Boundedness of the Weyl fractional integral on one-sided weighted Lebesgue and Lipschitz spaces

Author: Ombrosi, S.; De Rosa, L.
Publisher: Dipòsit Digital de Documents de la UAB
Year: 2003
DOI: 10.5565/PUBLMAT_47103_04
Source: https://ddd.uab.cat/pub/pubmat/02141493v47n1/02141493v47n1p71.pdf
Publ. Ma . 47 (2003), 71–102
BOUNDEDNESS OF THE WEYL FRACTIONAL
INTEGRAL ON ONE-SIDED WEIGHTED LEBESGUE
AND LIPSCHITZ SPACES
S. Omb osi and L. de Rosa
Abs ac
In his pape we in oduce he one-sided weigh ed spaces L−
w(β),
−1<β<1. The pu pose o his deﬁni ion is o ob ain an
ex ension o he Weyl ac ional in eg al ope a o I+
α om Lp
w
in o a sui able weigh ed space.
Unde ce ain condi ion on he weigh w,weha e ha L−
w(0)
coincides wi h he dual o he Ha dy space H1
−(w). We p o e
o 0 <β<1, ha L−
w(β) consis s o all unc ions sa is ying a
weigh ed Lipschi z condi ion. In o de o gi e ano he cha ac e -
iza ion o L−
w(β), 0 ≤β<1, we also p o e a one-sided e sion o
John-Ni enbe g Inequali y.
Finally, we ob ain necessa y and suﬃcien condi ions on he
weigh w o he boundedness o an ex ension o I+
α om Lp
w
in o L−
w(β), −1<β<1, and i s ex ension o a bounded ope a o
om L−
w(0) in o L−
w(α).
1. No a ions, deﬁni ions and p e equisi es
Le E⊆Rbe a Lebesgue measu able se . We shall deno e i s
Lebesgue measu e by |E|and he cha ac e is ic unc ion o Eby χE.
As usual, a weigh wis a measu able, non-nega i e and locally in e-
g able unc ion deﬁned on R.
Le wbe aweigh . Gi en a Lebesgue measu able se E⊆R, i s
w-measu e will be deno e by w(E)=Ew( )d .
2000 Ma hema ics Subjec Classiﬁca ion. P ima y: 26A33; Seconda y: 42B25.
Key wo ds. Weyl ac ional in eg al, weig hs, weigh ed Lebesgue and Lipschi z
spaces, weigh ed BMO.
This esea ch has been pa ially suppo ed by UBACYT 2000-2002 and CONICET.
72 S. Omb osi, L. de Rosa
Le 1 <p<∞. The weigh wbelongs o he class A−
pi he e exis s
a cons an Csuch ha
sup
h>01
hpa+h
a
w(x)dx a
a−h
w(x)−1
p−1dxp−1≤C,
o all eal numbe a.Inasimila way, wbelongs o A+
pi
sup
h>0
1
hpa
a−h
w(x)dx a+h
a
w(x)−1
p−1dxp−1
≤C,
o all eal numbe a. The class A−
1is deﬁned by he condi ion
sup
h>01
ha+h
a
w(x)dx≤Cw(a),
o almos e e y eal numbe a. The weigh wbelongs o A+
1i
sup
h>01
ha
a−h
w(x)dx≤Cw(a),
o almos e e y a. These classes A−
pand A+
pwe e in oduced by
E. Sawye in [12]. We ecall h ee basic esul s on hese weigh s.
(i) Fo 1 <p<∞,aweigh wbelongs o A−
pi and only i w1−p
belongs o A+
p, whe e 1
p+1
p=1.
(ii) I 1 ≤p<q<∞, hen A−
p⊂A−
q.
(iii) I 1 <p<∞and wbelongs o A−
p, hen wbelongs o A−
p− o
some >0.
The p oo o (i) and (ii) a e e y simple and (iii) can be ound in P opo-
si ion 3 in [3].
In he sequel, o each bounded in e al I=[a, b]weshall deno e
I−=[a−|I|,a] and I+=[b, b +|I|].
Le 1 ≤q<∞.Aweigh wsa isﬁes he condi ion RH−(q)i he e
exis s a cons an Csuch ha o e e y bounded in e al I.
1
|I|I
w(x)qdx1/q
≤C1
|I|I−
w(x)dx.
We shall say ha a weigh wbelongs o D−i he e exis s a cons an C
such ha o e e y bounded in e al I,
w(I∪I+)≤Cw(I).
Weyl F ac ional In eg al 73
I is well known ha i w∈A−
p,1≤p<∞, hen w∈D−.
Le wbe aweigh , 1 ≤p<∞and a measu able unc ion. We shall
say ha belongs o Lp
wi
 p
p,w =∞
−∞ | (x)|
w(x)p
dx
is ﬁni e. The unc ion belongs o 
Lp
wi
[ ]p
p,w = sup
>0
px∈R:| (x)|
w(x)>

