On the Coifman type inequality for the oscillation of one-sided averages

J. Math. Anal. Appl. 336 (2007) 577–592 www.elsevier.com/locate/jmaa

On the Coifman type inequality for the oscillation of one-sided averages ✩ María Lorente a,∗ , María Silvina Riveros b , Alberto de la Torre a a Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain b FaMAF, Universidad Nacional de Córdoba, CIEM (CONICET), 5000 Córdoba, Argentina

Received 27 September 2006 Available online 12 March 2007 Submitted by L. Grafakos

Abstract In this paper we study the Coifman type estimate for an oscillation operator related to the one-sided p discrete square function S + . We prove that for any A+ ∞ weight w, the L (w)-norm of this operator, and p + therefore the L (w)-norm of S , is dominated by a constant times the Lp (w)-norm of the one-sided Hardy–Littlewood maximal function iterated two times. For the kth commutator with a BMO function we show that k + 2 iterates of the one-sided Hardy–Littlewood maximal function are sufficient. © 2007 Elsevier Inc. All rights reserved. Keywords: One-sided weights; One-sided discrete square function

1. Introduction In [5], Coifman and Fefferman proved that if T is a Calderón–Zygmund operator, w is an A∞ weight and M is the Hardy–Littlewood maximal operator, then, for each p, 0 < p < ∞, there exists C such that   p |Tf | w  C (Mf )p w, ✩ This research has been supported by MEC Grant MTM2005-08350-C03-02, Junta de Andalucía Grant FQM354, Universidad Nacional de Córdoba and CONICET. * Corresponding author. E-mail addresses: [email protected] (M. Lorente), [email protected] (M.S. Riveros), [email protected] (A. de la Torre).

0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2007.03.005

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whenever the left-hand side is finite. Inequalities of the type   |Tf |p w  C (MT f )p w, where T is an operator and MT is a maximal operator which, in general, will depend on T , are known as Coifman type inequalities. Recently, de la Torre and Torrea [26] and Lorente, Riveros and de la Torre [14] have studied inequalities with weights for the one-sided discrete square function defined as follows: for f locally integrable in R and s > 0, let us consider the averages 1 As f (x) = s

x+s f (y) dy. x

The one-sided discrete square function of f is given by 1   2  A2n f (x) − A n−1 f (x)2 . S + f (x) = 2

n∈Z

We write S + instead of S to emphasize that this is a one-sided operator, i.e., S + f (x) = S + (f χ(x,∞) )(x). In [14] it was shown that if 0 < p < ∞ and w ∈ A+ ∞ , then    + p  + 3 p S f w M f w, f ∈ L∞ (1.1) c , R

R

whenever the left-hand side is finite, where (M + )k stands for the kth iteration of M + , and 1 M f (x) = sup h>0 h +

x+h  |f |. x

A natural question left open by this result is the following: can we improve the result using fewer iterates of M + in (1.1)? In this note we study a bigger operator for which two iterates are enough. Therefore the inequality (1.1) is improved in two ways: a bigger operator on the left and a smaller operator on the right. The operator that we will study is the oscillation of the averages, 1  2 2  sup A2n f (x) − As f (x) , O+ f (x) = n∈Z s∈Jn

where Jn = [2n , 2n+1 ). It is clear that S + f (x)  O+ f (x) for all x ∈ R. If we look at the definition of O+ f (x), we see that the sequence {τn (x)}, defined by τn (x) = sups∈Jn |A2n f (x) − As f (x)|, measures the oscillation of the As f (x) in the interval Jn . Then we take the 2 norm of this sequence. Operators of this kind are of interest in ergodic theory, [3,8] and singular integrals. The behavior of O+ with respect to the Lebesgue’s measure has already been studied. Concretely, in Lemma 2.1 in [4] it is proved that O+ is of weak type (1, 1) and strong type (p, p), 1 < p < ∞, with respect to the Lebesgue’s measure. It is natural to ask what happens if we change the measure, i.e., which are the good weights for the operator O+ ? It is worth mentioning that recently, in [9], it has been proved that the oscillation of the Hilbert Transform is bounded in Lp (w), for w ∈ Ap , 1 < p < ∞. In this paper we shall obtain that O+ is

