Spaces 619
P oo : The p oo is s anda d. By using he p ope ies o ηand condi-
ion (ii), we can w i e
X B(x,2ρ)Tη(y)Bdµ(y)
=X B(x,2ρ)
J

i=1 B(x,ρ)
(K(y,x)−K(y,x))aidηi(x)B
dµ(y)
≤
J

i=1 aiA
B(x,ρ)d(x,y)≥2d(x,x)
K(y,x)−K(y,x)L(A,B)dµ(y)d|ηi|(x)
≤A
J

i=1 aiAηi.
Rema k 2.3.Jus as be o e, he ollowing can be p o ed: i η=
J
i=1 aiηi+ dµ ∈A⊗M(X)+L1
A(µ) wi h supp η⊂B(x, ρ) and
η(X)=J
i=1 aiηi(X)+X dµ= 0, we also ob ain
X B(x,2ρ)Tη(y)Bdµ(y)≤AJ

i=1 aiAηi+ L1
A(µ).
2.1. Weak ype inequali y o elemen a y measu es.
An elemen a y measu e will be an elemen o A⊗M(X), whe e he
measu es in ol ed a e uni poin masses, namely
ν=
N

i=1
αiδxi∈A⊗M(X).
Theo em 2.4. Fo an elemen a y measu e as abo e, he ollowing in-
equali y holds
TνL1,∞
B(µ)≤C
N

i=1 αiA,
whe e Conly depends on he dimension n, he cons an Ain he defini-
ion o he ke nel Kand he no m TL2
A(µ)→L2
B(µ).
620 J. Ga c´
ıa-Cue a, J. M. Ma ell
Obse e ha he e, he e is no p oblem wi h he defini ion o Tν be-
cause he sum is fini e and
Tν(x)=
N

i=1
T(αiδxi)(x)=
N

i=1
K(x, xi)αi
makes sense e e ywhe e excep a fini ely many poin s.
P oo : We shall ollow he p oo o [NTV2, Theo em 5.1] paying special
a en ion o hose de ails ha diffe om he scala case. We can assume
ha N
i=1 αiA= 1 and p o e ha TνL1,∞
B(µ)≤C. Fix some >0,
and suppose ha µ(X)>1
. Following he “scala ” case p oo —wi h
αiAins ead o αi— we a e able o find some Bo el se s E1,...,E
N
such ha
B(xi,ρ
i)
i−1

=1
E⊂Ei⊂B(xi,ρ
i)
i−1

=1
Eand µ(Ei)=αiA
,
whe e B(xi,ρ
i)={y∈X:d(xi,y)<ρ
i}. I is clea ha he se s Ei
a e pai wise disjoin , i we pu E=iEi,

i
B(xi,ρ
i)⊂E⊂
i
B(xi,ρ
i) and µ(E)=1
.
Define
σ=
i
χX B(xi,2ρi)Tαi
αiA
χEi,
and
Tν− σ =
i
ϕi=
iT(αiδxi)− χ
X B(xi,2ρi)Tαi
αiA
χEi.
Since B(xi,ρ
i)⊂E,weha e
X EϕiBdµ ≤X B(xi,2ρi)Tαiδxi− αi
αiA
χEidµB
dµ
+B(xi,2ρi) B(xi,ρi)T(αiδxi)Bdµ
≤2AαiA+2
nAαiA,
whe e we ha e used Lemma 2.2 o he fi s e m and condi ion (i) o
he ke nel o he second. Thus
X ETν− σBdµ ≤
N