is ﬁni e.
Le 0 <α<1. Gi en a measu able unc ion on R, i s Weyl
ac ional in eg al is deﬁned by
I+
α (x)=∞
x
(y)
(y−x)1−αdy,
whene e his in eg al is ﬁni e.
In he sequel, he le e Cwill deno e a posi i e ﬁni e cons an no
necessa ily he same a each occu ence. I 1 ≤p≤∞ hen pwill be
i s conjuga e exponen , ha is, 1/p +1/p=1.
Le wbe aweigh and −1<β<1.
Deﬁni ion 1.1. We say ha a locally in eg able unc ion deﬁned on R
belongs o Lw(β), i he e exis s a cons an Csuch ha
1
w(I)|I|βI
| (y)− I|dy ≤C,
o e e y bounded in e al I, whe e I=1
|I|I . The leas cons an C
will be deno ed  Lw(β).
The spaces Lw(β)we e in oduced by E. Ha bou e, O. Salinas and
B. Vi iani in [1]. They a e a weigh ed e sion o he spaces Lλ,p, o
p=1,deﬁned by J. Pee e in [8]. I wbelongs o A−
q,1≤q<2, hen
Lw(0) is he dual space o he one-sided weigh ed Ha dy space H1
−(w),
see [10] and [11].
74 S. Omb osi, L. de Rosa
Deﬁni ion 1.2. We say ha a locally in eg able unc ion deﬁned on R
belongs o L−
w(β), i he e exis s a cons an Csuch ha
1
w(I−)|I|βI
| (y)− I|dy ≤C,
o e e y bounded in e al I. The leas cons an Csa is ying his in-
equali y will be deno ed  L−
w(β).
In he ollowing deﬁni ion, we conside a one-sided e sion o he
classes H(α, p) deﬁned in [1].
Deﬁni ion 1.3. Le 0 <α<1 and 1 <p≤∞.Wesay ha a weigh w
belongs o H−(α, p)i he e exis s a cons an Csuch ha o e e y
bounded in e al I=[a, b], he inequali y
|I|1
p−α+1 ∞
b
w(y)p
(y−a)(2−α)pdy1/p
≤Cw(I)
|I|,
holds.
2. S a emen o he main esul s
Lemma 4.1(iii) shows ha i wbelongs o H−(α, p), 1 <p≤∞,
hen wbelongs o D−and he e o e Lw(β)⊆L
−
w(β) o e e y β:−1<
β<1. The nex heo em s a es ha wbelonging o D−is a suﬃcien
condi ion o he equali y o hese spaces, whene e 0 ≤β<1.
Theo em 2.1. Le 0≤β<1and le wbelong o D−. Then, he
spaces Lw(β)and L−
w(β)a e equal, and hei no ms a e equi alen .
The nex heo em gi es us a cha ac e iza ion o he spaces Lw(β),
0≤β<1, whene e wbelongs o A−
p.In he case β=0,weshall
p o e his esul using P oposi ion 3.6, which s a es a one-sided weigh ed
e sion o John-Ni enbe g Inequali y.
Theo em 2.2. Le 0≤β<1and 1≤p<∞.Le wbeaweigh such
ha wbelongs o A−
p. Then, ∈L
w(β)i and only i he e exis s a
cons an Csuch ha
I−
| (x)− I+|qw(x)1−qdx ≤Cw(I−)|I|βq,(2.1)
o all bounded in e al Iand e e y q:1≤q≤p,q<∞.
The ollowing wo heo ems s a e a suﬃcien and necessa y condi ion
on he weigh w o ob ain ex ensions o I+
αdeﬁned on ce ain spaces.
Weyl F ac ional In eg al 75
Theo em 2.3. Le 0<α<1,1<p<∞and β=α−1/p. The
ollowing s a emen s a e equi alen .
(i) The weigh wbelongs o H−(α, p).
(ii) The ope a o I+
αcanbeex ended o a linea bounded ope a o 
I+
α
om 
Lp
win o L−
w(β)by means o
(2.2) 
I+
α( )(x)=−x
x0
(y)dy
|y−x|1−α
+∞
x01
|y−x|1−α−1−χ[x0,x0+1](y)
(y−x0)1−α (y)dy,
o any x0∈R.
(iii) The ope a o I+
αcanbeex ended o a linea bounded ope a o 
I+
α
om Lp
win o L−
w(β), whe e 
I+
αis deﬁned as in (2.2).
Theo em 2.4. Le wa weigh and 0<α<1. The ollowing s a emen s
a e equi alen .
(i) The weigh wbelongs o H−(α, ∞).
(ii) The ope a o I+
αcanbeex ended o a linea bounded ope a o

