WEIGHTED LOCAL BMO SPACES AND THE LOCAL HARDY-LITTLEWOOD MAXIMAL OPERATOR
AN´IBAL CHICCO RUIZ AND ELEONOR HARBOURE
Abstract. We define a local type of a weighted BMO space on R+ and
prove the boundedness of the local Hardy-Littlewood maximal function in that space, provided that the weight belongs to the local classA1loc.
1. Introduction
Let us fix κ > 1. We say that I = (a, b) is a κ-local interval whenever 0< a < b < κaand we will callcritical intervalsto those of the form (a, κa) for a >0. Also we shall denote withIκthe family of all local intervals with respect to
κ. With this notation we introduce the definition of theκ-local Maximal Operator onR+= (0,∞) as follows: Given any measurable functionf :R+→R, we set
Mκlocf(x) = sup x∈I∈Iκ
1
|I|
Z
I
|f(y)|dy,
for anyx∈R+. This operator, being smaller than the regular Hardy-Littlewood maximal function, is bounded on Lebesgue spacesLp(R+) for 1 < p <∞ and of weak type forp= 1. However, as it was shown in [4],Lp-weighted inequalities hold
for a wider class than Muckenhoupt’sAp weigths, theAploc,κclasses, which require
control of averages only for local intervals. Nowak and Stempak studied this prob-lem in connection with transplantation theorems associated to Hankel Transforms. Such classes of weights were also used in [2] to prove weighted inequalities for the maximal operator of the diffusion semigroup associated to Laguerre functions systems.
To be precise, we call a weightonR+ to any nonnegative andR+-locally inte-grable function. We shall denote byAploc,κ the class of all weightsω onR+ such that there existsCκ>0 satisfying, for allI∈ Iκ,
Z
I
ω(x)dx
1/p Z
I
ω(x)−p′/pdx
1/p′
≤Cκ|I|, (1.1)
whenp >1, and
ω(I)≤Cκ|I|ess infx∈Iω(x), (1.2)
whenp= 1.
The semi-norm [ω]p,κis the least constantCκfor which (1.1) or (1.2) holds.
In [4], the authors prove that, for a fixed p, the classes are independent of κ, namely, thatAploc,κ=Aploc,2, for anyκ >1. Therefore we will denote that class just withAploc. Nevertheless let us observe that the semi-norms [ω]p,κ actually depend
on κand could happen, for some weight ω ∈ Aploc, that [ω]p,κ → ∞ as κ → ∞.
Such is the case ofω(x) = 1/x.
In the same article, the authors also show that Mκ
loc is bounded onLp(ω) if and
only ifω∈Aploc, for 1< p <∞, and that Mκ
loc is of weak type (1,1) with respect
toω(x)dxif and only ifω∈A1
loc, with boundedness constants depending onκonly
by [ω]p,κ.
On the other hand, it is well known thatM, the usual Hardy-Littlewood maximal operator, is not bounded on BM O, the space of John and Nirenberg. In fact, it was shown in [1] that for aBM O(Rn) function,M(f) is either identically∞or it does belong toBM O. They also show that if the underlying space is a cube then M is actually bounded onBM O.
The purpose of this work is to investigate the behavior of the Local Maximal Operator on appropriate weightedBM Ospaces. We believe that our result is new even in the unweighted case.
The weighted local BMO space,BM Oκ
loc(ω), will be defined in section 3. In the
theorem 3.4, we will show the boundedness of Mκ
loc onBM Oκloc(ω) for weights ω
such thatω ∈A1
loc. Let us note that if we define L∞ω−1 ={f :f ω−1∈L∞(R+)}, it is clear that Mκ
loc : L∞ω−1 −→Lω∞−1 for weights ω ∈A1loc. Such class of weights can be seen as the limit case of the weighted Lp-inequalities Mκ
loc : Lp(ωp) −→
Lp(ωp), since the condition required, ωp ∈ Ap
loc, is equivalent to ω−p ′
∈ Aploc′ . Since L∞
ω−1 ⊂BM O(ω)⊂ BM Olocκ (ω), it is natural to ask ω ∈ A1loc in order to obtain boundedness of Mκ
loc onBM Olocκ (ω). Observe that for the classical
Hardy-Littlewood maximal function (see Theorem 2.3 below), that boundedness does not hold.