i=1 X EϕiBdµ ≤2n+1A
N

i=1 αiA=2
n+1A,
Two-Weigh Theo y on Non-Homogeneous Spaces 621
and µ{x∈X:(Tν− σ)(x)B>2n+1A }≤2
, since µ(E)=1
. Then,
i migh be enough o find some big cons an A0such ha
µ{σB>A
0}≤2
.(3)
In his case, µ{x∈X:Tν(x)B>(2n+1A+A0) }≤4
. In o de o
finish we only ha e o obse e ha he abo e inequali y is ob ious when
µ(X)≤1
. Then, i we ake C=4(2
n+1A+A0), we ha e jus ob ained
TνL1,∞
B(µ)≤C.
Le us show how can we ge (3) in his ec o - alued amewo k.
Fi s , we p o e his inequali y unde he assump ion ha Ais a eflexi e
Banach space. Fo a fixed A0, o be chosen la e , suppose ha µ{σB>
A0}>2
. Then, he e exis s a Bo el se F,F⊂{σB>A
0}, such ha
µ(F)=1
.Thusσχ
F∈L1
B(µ), because
Xσχ
FBdµ ≤µ(F)1/2σL2
B(µ)≤TL2
A(µ)→L2
B(µ)
1
N

i=1 αi1/2
A<∞.
Since L1
B(µ) is isome ically con ained in (L∞
B∗(µ))∗, he Hahn-Banach
heo em implies he exis ence o some β∈L∞
B∗(µ), βL∞
B∗(µ)= 1, such
ha
β,σχF=σχ
F(L∞
B∗(µ))∗=Fσ(x)Bdµ(x)>A
0µ(F)=A0
.(4)
On he o he hand, βχ
F∈L2
B∗(µ) wi h βχ
FL2
B∗(µ)≤ −1/2and we
can use he adjoin ope a o o ob ain
β,σχF=Xσ(x)χF(x),β(x)dµ(x)
=
N

i=1 Xαi
αiA
χEi(x),T∗(βχ
F B(xi,2ρi))(x)dµ(x)
≤
N

i=1 X
χEi(x)T∗(βχ
F B(xi,2ρi))(x)A∗dµ(x).
Fo e e y x∈Ei⊂B(xi,ρ
i), by condi ion (i) o he ke nel,
T∗(βχ
F B(xi,2ρi))(x)−T∗(βχ
F B(x,ρi))(x)A∗
≤B(xi,2ρi) B(x,ρi)
K(x, y)L(B∗,A∗)β(y)B∗dµ(y)≤2nA.
628 J. Ga c´
ıa-Cue a, J. M. Ma ell
ec o - alued heo y, and we can use he sel -imp o emen esul (The-
o em 2.9) in o de o ob ain his sequence- alued ex ension.
Co olla y 2.10. Le Tbe an ope a o as abo e and ake q,1<q<∞.
Then
(i) µ



x:


j|T j(x)|q


1
q
>λ




≤C
λX


j| j(x)|q


1
q
dµ(x).
(ii) 


j|T j|q


1
qLp(µ)
≤C


j| j|q


1
qLp(µ)
,i 1<p<∞.
Rema k 2.11.These ec o - alued esul s will be u he used in [GM]
o ob ain simila es ima es o he maximal ope a o associa ed wi h T,
which, unde he app op ia e condi ions, fi s in o his ec o - alued he-
o y. In pa icula , we shall p o e he p e ious inequali ies o he sup e-
mum o he unca ed Cauchy in eg als. By means o hem, weigh ed
inequali ies o his maximal ope a o will be ob ained and we shall be
able o s udy he exis ence o p incipal alues in weigh ed Lebesgue
spaces.
3. Vec o - alued inequali ies and weigh s
The ela ion be ween weigh ed inequali ies and ec o - alued inequal-
i ies was disco e ed by J. L. Rubio de F ancia in [R] and i can be also
ound in Chap e VI o [GR]. The wo-weigh p oblem o an ope a-
o Tconsis s in finding all pai s (u, ) o posi i e unc ions o which
he inequali y
X|T (x)|pu(x)dµ(x)≤C(u, )X| (x)|p (x)dµ(x),(5)
( ∈Lp( )) holds ue. We a e going o conside he ollowing weak
a ian o his gene al p oblem:
Find condi ions on 0≤ <∞µ-a.e. ( esp. u>0
µ-a.e.) such ha (5) is sa isfied by some u>0µ-a.e.
( esp. 0≤ <∞µ-a.e.).
To s a , we need he ollowing esul , p o ed in [FT], which es ab-
lishes he conc e e ela ionship be ween ec o - alued inequali ies and
weigh s. This heo em is closely ela ed o hose con ained in [GR,
pp. 549–554].