I+
α:Lw(0) →L
w(α)by means o

I+
α( )(x)=∞
−∞ χ[x0,∞)(y)
|y−x0|1−α−χ[x,∞)(y)
|y−x|1−α (y)dy,
o an app op ia e choice o x0∈R.
Rema k 2.5.Le 1 <p< 1
αand β=α−1/p<0.
(i) I is easy o see ha i wbelongs o RH−(1
1+β), hen L−1/β
w⊆
L−
w(β).
(ii) By Lemma 4.4 in [9], i wpbelongs o A−
−βp+1 hen wsa isﬁes
he condi ion RH−(p), and aking in o accoun ha 1
1+β<p
,i
ollows ha wbelongs o RH−(1
1+β).
(iii) Theo em 6 in [4] s a es he ac ha wpbelongs o A−
−βp+1 is
a necessa y and suﬃcien condi ion o he boundedness o I+
α
om Lp
win o L−1/β
w⊆L
−
w(β).
(i ) I wpbelongs o A−
−βp+1, since wp∈A−
p+1,weha e ha w
belongs o H−(α, p). Howe e , he e exis weigh s wbelonging
o H−(α, p) such ha wpdoes no belong o A−
p+1, o example,
w(x)=|x|γ o −β≤γ<1−β, see Rema k 4.3.

76 S. Omb osi, L. de Rosa
In consequence, i −1<β<0 and wpbelongs o A−
−βp+1, he
ex ension o I+
αin Theo em 2.3 can be ob ained om Theo em 6 in [4].
Bu , (i ) shows ha Theo em 2.3 can be applied o a la ge class o
weigh s.
Rema k 2.6.Le wbe aweigh . We shall say ha a locally in eg able
unc ion deﬁned on R,belongs o MW−(w)i he e exis s a cons an C
such ha
1
|I|
1
ess in I−wI
| (y)− I|dy ≤C,
o e e y bounded in e al I.
(i) By Deﬁni ion 1.2, i ollows ha MW−(w)⊆L
−
w(0). Mo eo e ,
i wbelongs o A−
1 hen Lw(0) ⊆MW−(w), and as a consequence
o Theo em 2.1, L−
w(0) = MW−(w).
(ii) Following he same lines o Theo em 7 in [7], i can be seen ha ,
in he case α=1/p, he weigh wpbelongs o A−
1i and only i
he ope a o I+
αis bounded om Lp
win o MW−(w). Also see [2].
(iii) I wpbelongs o A−
1 hen, by Rema k 4.3, wbelongs o H−(α, p).
In consequence, he ac ha wpbelongs o A−
1implies he bound-
edness o I+
α om Lp
win o MW−(w), is con ained in Theo em 2.3.
3. The spaces L
w
(β) and L
−
w
(β)
The nex lemma will be used in he p oo o Theo em 2.1.
Lemma 3.1. Le −1<β<1, alocally in eg able unc ion deﬁned
on R, and w∈D−. The ollowing s a emen s a e equi alen .
(i) ∈L
−
w(β).
(ii) The e exis s a cons an Csuch ha o e e y a∈Rand h>0,
1
w([a−h/2,a])hβa+h
a
| (y)− [a+h/2,a+h]|dy ≤C.
(iii) The e exis s a cons an Csuch ha o e e y a∈Rand h>0,
1
w([a−h/2,a])hβa+h
a
| (y)− [a+h,a+3h]|dy ≤C.
The cons an s Cin (ii) and (iii) a e equi alen o  L−
w(β).
Weyl F ac ional In eg al 77
P oo : (i) ⇒(ii). Using (i) and aking in o accoun ha w∈D−,we
ha e
a+h/2
a
| (y)− [a+h/2,a+h]|dy
≤a+h/2
a
| (y)− [a+h/4,a+h/2]|dy+2a+h
a+h/4
| (y)− [a+h/2,a+h]|dy
≤3a+h/2
a
| (y)− [a,a+h/2]|dy +5a+h
a+h/4
| (y)− [a+h/4,a+h]|dy
≤C L−
w(β)w([a−h/2,a])hβ+C L−
w(β)w([a−h/2,a+h/4])hβ
≤C L−
w(β)w([a−h/2,a])hβ.
F om hese inequali ies and using (i) again, we ha e he es ima e
a+h
a
| (y)− [a+h/2,a+h]|dy
=a+h/2
a
| (y)− [a+h/2,a+h]|dy +a+h
a+h/2
| (y)− [a+h/2,a+h]|dy
≤C L−
w(β)w([a−h/2,a])hβ+C L−
w(β)w([a, a +h/2])hβ
≤C L−
w(β)w([a−h/2,a])hβ,
which shows ha (ii) holds. In a simila way i can be p o ed ha
(ii) ⇒(iii) and (iii) ⇒(i).
As we ha e al eady mencioned i wbelongs o D− hen, o e e y
−1<β<1weha e he inclusion Lw(β)⊆L
−
w(β). In o de o p o e
Theo em 2.1, i will be suﬃcien o show ha L−
w(β)⊆L
w(β).
78 S. Omb osi, L. de Rosa
P oo o Theo em 2.1: We suppose ha ∈L
−
w(β). Le a∈Rand
h>0. Fo each j≥0wedeﬁne aj=a+h/2j. Then,
a+h/2
a
| (y)− [a+h/2,a+h]|dy
=
∞