2. Some preliminary results
From their definition it is clear thatAploc classes satisfy
(1) Aploc⊂Aqloc,1≤p≤q; (2) ω∈Aploc if and only ifω1−p′
∈Aploc′ .
As usual, we denoteA∞
loc = S
p≥1A p
loc. The following property, that we borrow
from [4], will be useful in the sequel.
Lemma 2.1. Let ω ∈ Aploc, 1 ≤ p < ∞. Then, for every κ > 1, there exists a constant Cκ depending onκ,pand the Aploc-norm ofω, such that
ω(I) ω(S)≤Cκ
|
I| |S|
p
,
Remark2.2. The casep= 1 of this Lemma arises directly fromA1
locclass definition,
since for allS⊂Iwe have ess infx∈Iω(x)≤ess infx∈Sω(x)≤ |S|ω(S). Thus, (1.2)
give us
ω(I)≤Cκ
|I|
|S|ω(S) (2.1)
for anyI∈ Iκ and any setS⊂I.
Next we introduce the precise definitions of the Hardy-Littlewood maximal op-erator supported on a given cube and the correspondingBM Ospace.
Let Q a fixed cube in Rn. The Hardy-Littlewood maximal functionMQ
sup-ported onQis given, for anyQ-locally integrable functionf and anyx∈Q, by
MQf(x) = sup 1
|Q′|
Z
Q′
|f(y)|dy,
where the supremum is taken over all cubesQ′ contained inQand containingx.
Given a weightω defined onQ, the weighted Bounded Mean Oscillation Space on Q, BM OQ(ω), is defined as the set of Q-locally integrable functions f that
satisfies
1 ω(I)
Z
I
|f(x)−fI|dx≤C, (2.2)
for all cubesI ⊂Q, wherefI = |1I|RIf(x)dx. The semi-norm kfkBMOQ(ω) is the least constantC that satisfies this condition.
WithBM On(ω) we denote the space whenQ=Rn and in that case we required
f to be locally integrable and satisfying (2.2) for any cubeI⊂Rn.
In [1], the unweighted version of the following result is established (see theorem 4.2 there). We claim that the same proof, with some obvious modifications, can be adapted to this setting.
Theorem 2.3. (1) Let Q a fixed cube in Rn and ω a weight of A1(Q) class.
If f belongs toBM OQ(ω)thenMQf belongs toBM OQ(ω)and
kMQfkBMOQ(ω)≤CkfkBMOQ(ω),
where C depends only on the dimensionnand the A1(Q)constant ofω.
(2) Let ω ∈ A1(Rn). If f belongs toBM O
n(ω) and if M f is not identically
infinity, then M f belongs toBM On(ω)and
kM fkBMOn(ω)≤CkfkBMOn(ω),
3. Local BMO space
For κ > 1 and a R+ weight ω, we denote with BM Oκ
loc(ω) the family of all
functionsf ∈L1
loc(R+) that satisfy the bounded oscillation condition
1 ω(I)
Z
I
|f(x)−fI|dx≤Cκ, for allI∈ Iκ, (3.1)
and thebounded mean condition 1
ω(I)
Z
I
|f(x)|dx≤Cκ, for allI∈ Iκc, (3.2)
where we considerIc
κ={(a, b) :a >0, b≥κa}.
TheBM Oκ
loc(ω) norm off is the least constant that satisfies both conditions
and will be denoted withkfkBMOκ loc(ω). Observe that, since 1
ω(I) R
I|f(x)−fI|dx≤2 1 ω(I)
R
I|f(x)|dx for any measurable
set I, we have that, for f ∈ BM Oκ
loc(ω), the bounded oscillation condition (3.1)
actually holds for any interval I⊂⊂R+. Also, if 1< κ < κ′ then BM Ok
loc(ω)֒→
BM Ok′
loc(ω). Moreover, we have:
Lemma 3.1. If ω∈A∞
loc then BM Olock (ω) =BM Ok ′
loc(ω)for any κ, κ′ >1, with
norms and equivalence constants depending onω,κandκ′.