Two-Weigh Theo y on Non-Homogeneous Spaces 629
Theo em 3.1. Le (Y,dν)be a measu e space; F,GBanach spaces,
and {Ak}k∈Za sequence o pai wise disjoin measu able subse s o Y
such ha Y=kAk. Conside 0<s<p<∞and Ta sublinea
ope a o which sa isfies he ollowing ec o - alued inequali y



jT jp
G


1
pLs(Ak,d ν)
≤Ck


j jp
F


1
p
,k∈Z,(6)
whe e, o e e y k∈Z,Ckonly depends on F,G,pand s. Then, he e
exis s a posi i e unc ion u(x)on Ysuch ha
YT (x)p
Gu(x)dν(x)1
p
≤C F
whe e Cdepends on F,G,pand s. Mo eo e , gi en a sequence o posi i e
numbe s {ak}k∈Zwi h kap
k<∞, and σ=p
s,u(x)can be ound in
such a way ha u−1χAkLσ−1(Ak,dµ)≤(a−1
kCk)p.
In ou con ex (Y,dν)=(X,dµ) which is a σ-fini e measu e space.
Then, a simple a gumen shows ha he weigh ucan be also aken so
ha u<∞a.e.
Gi en 1 <p<∞and some x0∈X, emembe he defini ion o he
classes o weigh s in X:
Dp=0≤w<∞µ-a.e. : X
w(x)1−p(1 + d(x, x0))−np
dµ(x)<∞
Zp=w>0µ-a.e. : X
w(x)(1 + d(x, x0))−npdµ(x)<∞.
No e ha hese classes o weigh s do no depend on he poin x0.
Rema k 3.2.In he case ha he diame e o he space is fini e, (o
equi alen ly, he dis ance is bounded), he e exis s Rla ge enough such
ha X⊂B(x0,R) and so µ(X)≤Rn<∞. Thus, he p e ious classes
can be gi en by he equi alen defini ion:
Dp=0≤w<∞µ-a.e. : X
w(x)1−pdµ(x)<∞
Zp=w>0µ-a.e. : X
w(x)dµ(x)<∞.
630 J. Ga c´
ıa-Cue a, J. M. Ma ell
I he suppo o he measu e is a bounded se , we can es ic he whole
space o his se , and we would be in he p e ious case. So, when we
alk abou spaces wi h fini e diame e , we shall be conce ned wi h bo h
cases.
We would like o apply he las heo em o ou ope a o s. In wha
ollows Twill be a “scala ” Calde ´on-Zygmund ope a o T, ha is, an
ope a o like hose in Defini ion 1.1.
P oposi ion 3.3. Take 0<s<1<p<∞and ∈Dp. Then, i he
diame e o Xis equal o infini y, we ha e



j|T j|p


1
pLs(Sk,d µ)
≤Cs,p2kn
s


j jp
Lp( dµ)


1
p
,
o k=0,1,...,
whe e S0={x:d(x, x0)≤1}and Sk={x:2
k−1<d(x, x0)≤2k}, o
k=1,2,.... O he wise,



j|T j|p


1
pLs(µ)
≤Cs,p 


j jp
Lp( dµ)


1
p
.
P oo : Le us see wha happens in he fi s si ua ion. Fix k≥0 and
se Bk+1 =B(x0,2k+1). E e y unc ion is spli as = +  =
χ
Bk+1 + χ
X Bk+1 . Then, o x∈Skand y∈X Bk+1 we obse e ha
2d(x, y)>d(y,x0) and hus
|T (x)|≤X Bk+1
A
d(x, y)n| (y)|dµ(y)
≤4nAX
(1 + d(y,x0))−n| (y)| (y)1
p (y)−1
pdµ(y)
≤4nAX| (y)|p (y)dµ(y)1
pX
(y)1−p
(1 + d(y,x0))np
dµ(y)1
p
≤C Lp( dµ).
Two-Weigh Theo y on Non-Homogeneous Spaces 631
No e ha he las inequali y holds because ∈Dp. Then, since µ(Sk)≤
µ(Bk)≤2kn, we p o e



j|T 
j|p


1
pLs(Sk,dµ)
≤C2kn
s


j jp
Lp( dµ)