j=1 aj
aj+1
| (y)− [a+h/2,a+h]|dy
≤
∞

j=1 aj
aj+1
| (y)− [aj,aj−1]|dy+
∞

j=2
h
2j+1 | [aj,aj−1]− [a1,a0]|
=I+II.
(3.1)
Taking in o accoun ha o each j≥2,
| [aj,aj−1]− [a1,a0]|≤2j
haj−1
aj
| − [a+h/2,a+h]|
i ollows ha ,
II ≤
∞

j=2
1
2aj−1
aj
| − [a+h/2,a+h]|=1
2a+h/2
a
| (y)− [a+h/2,a+h]|dy.
Then, by (3.1)
a+h/2
a
| (y)− [a+h/2,a+h]|dy ≤2I.(3.2)
Now, using (iii) o Lemma 3.1 and keeping in mind ha β≥0weha e
ha ,
I≤C
∞

j=1 h
2jβ
w([aj+2,a
j+1]) ≤Chβw([a, a +h/4]).(3.3)
Weyl F ac ional In eg al 79
F om (3.2) and (3.3), and aking in o accoun ha ∈L
−
w(β), we ge
a+h
a
| (y)− [a+h/2,a+h]|dy
=a+h/2
a
| (y)− [a+h/2,a+h]|dy +a+h
a+h/2
| (y)− [a+h/2,a+h]|dy
≤Chβw([a, a +h/4]) + Chβw([a, a +h/2])
≤Chβw([a, a +h]).
The e o e,
a+h
a
| (y)− [a,a+h]|dy
≤3a+h
a
| (y)− [a+h/2,a+h]|dy ≤Chβw([a, a +h]),
which shows ha ∈L
w(β).
Rema k 3.2.Le −1<β<0 and w( )=e− . The weigh wbelongs
o A−
1howe e , we only ha e he s ic inclusion Lw(β)⊂L
−
w(β). Fo
example, gi en a>1weconside he unc ion
( )=e−a , ≥0
1, <0.
We obse e, using Rema k 2.5(i), ha ∈L
−
w(β). On he o he hand,
1
hβw([0,h])h
0
| − [h,2h]|=1
hβ(1 −e−h)1−e−ah
a−e−ah
a(1−e−ah)
=(1 −e−ah)2
hβ(1 −e−h)a,
which ends o inﬁni e whene e h ends o inﬁni e. This implies ha
/∈L
w(β).
The nex p oposi ion will be used in he p oo o Theo em 2.2.
86 S. Omb osi, L. de Rosa
By (3.15) and (3.14), we ha e
II ≤Cs
γp
i,k
w(H−
i,k)=Cs
γp
i
w(Hi)≤Cµsp−1
γpw(I−).
Then, (3.16) and (3.17) imply ha
A(λ, I)≤CµA(λ−γ)
s+sp−1
γpw(I−).
F om his inequali y, (ii) ollows as in Theo em 3 o [6].
P oposi ion 3.7. Le 0<β<1and 1<p<∞.Le wbeaweigh
such ha w1+ β
1−βpbelongs o A−
p. Then, ∈L
w(β)i and only i he e
exis s a cons an Csuch ha (2.1) holds o all bounded in e al Iand
e e y q:1≤q≤p/(1 −β).
P oo : Suppose ha (2.1) holds o e e y q:1≤q≤p/(1 −β). Taking
q=1i is easy o show ha ∈L
w(β). Con e sely, le belong
o Lw(β). We obse e ha i will be suﬃcien o conside q=p/(1−β),
because om his case and applying H¨olde ’s inequali y we ob ain (2.1)
o e e y 1 ≤q<p
/(1 −β). Gi en a bounded in e al Iand using
P oposi ion 3.3, we ha e ha
I−
| (x)− I+|qw(x)1−qdx
≤I−1
|I+|I+
| (x)− (y)|dyq
w(x)1−qdx
≤CI−
w(x)1−q1
|I+|I+x+|y−x|
2
x
w(z)
(z−x)1−βdz
+y+|y−x|
2
y
w(z)
(z−y)1−βdzdyq
dx
≤CI−
w(x)1−qx+3|I|
2
x
w(z)
(z−x)1−βdzq
dx
+C
|I+|qI−
w(x)1−qI+y+3|I|
2
y
w(z)
(z−y)1−βdz dyq
dx
=A+B.
(3.19)