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bounded from Lp (w) to Lp (w), for w ∈ A+ p , 1 < p < ∞, as a consequence of the Coifman type inequality stated in the following theorem: Theorem 1.1. Let w ∈ A+ ∞ and 0 < p < ∞. Then, there exists C > 0 such that     + p  + p  + 2 p S f w O f wC M f w, f ∈ L∞ c , R

R

R

whenever the left-hand side is finite. Remark 1.2. As a consequence of the above theorem, if 1 < p < ∞ and w ∈ A+ p , we obtain that + + p + O and S are bounded in L (w). For S this was first proved in [26]. Remark 1.3. This theorem improves inequality (1.1) for S + substituting (M + )3 by the smaller operator (M + )2 . It is an open question if (1.1) holds with M + instead of (M + )2 , which holds for standard one-sided Calderón–Zygmund operators. The paper is organized as follows: In Section 2 we introduce notation and recall some basic results about one-sided weights and maximal operators associated to Young functions. In Section 3 we prove Theorem 1.1 and in Section 4 we study the commutators of S + and O+ with a BMO function b. For p > 1 we prove Coifman type inequalities that imply the boundedness of the commutators in Lp (w) whenever w ∈ A+ p . For p = 1 we also obtain a weak type inequality for the kth order commutator. 2. Definitions and basic facts about one-sided operators Definition 2.1. The one-sided Hardy–Littlewood maximal operators M + and M − are defined for locally integrable functions f by 1 M f (x) = sup h>0 h +

x+h 

|f | x

1 and M f (x) = sup h>0 h −

The one-sided weights are defined as follows,

p−1 b c 1 1−p  sup w w < ∞, p a
M − w(x)  Cw(x)

b

a.e.

+ A+ ∞ is defined as the union of the Ap classes,  A+ A+ ∞= p.

x |f |. x−h

1 < p < ∞,

 + Ap  + A1 

A+ ∞



p1

The A− p classes are defined reversing the orientation of R. It is interesting to note that Ap = − , A  A+ and A  A− . (See [23,15–17] for more definitions and results.) ∩ A A+ p p p p p p + p It was proved in [26], that w ∈ A+ p , 1 < p < ∞, if and only if S is bounded from L (w) to + Lp (w), and that w ∈ A+ 1 , if and only if S is of weak-type (1, 1) with respect to w.

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Definition 2.2. Let b be a locally integrable function. We say that b ∈ BMO if  1 bBMO = sup |b − bI | < ∞, I |I | I

where I denotes any bounded interval and bI =



1 |I | I

b.

Definition 2.3. Let f be a locally integrable function. The one-sided sharp maximal function is defined by

+ x+h x+2h   1 1 f dy. M +,# (f )(x) = sup f (y) − h h>0 h x

x+h

For δ > 0 we define  1 Mδ+,# f (x) = M +,# |f |δ (x) δ . It is proved in [18] that M

+,#

1 (f )(x)  sup inf h>0 a∈R h

x+h 

 + 1 f (y) − a dy + h

x

x+2h 

 + a − f (y) dy

x+h

 Cf BMO . Now we give definitions and results about Young functions. A function B : [0, ∞) → [0, ∞) is a Young function if it is continuous, convex, increasing and satisfies B(0) = 0 and B(t) → ∞ as t → ∞. A Young function B is said to satisfy Δ2 -condition (or B ∈ Δ2 ) if there exists a constant C such that B(2t)  CB(t) for t  0. The Luxemburg norm of a function f , given by B is  

  |f | 1 , f B = inf λ > 0: B λ and so the B-average of f over I is  

  1 |f | 1 . f B,I = inf λ > 0: B |I | λ I

We will denote by B the complementary function associated to B (see [1]). The following version of Hölder’s inequality holds,  1 |f g|  2f B,I gB,I . |I | I

This inequality can be extended to three functions (see [19]). If A, B, C are Young functions such that A−1 (t)B −1 (t)  C −1 (t), then f gC,I  2f A,I gB,I .