1
p
.
On he o he hand, due o ha ac ha 0 <s<1, we can use Kol-
mogo o inequali y (see [GR, p. 485]) and Co olla y 2.10 o ob ain



j|T 
j|p


1
pLs(Sk,dµ)
≤Csµ(Sk)1
s−1


j|T 
j|p


1
pL1,∞(Sk,dµ)
≤Cµ(Sk)1
s−1Bk+1 


j| j(x)|p


1
p
(x)1
p (x)−1
pdµ(x)
≤Cµ(Sk)1
s−1

X
j| j(x)|p (x)dµ(x)


1
pBk+1
(x)−p
pdµ(x)1
p
=Cµ(Sk)1
s−1


j jp
Lp( dµ)


1
pBk+1
(x)1−pdµ(x)1
p
.
As 1
s−1>0, we obse e µ(Sk)1
s−1≤µ(Bk)1
s−1≤(2kn)1
s−1. Fu he -
mo e,
Bk+1
(x)1−pdµ(x)1
p
=Bk+1
(x)1−p
(1 + d(x, x0))np
(1 + d(x, x0))np
dµ(x)1
p
≤C2n2(k+1) n,
since ∈Dp. Then,



j|T 
j|p


1
pLs(Sk,dµ)
≤C2kn
s


j jp
Lp( dµ)


1
p
.
Collec ing hese inequali ies, we ge he desi ed es ima e.
632 J. Ga c´
ıa-Cue a, J. M. Ma ell
When he space has fini e diame e , i measu e will be fini e as well.
Thus, we p oceed like we did wi h he unc ions 
j. Since 0 <s<1, we
can apply Kolmogo o inequali y (see [GR, p. 485]) and Co olla y 2.10
o ob ain



j|T j|p


1
pLs(µ)
≤Csµ(X)1
s−1


j|T j|p


1
pL1,∞(µ)
≤Cµ(X)1
s−1X


j| j(x)|p


1
p
(x)1
p (x)−1
pdµ(x)
≤Cµ(X)1
s−1

X
j| j(x)|p (x)dµ(x)


1
pX
(x)−p
pdµ(x)1
p
≤C


j jp
Lp( dµ)


1
p
,
because Xhas fini e measu e and ∈Dp(which, in his case, means
1−p∈L1(µ)).
Once we ha e he ec o - alued inequali ies we can use Theo em 3.1
o ob ain weigh ed inequali ies.
Theo em 3.4. Take p,1<p<∞.I u∈Zp( esp. ∈Dp), hen
he e exis s some weigh 0< <∞µ-a.e. ( esp. 0<u<∞µ-a.e.)
such ha (5) holds. Mo eo e , ( esp. u) can be ound in such a way
ha α∈Zp( esp. uα∈Dp), p o ided ha 0<α<1.
P oo : Assume ha he case ∈Dpis p o ed. I u∈Zp, hen u=
u1−p∈Dp. Apply his assump ion o he adjoin ope a o T∗(which
is an ope a o wi h he same p ope ies as T) in o de o ob ain some
weigh  ,0< <∞µ-a.e., such ha
X|T∗ (x)|p (x)dµ(x)≤CX| (x)|pu(x)dµ(x).
Take so ha  = 1−p. Then, since 0 < <∞µ-a.e., an s anda d
a gumen yields ha he las inequali y implies (5). Fu he mo e, we
can choose  such ha  α∈Dp, p o ided ha 0 <α<1. Tha is, we
can find in such a way ha α∈Zp.
Two-Weigh Theo y on Non-Homogeneous Spaces 633
Le us p o e he case ∈Dp. Fix 0 <α<1 and pu q=1+α(p−1).
Then 1 <q<p
and we can find some s,0<s<1, such ha σ=p
s>
q.
When Xhas infini e diame e , we use Theo em 3.1 wi h (Y,dν)=
(X,dµ), F=Lp( dµ), G=C,{Ak}k={Sk}∞
k=0 and Ck=C2kn
s.
The ec o - alued inequali y (6) is supplied by P oposi ion 3.3. Then,
we know ha he e exis s a weigh usuch ha (5) holds. Mo eo e , u
can be aken in such a way ha u−1Lσ−1(Sk,dµ)≤C(a−1
k2kn
s)p, wi h
ak>0 and kap
k<∞. The e o e,
X
u(x)1−q
(1 + d(x, x0))np
dµ(x)=
∞