Proof. Consider 1< κ < κ′. By the observation made before, it will be enough to
prove, forf ∈BM Oκ′
loc(ω) and I∈ Iκc, that ω(I)1 R
I|f(x)|dx ≤CkfkBMOκ′
loc, with C=C(κ, κ′, ω).
If I ∈ Ic
k′, then there is nothing to prove. If I = (a, b) ∈ Ik′ ∩ Iκc we have
κa≤b < κ′a. Then, using Lemma 2.1, we obtain
Z
I
|f(x)|dx ≤
Z κ′a
a
|f(x)|dx
≤ kfkBMOκ′
locω((a, κ
′a))
≤ CkfkBMOκ′
locω((a, κa))
≤ CkfkBMOκ′
locω(I),
and this complete the proof.
The following Lemma says that is enough to prove the bounded mean condition (3.2) only for critical intervals to conclude that a function is inBM Oκ
loc(ω).
Lemma 3.2. Letω∈A∞
loc. If a function f satisfies (3.2) for anyI= (a, κa)with
Proof. Let I= (a, b) withb > κa and letj0 ≥1 an integer such thatκj0a < b≤
κj0+1a. Then
Z
I
|f(x)|dx≤
j0
X
j=0 Z κj+1a
κja
|f(x)|dx.
Since each (κja, κj+1a) are critical intervals, by hypothesis we have Z
I
|f(x)|dx ≤ Cκ j0
X
j=0
ω (κja, κj+1a)
≤ Cκω(I) +ω (κj0a, κj0+1a).
Since the interval (κj0−1a, κj0+1a) belongs toI
κ3, Lemma2.1implies
ω (κj0a, κj0+1a) ≤ ω (κj0−1a, κj0+1a)
≤ Cκ ω((κj0−1a, κj0a))
≤ Cκω(I).
Thus, we have obtainedRI|f(x)|dx≤Cκ ω(I) for anyI∈ Iκc.
Another useful property is the following one. Note that, in the classic BMO context, this is a consequence of the John-Nirenberg inequality.
Lemma 3.3. Equivalence of norm’s property. Let ω ∈ Aploc and κ > 1. For 1≤r≤p′, there exists a constantC
κ=C(r, κ,[ω]p,κ)such that iff ∈BM Oκloc(ω)
then
1 ω(I)
Z
I
|f(x)−fI|rω1−r(x)dx 1/r
≤Cκ kfkBMOκ
loc(ω) (3.3) for allI∈ Iκ, and
1 ω(I)
Z
I
|f(x)|rω1−r(x)dx 1/r
≤Cκ, kfkBMOκ
loc(ω) (3.4)
for allI∈ Ic κ.
Proof. Let ω ∈ Aploc and f ∈ BM Oκ
loc(ω). First, we will prove that (3.3) holds.
For anyi∈Z, letJi= (κi, κi+3). Then,ω∈Aploc andJi ∈ Iκ4 impliesω∈Ap(Ji), with [ω]Ap(Ji)≤[ω]p,κ, for anyi∈Z.
Since BM Oκ
loc(ω) ⊂ BM O(ω), the BM O space supported on R+, f|Ji ∈ BM OJi(ω) and from the known equivalence of norm’s inequality for BM OJi(ω)
we have
1 ω(I)
Z
I
|f(x)−fI|rω1−r(x)dx 1/r
≤CikfkBMOκ loc(ω),
for anyI⊂Ji. Since the constantCi depend ofionly by [ω]Ap(Ji), we can replace it by a constant Cκ independent of Ji. Thus, since every I ∈ Iκ is contained in
someJi,i∈Z, we obtain the desired result (3.3).
1 ω(I)
Z
I
|f(x)|rω1−r(x)dx 1/r
≤
1 ω(I)
Z
I
|f(x)−fI|rω1−r(x)dx 1/r
+
ω1−r(I)
ω(I)
1/r
|fI|.