Weyl F ac ional In eg al 87
I we deno e J=I−∪I∪I+ hen we ha e he es ima e
A≤CI−
w(x)1−qI+
β(wχJ)(x)qdx.
Ou hypo hesis w1+ β
1−βp∈A−
pis equi alen o
w1−p
1−β∈A+
p,(3.20)
whe e p=1+ q
sand 1
s=1
q+β. Then, by Theo em 6 in [4]i ollows
ha
A≤C∞
−∞
w(x)−s
q|wχJ(x)|sdxq/s
=CJ
w(x)s/q dxq/s
.
Since q/s =qβ +1 >1, applying H¨olde ’s inequali y and aking in o
accoun ha w∈D−we ob ain
A≤CJ
w(x)dx |J|q
s−1≤Cw(I−)|I|βq.(3.21)
Le us es ima e B.I wese J=I+∪I++ ∪I+++, hen
B≤C
|I+|qI−
w(x)1−qI+
I+
β(wχJ)(y)dyq
dx.
Applying H¨olde ’s inequali y,
B≤C
|I+|qI−
w(x)1−qdxI+
w(y)dyq/q
I+
w(y)1−qI+
β(wχJ)(y)qdx.
F om (3.20), i ollows ha w1−q∈A+
q hen, we ha e ha
B≤CI+
w(y)1−qI+
β(wχJ)(y)qdx.
P oceeding as in he es ima ion o Aand aking in o accoun ha w∈
D−we ob ain
B≤Cw(I−)|I|βq.(3.22)
As consequence o (3.19), (3.21) and (3.22) we ge (2.1) and he p oo
o his p oposi ion is comple e.
P oo o Theo em 2.2: We shall p o e ha belonging o Lw(β)isa
suﬃcien condi ion o (2.1) holds. The ac ha (2.1) is a necessa y
condi ion ollows as in he p e ious p oposi ion. Fo ha , we shall con-
side diﬀe en cases.
Fi s o all, we assume ha β=0and ∈L
w(0). I w∈A−
1we
ha e ha (2.1) is an immedia e consequence o P oposi ion 3.6(i). I
88 S. Omb osi, L. de Rosa
w∈A−
p,1<p<∞,weha e ha w∈A−
p− o some >0. Then,
by P oposi ion 3.6(ii), and p oceeding as in Theo em 4 o [6], we ob ain
ha sa isﬁes (2.1).
Le 0 <β<1 and 1 <p<∞. Since he weigh wbelongs o A−
p
he e exis s 0 <α<βsuch ha w1+ α
1−αpbelongs o A−
p. P oceeding
as in (3.19), we ha e ha
I−
| (x)− I+|qw(x)1−qdx
≤CI−
w(x)1−q1
|I+|I+x+|y−x|
2
x
w(z)
(z−x)1−βdz
+y+|y−x|
2
y
w(z)
(z−y)1−βdzdyq
dx
≤C|I|(β−α)qI−
w(x)1−qx+3|I|
2
x
w(z)
(z−x)1−αdzq
dx
+C
|I|(β−α−1)qI−
w(x)1−qI+y+3|I|
2
y
w(z)
(z−y)1−αdz dyq
dx
=|I|(β−α)q(A+B).
Subs i u ing in he p oo o he p e ious p oposi ion α o βin he
es ima ion o Aand Bwe ob ain his case.
Finally, we suppose ha 0 <β<1 and p=1. Since he weigh w
belongs o A−
1i ollows ha wbelongs o A−
s o e e y 1 <s<
∞. Then, by he p e ious case we ob ain ha (2.1) holds o e e y
1≤q<∞.
4. The classes H
−
(α, p)
The nex lemma s a es necessa y condi ions o ha a weigh wbe-
longs o H−(α, p).
Lemma 4.1. Le 1<p≤∞.I w∈H−(α, p) hen,
(i) wpbelongs o ∈D−,
(ii) wbelongs o ∈RH−(p),
(iii) wbelongs o ∈D−.
Weyl F ac ional In eg al 89
P oo : The p oo o (i) and (ii) a e simila o ones o Lemma 3.7 and
Lemma 3.8, in [1], espec i ely. Applying H¨olde ’s inequali y and (ii),
we ob ain (iii).
Lemma 4.2. Le wbe a weigh . The ollowing condi ions a e equi a-
len .
(a) w∈H−(α, p).
(b) w∈RH−(p)and he e exis posi i e cons an s Cand such ha ,
wp([a, a +θ ]) ≤Cθ(2−α)p−wp([a, a + ]),
o e e y a∈R, >0and θ≥1.
(c) The e exis posi i e cons an s Cand such ha ,
wp([a, a +θ ])
θ 1/p
≤Cθ1
p+1−α−
pw([a− , a])
,
o e e y a∈R, >0and θ≥1.
P oo : (a) ⇒(b). By Lemma 4.1(ii) we ha e ha w∈RH−(p).
Le I=[a, a + ]. Applying H¨olde ’s inequali y and keeping in mind
ha w∈H−(α, p),
wp(I)
|I|≥w(I)
|I|p
≥C|I|(1
p−α+1)p∞
a+
w(y)p
(y−a)(2−α)pdy
≥C|I|(1
p−α+1)p
k≥0
1
(2k+1 )(2−α)pa+2k+1
a+2k
w(y)pdy.
(4.1)
Since i≥k1
2(2−α)pi=C1
2(2−α)pk,by(4.1) and applying Fubini’s
Theo em,
wp(I)
|I|≥C|I|(1
p−α+1)p1
(2−α)p
k≥0a+2k+1
a+2k
w(y)pdy 
i≥k1
2(2−α)pi
=C|I|(1
p−α+1)p
i≥0
1
(2i )(2−α)p
i