k=0 Sk
u(x)1−q
(1 + d(x, x0))np
dµ(x)
≤2np

∞

k=0
2−knp
Sk
u(x)1−σdµ(x)q−1
σ−1
µ(Sk)
1
(σ−1
q−1)
≤2np
C
∞

k=0
a−p(q−1)
k2
kn
−p+p(q−1)
s+1
(σ−1
q−1),
whe e we ha e used H¨olde ’s inequali y wi h exponen σ−1
q−1>1. Obse e
ha
−p+p(q−1)
s+1
σ−1
q−1=q−p<0,
so, we can choose ε>0 such ha q−p+ε<0. Take he sequence {ak}k
e i ying a−p(q−1)
k=2
knε. Then
∞

k=0
ap
k=
∞

k=0
2−knε
q−1<∞
and X
u(x)1−q
(1 + d(x, x0))np
dµ(x)≤C
∞

k=0
2kn(q−p+ε)<∞.
In o de o finish i is enough o no e ha α=1−q
1−pand hus uα∈Dp.
When he space has fini e diame e , as well as be o e, we use Theo-
em 3.1 wi h (Y,dν)=(X,dµ), F=Lp( dµ), G=C. In his case, we
do no decompose he space, ha is, we jus ake A0=Xand Ak=∅i
k= 0. The ec o - alued inequali y (6) is p o ided by he second pa
o P oposi ion 3.3. Then, he e exis s a weigh usuch ha (5) holds.

634 J. Ga c´
ıa-Cue a, J. M. Ma ell
Fu he mo e, ucan be aken in such a way ha u−1Lσ−1(X,dµ)≤C.
Since he measu e o Xis fini e and σ−1
q−1>1, we use H¨olde ’s inequali y
o his exponen o conclude
X
u(x)1−qdµ(x)≤X
u(x)1−σdµ(x)q−1
σ−1
µ(X)
1
(σ−1
q−1)
<∞.
Obse e ha α=1−q
1−pand we ha e uα∈Dp.
4. Cauchy in eg al ope a o
Fo a non-nega i e Bo el measu e µin he complex plane C, he
Cauchy in eg al ope a o o a compac ly suppo ed unc ion ∈Lp(µ),
1≤p≤∞, is defined as
C (z)=Cµ (z)=C
(ξ)
z−ξdµ(ξ), o µ-a.e. z∈C supp .
Assume ha µis such ha he unca ed Cauchy in eg als a e uni o mly
bounded in L2(µ). By [To1], µwill be in pa icula “1-dimensional”. In
ha case, we know ha he exis ence o he p incipal alue o compac ly
suppo ed unc ions holds (see [To2]). Then, a bounded ex ension o
he whole L2(µ) a ises om hese ac s. Thus, we ha e a me ic space C
wi h he euclidean me ic and µa “1-dimensional” measu e o which he
Cauchy in eg al ope a o is bounded in L2(µ). We obse e ha his op-
e a o alls in o he heo y de eloped by [NTV2]. The Cauchy in eg al
ope a o is defined o compac ly suppo ed unc ion in L2(µ) by means
o i s ke nel K(z,ξ)= 1
z−ξ, ha is clea ly a “1-dimensional” Calde ´on-
Zygmund ke nel. Then we can apply he esul s we ha e ob ained o ge
ec o - alued inequali ies o C. By Co olla y 2.10, he ollowing esul
is es ablished.
Theo em 4.1. Unde he abo e assump ions and o 1<p,q<∞we
ha e
(i) µ