The first term of the right side is bounded by kfkBMOκ
loc(ω), we can prove this following the same argument as in the proof of (3.3). For the second term, observe that Ibelong to Iκ2 andω1−r belong toArloc, sinceω∈Ap
loc andp≤r′, and this
impliesω1−r(I)1/rω(I)1/r′
≤Cκ |I|. Then
ω1−r(I)
ω(I)
1/r
|fI| ≤ Cκ
1 ω(I)
Z
I
|f(x)|dx
≤ Cκ kfkBMOκ loc(ω).
To extend this result to intervalsI = (a, b) with b > κa, we proceed as we did
in the proof of Lemma3.2.
We now state our main result.
Theorem 3.4. If κ >1 andω∈A1
loc then there exist a constantC=C(κ,[ω]1,κ)
such that
kMκ
locfkBMOκ
loc(ω)≤CkfkBMOlocκ (ω) for allf ∈BM Oκ
loc(ω).
Proof. Letf ∈BM Oκ loc(ω).
We will prove first that the bounded oscillation condition (3.1) holds for Mκ locf.
ConsiderI= (a, b)∈ Iκ, i.e., 0< a < b < κa. We want to prove
1 ω(I)
Z
I
|Mκ
locf(x)−c|dx≤CkfkBMOκ
loc(ω), (3.5)
for some constantcdepending onf andIandC=C(κ,[ω]1,κ).
Let j0 ∈ Zsuch that κj0 < a≤κj0+1 and call I0 = (κj0−1, κj0+3). Then, for
anyx∈Iand anyJ= (a′, b′)∈ I
κwithx∈J, we haveJ ⊂I0. That is true since
I∩J6=∅impliesa′< bandb′ > aand thereforeb′< κa′< κb < κ2a≤κj0+3 and a′> b′/κ > a/κ > κj0−1. Then, for anyx∈I,
Mκ
locf(x)≤MI0f(x),
whereMI0 is the Hardy Littlewood maximal operator supported inI0. That is, for x∈I we take averages only over intervals contained inI0.
We bound the left side of (3.5) by the sum ofA andB, where
A= 1 ω(I)
Z
I
and
B = 1 ω(I)
Z
I
|MI0f(x)−c|dx.
We first consider A. Since for all x ∈ I we have Mκ
locf(x) ≤ MI0f(x) ≤ Mκ
locf(x) +Meκlocf(x), where
e
Mκlocf(x) = sup x∈J⊂I0,J∈Icκ
1
|J|
Z
J
|f(y)|dy,
we have
A≤ 1
ω(I)
Z
I e
Mκlocf(x)dx.
If J ∈ Ic
κ = {(α, β) : α > 0, β ≥ κα} and J ⊂ I0 = (κj0−1, κj0+3), then
|J|>(κ−1)κj0−1= κ−1
κ4−1|I0|. This implies that e
Mκlocf(x) ≤ Cκ 1
|I0| Z
I0
|f(y)|dy
≤ Cκ kfkBMOκ loc(ω)
ω(I0)
|I0|
,
for anyx∈I, where the last inequality arises from (3.2) sincef ∈BM Oκ
loc(ω) and
I0∈ Iκc.
Then
A≤Cκ kfkBMOκ loc(ω)
|I|
ω(I) ω(I0)
|I0|
. (3.6)
Sinceω ∈A1
loc,I0∈ Iκ5 andI⊂I0, (2.1) implies ω(I0)≤Cκ
|I0|
|I|ω(I)
and thenAis bounded bykfkBMOκ
loc(ω)times a constantC=C(κ,[ω]1,κ). In order to obtain the same for
B = 1 ω(I)
Z
I
|MI0f(x)−c|dx,
consider c = (MI0f)I. Observe that MI0f < ∞ a.e., since f ∈BM O(ω). Also, ω∈A1
locandI0∈ Iκ5 impliesω∈A1(I0), with theA1(I0) constant depending only of [ω]1,κ5, that is, independent ofI0and hence ofI. Then, we use Theorem2.3with Q=I0 to obtain B ≤Ckf0kBMOI0(ω), withC =C(κ, ω). Sincekf0kBMOI0(ω) ≤
kfkBMO(ω)≤ kfkBMOκ
loc(ω), we obtain the desired inequality.