k=0 a+2k+1
a+2k
w(y)pdy
=C|I|(1
p−α+1)p
i≥0
1
(2i )(2−α)pa+2i+1
a+
w(y)pdy.
90 S. Omb osi, L. de Rosa
The e o e,
wp(I)
|I|≥C|I|(1
p−α+1)p
i≥0
1
(2i )(2−α)pa+2i+1
a
w(y)pdy
≥C|I|(1
p−α+1)p
i≥02i+1
2i
wp([a, a +s])
s(2−α)p
ds
s
=C|I|(1
p−α+1)p∞
wp([a, a +s])
s(2−α)p
ds
s.
In consequence,
∞
wp([a, a +s])
s(2−α)p
ds
s≤Cwp([a, a + ])
(2−α)p.
Now, using Lemma 3.3 in [1] wi h ϕ(s)=wp([a, a+s]) and =(2−α)p,
he e exis Cand such ha
ϕ(θ )≤Cθ −ϕ( ),
o e e y >0 and θ≥1. Tha is,
wp([a, a +θ ]) ≤Cθ(2−α)p−wp([a, a + ]),
o e e y >0 and θ≥1, This comple es he p oo o (a) ⇒(b).
(b) ⇒(a). Le I=[a, a + ]. I (b) holds, we ha e ha
∞
a+
w(y)p
(y−a)(2−α)pdy1/p
=∞

k=0 a+2k+1
a+2k
w(y)p
(y−a)(2−α)pdy1/p
≤∞

k=0
1
(2k )(2−α)pwp([a+ , a + +2
k+1 ])1/p
≤C∞

k=0
(2k+1)(2−α)p−
(2k )(2−α)pwp([a+ , a +2 ])1/p
≤C1
a+2
a+
w(y)pdy1/p
1
p−2+α.
(4.2)
Weyl F ac ional In eg al 91
Using he hypo hesis w∈RH−(p)weob ain ha (4.2) is bounded by
C1
a+
a
w(y)dy 1
p−2+α=Cw([a, a + ])
1
p+2−α,
which shows ha w∈H−(α, p).
The p oo o (b) ⇒(c) is e y simple and we shall omi i .
(c) ⇒(b). Taking θ=1in (c) we ha e ha w∈RH−(p). Using (c)
and H¨olde ’s inequali y,
wp([a− , a +θ ])
θ 1/p
=wp([a− , a])
θ +wp([a, a +θ ])
θ 1/p
≤wp([a− , a])
θ 1/p
+Cθ1
p+1−α−
pwp([a− , a])
1/p
.
We can suppose ha 1
p+1−α−
p>0, hen aking in o accoun ha
θ≥1
wp([a− , a − +θ ])
θ 1/p
≤wp([a− , a +θ ])
θ 1/p
≤Cθ1
p+1−α−
pwp([a− , a])
1/p
.
F om hese inequali ies wi h a=b+ we ob ain ha
wp([b, b +θ ]) ≤Cθ(2−α)p−wp([b, b + ]),
which comple es he p oo .
Rema k 4.3.I is easy o see ha i wpbelongs o A−
1 hen, w∈
H−(α, p). On he o he hand, applying Lemma 4.2 (b) ⇒(a), i ol-
lows ha i w(x)=|x|γwi h 0 <γ<1/p −α+1, hen wbelongs o
H−(α, p), bu wdoes no belong o A−
1.Fo 0<α<1/p,asanimme-
dia e consequence o Lemma 4.2 (c) ⇒(a) i ollows ha i wpbelongs
o A−
p+1 hen, wbelongs o H−(α, p).
The nex wo lemmas show ha i wbelongs o H−(α, p), 1 <p<
∞, hen he e exis s η>0 such ha wbelongs o H−(α, q) o e e y
q:p−η<q<p+η.