z∈C:


j|C j(z)|q


1
q
>λ




≤C
λC


j| j(z)|q


1
q
dµ(z).
(ii) 


j|C j|q


1
qLp(µ)
≤C


j| j|q


1
qLp(µ)
.
Two-Weigh Theo y on Non-Homogeneous Spaces 635
In his amewo k, o 1 <p<∞, he classes o weigh s will be
Dp=0≤w<∞µ-a.e. : C
w(z)1−p(1 + |z|)−pdµ(z)<∞
Zp=w>0µ-a.e. : C
w(z)(1 + |z|)−pdµ(z)<∞.
I he measu e has bounded suppo , hese classes admi he equi alen
defini ion gi en in Rema k 3.2. In ac , se e al esul s will be easie
when i happens. Fo w≥0 a.e. we deno e w(A)=Aw(z)dµ(z), o
any measu able se A⊂C.
We would like o apply o his ope a o he esul s abou weigh s we
ha e p o ed. The poin is ha he e we can ob ain ha hese classes a e
sha p o his weak a ian o he wo-weigh p oblem o he Cauchy
in eg al ope a o : suppose ha o some fixed 0 <u, <∞µ-a.e., he
ollowing wo-weigh inequali y holds
C|C (z)|pu(z)dµ(z)≤C(u, )C| (z)|p (z)dµ(z),(7)
o any ∈Lp( dµ). We a e going o p o e ha , in his case, he
weigh s belong o he gi en classes.
I z=z1+iz
2,ξ=ξ1+iξ
2and is a eal- alued unc ion, o µ-a.e.
z∈C supp , we obse e
C (z) = Re(C (z)) + iIm(C (z))
=C
z1−ξ1
|z−ξ|2 (ξ)dµ(ξ)−iC
z2−ξ2
|z−ξ|2 (ξ)dµ(ξ).
Lemma 4.2. Assume ha (7) holds. Then o any z∈supp µ he e
exi s a adius z>0, such ha , u(B(z, z)) <∞.
P oo : Fix z0=z0
1+iz0
2∈supp µ, hen µ(B(z0, )) >0 o all >0.
Fo z=z1+iz
2, we w i e |z|∞= max{|z1|,|z2|} and
F1={z∈C:|z−z0|∞=z1−z0
1},F
2={z∈C:|z−z0|∞=z2−z0
2},
F3={z∈C:|z−z0|∞=z0
1−z1},F
4={z∈C:|z−z0|∞=z0
2−z2}.
Se Bk=B(z0,2−k) and Sk=Bk Bk+1. Then, he e is some k0≥0
such ha Sk0has posi i e measu e (o he wise µ(B0) = 0). Assume
o ins ance ha µ(Sk0 F1)>0 (in he o he cases we p oceed in a
simila way). Thus, he e will exis A⊂Sk0 F1so ha µ(A)>0 and
(A)<∞.Fo z∈Bk0+2 and ξ∈A, we ha e |z−ξ|≤5·2−k0−2. Since
636 J. Ga c´
ıa-Cue a, J. M. Ma ell
ξ∈A⊂F1,
2−k0−1≤|ξ−z0|≤√2 max{|ξ1−z0
1|,|ξ2−z0
2|} =√2(ξ1−z0
1).
Besides, z1−z0
1≥−|z−z0|≥−2−k0−2and
ξ1−z1=ξ1−z0
1+z0
1−z1≥1
√2|ξ−z0|−2−k0−2≥(√2−1)2−k0−2.
The e o e, o z∈Bk0+2,ξ∈A
ξ1−z1
|z−ξ|2≥(√2−1)2−k0−2
(5 ·2−k0−2)2=√2−1
25 2k0+2 =Ck0.
Then, i z∈Bk0+2,
−Re(C(χA)(z)) = A
ξ1−z1
|z−ξ|2dµ(ξ)≥Ck0µ(A)=C>0.