Now we will prove that the bounded mean condition (3.2) for Mκlocf. By Lemma
3.2, it will be enough to prove 1 ω(I)
Z
I
|Mκlocf(x)|dx≤CκkfkBMOκ
loc(ω) (3.7)
LetI∗ = (a/κ, κ3
2a) and write f =f
1+f2, where f1 =f χI∗ and f2 =f χI∗c, where the complement is taken onR+. We will prove (3.7) for Mκ
locf1 and Mκlocf2
separately.
Consider first Mκ
locf1. From H¨older’s inequality we have
1 ω(I)
Z
I
|Mκlocf1(x)|dx≤
1 ω(I)
Z
I
|Mκlocf1(x)|2ω−1(x)dx 1/2
. (3.8)
Sinceω∈A1
loc⊂A2loc and henceω−1∈A2loc, we have, by Proposition 6.3 of [4],
that Mκ
loc is of strong type (2,2) with weight ω−1. Then, the right side of (3.8) is
bounded by a constant times
1
ω(I)
Z
I
|f1(x)|2ω−1(x)dx 1/2
. (3.9)
SinceI∗ ∈ I
κ3, I⊂I∗ and |I∗|=Cκ|I|, (2.1) impliesω(I∗)≤Cκ ω(I), and then (3.9) is bounded by
Cκ
1 ω(I∗)
Z
I∗
|f(x)|2ω−1(x)dx
1/2
.
Finally, sinceI∗∈ Ic
κ, we use the equivalence of norm’s inequality (3.4) withr= 2
and we obtain that the left side of (3.8) is bounded by a constantC=C([ω]1,κ, κ)
timeskfkBMOκ loc(ω). Consider now Mκ
locf2(x), with x ∈ I = (a, κa). Let us observe that here is
enought to take the supremum of the averages over thoseJ ∈ Iκ such thatx∈J
andJ∩I∗c 6=∅. Remember thatI∗= (a κ,(κ
3
2a)). If an intervalJ = (a′, b′) satisfies
J∩I6=∅, thena′ < κaanda < b′. If it alsoJ ∈ I
κ, thena′> a/κandb′ < κ2a.
Then we have J ⊂ I∗∗, where I∗∗ = (a/κ, κ. 2a). Otherwise that I∗ ⊂I∗∗, since
κ >1. Also, if J∩I∗c6=∅ thenb′ ≥κ3
2aand this, together witha′< κa, implies
|J|> Cκ|I∗∗|. Thus, for every x∈I we have
Mκ
locf2(x) ≤ Cκ
1
|I∗∗|
Z
I∗∗
|f(y)|dy
≤ CκkfkBMOκ loc(ω)
ω(I∗∗)
|I∗∗| , (3.10)
where the last inequality arises sincef ∈BM Oκ
loc(ω) andI∗∗∈ Iκc.
Finally, sinceω∈A1
loc, I∗∗∈ Iκ4 andI⊂I∗∗, (2.1) implies
1 ω(I)
Z
I
|Mκlocf2(x)|dx≤CκkfkBMOκ loc(ω). Therefore, the proof of Theorem3.4is complete.
4. A necessary condition
In [3], Muckenhoupt and Wheeden introduced another version of weighted BMO. More precisely, for a given intervalI, ω(I) is replaced by ess infx∈Iω(x)|I|.
Sim-ilarly, we consider now the corresponding local versionBM Oκ,loc∗(ω), the space of allR+-locally integrable functionf that satisfy
1
ess infx∈Iω(x)|I| Z
I
|f(x)−fI|dx≤Cκ, for allI∈ Iκ, (4.1)
and
1
ess infx∈Iω(x)|I| Z
I
|f(x)|dx≤Cκ, forI= (a, κa), a >0, (4.2)
and the normkfkBMOκ,∗
loc(ω) will be the least constant satisfying both conditions. It is clear that BM Oκ,loc∗(ω)⊂BM Oκ
loc(ω), since for any weightω and any ball
I we haveω(B)≥ infx∈Iω(x)|I|, and, by Lemma 3.2, a function need to satisfy
the bounded mean condition only for critical balls in order to be inBM Oκ loc(ω).