92 S. Omb osi, L. de Rosa
Lemma 4.4. Le 1<p<∞and w∈H−(α, p). Then, he e exis s
δ0∈(0,1) such ha w∈H−(α, (pδ)) o any δ:δ0<δ≤1.
P oo : I is a simple a ian o Lemma 3.13 in [1].
Lemma 4.5. Le 1<p<∞and w∈H−(α, p). Then, he e exis s
τ0>1such ha w∈H−(α, (pτ)) o any 1≤τ≤τ0.
P oo : Since w∈RH−(p) applying Theo em 5.3 in [9], he e exis s
τ0>1 such ha o e e y τ:1≤τ≤τ0 he e exis s a cons an Csuch
ha
1
c−bc
b
w(y)pτdy1
pτ
≤C1
b−ab
a
w(y)dy
≤C1
b−ab
a
w(y)pdy1
p
(4.3)
o e e y a<b<cwi h c−b=2(b−a). Le I=[a, b]. Using (4.3) we
ha e ha ,
∞
b
w(y)pτ
(y−a)(2−α)pτdy
=
k≥02k|I|≤y−a≤2k+1|I|
w(y)pτ
(y−a)(2−α)pτdy
≤
k≥0
1
(2k|I|)(2−α)pτ2k|I|≤y−a≤2k+1|I|
w(y)pτdy
≤C
k≥0
1
(2k|I|)(2−α)pτ−11
2k|I|2k−1|I|≤y−a≤2k|I|
w(y)pdyτ
.
(4.4)
Taking in o accoun ha τ>1, (4.4) is bounded by
C
k≥0
2k|I|1
2k|I|2k−1|I|≤y−a≤2k|I|
w(y)p
(y−a)(2−α)pdyτ
≤C|I|1−τ|I|
2≤y−a
w(y)p
(y−a)(2−α)pdyτ
.
Weyl F ac ional In eg al 93
Keeping in mind ha w∈H−(α, p)weha e,
∞
b
w(y)pτ
(y−a)(2−α)pτdy ≤C|I|1−τw([a, a +|I|/2])
|I||I|−1/p+α−1pτ
=Cw(I)
|I|
1
|I|1
(pτ)−α+1 pτ
,
which implies ha w∈H−(α, (pτ)).
Lemma 4.6. Le 1<p
1<p
2<∞. Suppose ha w∈H−(α, pi) o
i=1,2. Then w∈H−(α, p) o e e y p:p1<p<p
2.
P oo : This is an one-sided e sion o Lemma 3.15 in [1].
Lemma 4.7. Le 1<p<∞and w∈RH−(p). The e exis s a con-
s an Csuch ha o e e y ∈
Lp
wand e e y bounded in e al I=[a, b],
i we deno e 
I−=[a−|I|
2,a] hen,
I
| (x)|dx ≤Cw(
I−)
|I|1/p [ ]p,w.
P oo : Since w∈RH−(p)byTheo em 5.3 in [9], he e exis s s>p

such ha w∈RH−(s), ha is, he e exis s a cons an Csuch ha o
e e y bounded in e al I,
1
|I|I
w(x)sdx1/s
≤Cw(
I−)
|I|.
F om his ac , he p oo ollows as in Lemma 4.1 o [1].
Lemma 4.8. Le 1<p<∞and w∈H−(α, p). Then he e exis s a
cons an Csuch ha o e e y ∈
Lp
wand e e y bounded in e al I=
[a, b],∞
b
| (y)|
(y−a)2−αdy ≤Cw(I)
|I|2+ 1
p−α[ ]p,w.
P oo : Taking in o accoun Lemma 4.4 and Lemma 4.5, he p oo o his
lemma is simila o one in Lemma 4.4 o [1].
94 S. Omb osi, L. de Rosa
Lemma 4.9. Le α>0and δ≥0such ha 0<α+δ<1.Le w∈D−.
Fo a<b,wedeno e c=a+b
2and I=[c, b]. Then, o e e y ∈L
w(δ),
he e exis s a cons an Csuch ha ,
∞
b
| (y)− I|
(y−a)2−αdy ≤C Lw(δ)∞
c
w(y)
(y−a)2−α−δdy.(i)
b
a
| (y)− I|
(y−a)1−αdy ≤C Lw(δ)c
a
w(y)
(y−a)1−α−δdy.(ii)
P oo : The p oo o (i) and (ii) a e simila , hen we only p o e (i).
Fo e e y j≥0, le Ij=[a+2
j|I|,a+2
j+1|I|]. We obse e ha
I0=[a+|I|,a+2|I|]=[c, b]=I. Since ∈L
w(δ)weha e ha ,
∞
b
| (y)− I|
(y−a)2−αdy=
∞