So, o he le hand side o (7) we ha e
C|C(χA)(z)|pu(z)dµ(z)≥Bk0+2
(−Re(C(χA)(z)))pu(z)dµ(z)
≥CpBk0+2
u(z)dµ(z).
Use his es ima e and (7), wi h =χA∈Lp( ), o ob ain u(Bk0+2)<
∞. Then, by aking z0=2
−k0−2 he p oo is finished.
Lemma 4.3. Assume ha (7) holds, hen he e exis s R>0such ha
C B(0,R)
u(z)
(1 + |z|)pdµ(z)<∞.
P oo : Fo j=1,...,4, se Ejby pu ing z0= 0 in he defini ion o Fj.
Then, i migh be enough o find some Rj>0, o each j, such ha ,
Ej B(0,Rj)
u(z)
(1 + |z|)pdµ(z)<∞.
We shall only do i o j= 1 and he o he cases can be pe o med in
he same manne . We can assume ha µ(E1)>0 (o he wise he e is
no hing o p o e). I E supp µis a bounded se , he es ima e is i ial
by choosing R1la ge enough. In he o he case, he e exis s R1such
ha µ(B(0,R
1/2) E1)>0. Take A⊂B(0,R
1/2) E1wi h µ(A)>0
and (A)<∞. Then, o z∈E1 B(0,R
1) and ξ∈A,|z|>2|ξ|and
|z−ξ|≤|z|+|ξ|≤3
2|z|. Mo eo e , since bo h poin s belong o E1,
|z|=!z2
1+z2
2≤√2 max{|z1|,|z2|} =√2z1,ξ
1=|ξ1|≤|ξ|<1
2|z|,
Two-Weigh Theo y on Non-Homogeneous Spaces 637
and hence
z1−ξ1≥1
√2|z|−1
2|z|=√2−1
2|z|.
Then,
z1−ξ1
|z−ξ|2≥2(√2−1)
9
1
|z|≥2(√2−1)
9
1
1+|z|,
and o z∈E1 B(0,R
1),
Re(C(χA)(z)) = A
z1−ξ1
|z−ξ|2dµ(ξ)≥2(√2−1)
9
1
1+|z|µ(A)
=C
1+|z|>0.
The e o e, o he le hand side o (7) we ge
C|C(χA)(z)|pu(z)dµ(z)≥E1 B(0,R1)
(Re(C(χA)(z)))pu(z)dµ(z)
≥CpE1 B(0,R1)
u(z)
(1 + |z|)pdµ(z).
Since (A)<∞, (7) can be used. Then he igh hand side o his
inequali y is fini e and he p oo is finished.
Now, we a e able o p o e he ollowing esul , which, oge he wi h
Theo em 3.4, gi es us necessa y and sufficien condi ions on he weigh s
in o de o sol e, o he Cauchy in eg al ope a o , he weak a ian o
he wo-weigh p oblem we a e dealing wi h.
Theo em 4.4. Take p,1<p<∞. Gi en 0<u<∞µ-a.e. ( esp. 0<
<∞µ-a.e.), i he e exis s some weigh 0< <∞µ-a.e. ( esp. 0<
u<∞µ-a.e.) such ha (7) holds, hen u∈Zp( esp. ∈Dp).
P oo : We shall use he p e ious lemmas. Fix 0 <u, <∞µ-a.e. such
ha (7) holds. By aking he adius R>0 supplied by Lemma 4.3, we
only ha e o see wha happens on he ball. Lemma 4.2 and a compac ness
a gumen lead o
B(0,R)
u(z)
(1 + |z|)pdµ(z)<∞.