Also, if we supposeω∈A1
loc, thenBM O κ,∗
loc(ω) =BM Oκloc(ω), with equivalence of
norms. Thus, from Theorem3.4, we have that Mκ
loc is bounded fromBM Oκloc(ω)
to BM Oκ,loc∗(ω), if ω ∈ A1
loc. We will see now that the converse statement also
holds.
Theorem 4.1. If κ > 1, thenMκ
loc : BM Olocκ (ω)−→BM O κ,∗
loc(ω) if and only if
ω∈A1 loc.
Proof. From the above remark we only need to prove the necessity of ω ∈ A1 loc.
Suppose then that Mκlocis bounded fromBM Olocκ (ω) intoBM O κ,∗
loc(ω) and consider
an interval I ∈ Iκ. Since L∞(ω−1) = {f : f ω−1 ∈ L∞(R+)} is continuously
contained inBM Oκ
loc(ω), we have
1
ess infx∈Iω(x)|I|2 Z
I Z
I
|Mκlocf(x)−Mκlocf(y)|dxdy≤Ckf ω−1k∞, (4.3)
for everyf ∈L∞(ω−1).
We divide the intervalI into six disjoint subintervals of equal measure, that is,
I=I1∪I2∪I3∪I4∪I5∪I6
where all the Ii are disjoint and |Ii| = |I6|. More precisely, if I = (a, b) then
Ii= (a+b−6a(i−1), a+b−6ai),i= 1, ...,6.
If we take f =ωχI1, then from (4.3) we get
Z
I4
Z
I1
|Mκlocf(x)−Mκlocf(y)|dxdy≤C|I|2inf
x∈Iω(x), (4.4)
If x ∈ I1 then clearly Mκlocf(x) ≥ |I11|
R I1|f| =
ω(I1)
I1 . If y ∈ I4 then for any interval J such that y ∈ J and J ∩I1 6=∅ we have|J| > |I2∪I3| = 2|I1|, thus
Mκ
locf(y) ≤ 12 ω(I1)
I1 . Then we have |M
κ
locf(x)−Mκlocf(y)| ≥ Cω(I1)/|I| for any
ω(I1)≤C|I|inf x∈Iω(x).
Analogously, we can obtain the same inequality for the other intervals Ii, i =
2, ...,6, consideringf =ωχIi and integrating xoverIi and y over Ij, whereIj is at least at a distance|I|/3 away fromIi. For example, we may compareI2withI5
andI3withI1 orI6and so on.
In this way we will arrive to ω(Ii)≤C|I|inf
x∈Iω(x), fori= 1, ...,6.
Finally, adding oniwe obtain theA1condition for the intervalI∈ I
κ. Therefore,
ω∈A1 loc.
References
1. Colin Bennett, Ronald A. DeVore, and Robert Sharpley,Weak-L∞
and BMO, Ann. of Math. (2)113(1981), no. 3, 601–611.48,49
2. An´ıbal Chicco Ruiz and Eleonor Harboure,Weighted norm inequalities for heat-diffusion La-guerre’s semigroups, Math. Z.257(2007), no. 2, 329–354.47
3. Benjamin Muckenhoupt and Richard L. Wheeden,Weighted bounded mean oscillation and the Hilbert transform, Studia Math.54(1975/76), no. 3, 221–237.55
4. Adam Nowak and Krzysztof Stempak, Weighted estimates for the Hankel transform trans-plantation operator, Tohoku Math. J. (2)58(2006), no. 2, 277–301.47,48,54
An´ıbal Chicco Ruiz
Instituto de Matem´atica Aplicada del Litoral CONICET-Universidad Nacional del Litoral Santa Fe, Argentina.
anibalchicco@gmail.com
Eleonor Harboure
Instituto de Matem´atica Aplicada del Litoral CONICET-Universidad Nacional del Litoral Santa Fe, Argentina.
harbour@santafe-conicet.gov.ar