j=1 a+2j+1|I|
a+2j|I|
| (y)− I|
(y−a)2−αdy
≤
∞

j=1
1
(2j|I|)2−αa+2j+1|I|
a+2j|I|
| (y)− I0|dy
≤
∞

j=1
1
(2j|I|)2−αa+2j+1|I|
a+2j|I|
| (y)− Ij|dy
+2j|I|
j

k=1
| Ik− Ik−1|
≤
∞

j=1
1
(2j|I|)1−αC Lw(δ)w(Ij)(2j|I|)δ−1
+
j

k=1
1
|Ik−1|Ik−1
| (y)− Ik|dy.
(4.5)
Using ha ∈L
w(δ) and w∈D−we ob ain he es ima e,
1
|Ik−1|Ik−1
| (y)− Ik|dy ≤C Lw(δ)w(Ik−1)(2k−1|I|)δ−1.
Weyl F ac ional In eg al 95
Then applying Fubini’s Theo em, (4.5) is bounded by
C Lw(δ)
∞

j=1
1
(2j|I|)1−α
j

k=0
w(Ik)(2k|I|)δ−1
=C Lw(δ)
∞

k=0
w(Ik)(2k|I|)δ−1
∞

j=k
1
(2j|I|)1−α
=C Lw(δ)
∞

k=0
1
(2k|I|)2−α−δa+2k+1|I|
a+2k|I|
w(y)dy
≤C Lw(δ)∞
c
w(y)
(y−a)2−α−δdy,
as we wan ed o p o e.
5. P oo o Theo ems 2.3 and 2.4
P oo o Theo em 2.3: (i) ⇒(ii). Le w∈H−(α, p) and x0∈R. Gi en
∈
Lp
wle 
I+
α( ) deﬁne as in (2.2). Choose a bounded in e al I=
[a, a +h]. We conside I0=[a+2h, x0]i a+2h≤x0and I0=∅i
x0<a+2h, and we also deﬁne I1=[x0,a+2h]i x0<a+2hand
I1=∅in he o he case. We se
aI=I0
(y)
(y−a)1−αdy +∞
x01−χI1(y)
(y−a)1−α−1−χ[x0,x0+1](y)
(y−x0)1−α (y)dy.
We shall show ha aIis a ﬁni e cons an .
Suppose ha x0<a+2h. Le nbe aposi i e in ege such ha
a+2
nh>x
0+1and |a−x0|≤2n−1h. Then,
aI=a+2nh
x0
+∞
a+2nh1−χ[x0,a+2h](y)
(y−a)1−α−1−χ[x0,x0+1](y)
(y−x0)1−α (y)dy
=J1+J2.
Fo each y≥a+2nh,byMean Value Theo em, he e exis s θ:0<θ<1
such ha ,

1
(y−a)1−α−1
(y−x0)1−α≤C|x0−a|
|y−θa −(1 −θ)x0|2−α≤C|x0−a|
|y−a|2−α.
102 S. Omb osi, L. de Rosa
[6] B. Muckenhoup and R. L. Wheeden,Weigh ed bounded
mean oscilla ion and he Hilbe ans o m, S udia Ma h. 54(3)
(1975/76), 221–237.
[7] B. Muckenhoup and R. L. Wheeden,Weigh ed no m inequal-
i ies o ac ional in eg als, T ans. Ame . Ma h. Soc. 192 (1974),
261–274.
[8] J. Pee e,On he heo y o Lp,λ spaces, J. Func ional Analysis 4
(1969), 71–87.
[9] M. S. Ri e os and A. de la To e,On he bes anges o A+
p
and RH+
,Czechoslo ak Ma h. J. 51(126), no. 2 (2001), 285–301.
[10] L. de Rosa and C. Sego ia,Weigh ed Hpspaces o one sided
maximal unc ions, in: “Ha monic analysis and ope a o heo y”
(Ca acas, 1994), Con emp. Ma h. 189, Ame . Ma h. Soc., P o i-
dence, RI, 1995, pp. 161–183.
[11] L. de Rosa and C. Sego ia, Dual spaces o one-sided weigh ed
Ha dy spaces, Re . Un. Ma . A gen ina 40(3–4) (1997), 49–71.
[12] E. Sawye ,Weigh ed inequali ies o he one-sided Ha dy-
Li lewood maximal unc ions, T ans. Ame . Ma h. Soc. 297(1)
(1986), 53–61.
Depa amen o de Ma em´a ica
Facul ad de Ciencias Exac as y Na u ales
Uni e sidad de Buenos Ai es
Ciudad Uni e si a ia, Pabell´on I
1428 Ciudad de Buenos Ai es
A gen ina
E-mail add ess:[email p o ec ed]
E-mail add ess:[email p o ec ed]
P ime a e si´o ebuda el 12 de desemb e de 2001,
da e a e si´o ebuda el 8 de juliol de 2002.