(2.1)

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Definition 2.4. For each locally integrable function f , the one-sided maximal operators associated to the Young function B are defined by MB+ f (x) = sup f B,(x,b)

and MB− f (x) = sup f B,(a,x) . a
x
We will be dealing with the Young functions Bk (t) = et − 1 and B k (t) = t (1 + log+ (t))k , + k ∈ N. The maximal operator associated to B k , M + will be denoted by ML(1+log + L)k . It is 1/k

Bk

+ + k+1 . proved in [22] that ML(1+log + L)k is pointwise equivalent to (M ) It is convenient to look at our operators as vector valued. Let us consider the sequence

 1 1 H (x) = n χ(−2n ,0) (x) − n−1 χ(−2n−1 ,0) (x) 2 2 n∈Z

and let us define the operator U : f → Uf by  Uf (x) = H (x − y)f (y) dy. R

Then it is clear that S + f (x) = Uf (x)2 . If instead of the sequence H we consider for each s > 0 the sequence K(x) = {Kn,s (x)}n∈Z , where   1 1 n χ Kn,s (x) = χ (x) − (x) χJn (s), (−2 ,0) (−s,0) 2n s we can define the operator V acting on locally integrable functions f , as Vf (x) = R K(x − y)f (y) dy. If for functions h : R × Z → R we define the norm 1  2 2    hE = sup h(s, n) , n∈Z s∈Jn

then O+ f (z) = Vf (z)E . 3. Proof of Theorem 1.1 The key point in the proof of Theorem 1.1 is the following pointwise estimate: for each 0 < δ < 1, there exists C so that for any locally integrable function   + (3.1) Mδ+,# O+ f (x)  CM + f (x) + CML(1+log + L) f (x). Let us prove inequality (3.1). For 0 < δ < 1 we have

  Mδ+,# O+ f (x)  C sup inf h>0 c∈R

1 h

x+2h 

  +  O f (y)δ − |c|δ  dy

1 δ

.

(3.2)

x

Let x ∈ R and h > 0. Let us consider the unique i ∈ Z such that 2i  h < 2i+1 and let denote by J the interval J = [x, x + 2i+3 ). If we write f = f1 + f2 , where f1 = f χJ , and choose c = O+ (f2 )(x), we have

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1 h

x+2h 

     + O f (y)δ − O+ f2 (x)δ  dy

1 δ

x



x+2  i+2



x+2  i+2

  + O f (y) − O+ f2 (x)δ dy

1 C i 2

1 δ

x

  + O f1 (y)δ dy

1 C i 2

1 δ



1 +C i 2

x

x+2  i+2

  + O f2 (y) − O+ f2 (x)δ dy

1 δ

x

= I + II.

(3.3)

Using Lemma 2.1(1) in [4], that is, the fact that O+ is weak type (1, 1) with respect to the Lebesgue’s measure, and Kolmogorov’s inequality, we get 1 I C i 2

x+2  i+3

  f (y) dy  CM + f (x).

(3.4)

x

In order to estimate II, we first use Jensen’s inequality and obtain

1 II = C i 2

x+2  i+2

     Vf2 (y) − Vf2 (x) δ dy E E

1 δ

x

C

1 2i

x+2  i+2

  Vf2 (y) − Vf2 (x) dy. E

(3.5)

x

Let us estimate Vf2 (y) − Vf2 (x)E :   Vf2 (y) − Vf2 (x) E n   y+2  y+s

 1 1  = f2 − f2 χJn (s)  2n s y

 −

y



x+2n

1 2n

x

1 f2 − s



n∈Z, s∈R



x+s f2 χJn (s) x

n∈Z, s∈R E

n   y+2 x+2   n

 1 1   f2 − n f2 χJn (s)  2n 2

y

x

 x+s  y+s

 1 1  + f2 − f2 χJn (s)  s s x

= III + IV.

y

    

    

n∈Z, s∈R E

    

n∈Z, s∈R E

(3.6)

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Observe that since y ∈ (x, x + 2i+2 ) and f2 has support in (x + 2i+3 , ∞), it follows that, if n  i + 2, then x + 2n  x + 2i+2 and y + 2n  x + 2i+2 + 2n  x + 2i+2 + 2i+2 = x + 2i+3 . As a consequence the only nonzero terms in III are those with n > i + 2. Therefore ∞  y+2n 2 1  2   1   III  f . (3.7)  n 2  n=i+3

x+2n

Let us consider the Young function B1 (t) = et − 1. Then B 1 (t) = t (1 + log+ t) and B1−1 (t) = log+ (1 + t). Using the generalized Hölder’s inequality, we obtain that, for n  i + 3, n     1 y+2  1   f n  n 2  2

x+2n

x+2  n+1

|f |χ[x+2n ,y+2n ) x+2n

 CM + f (x)χ[x+2n ,y+2n ) B1 ,[x+2n ,x+2n+1 ) B1

= CMB+ f (x) 1

1 2n B1−1 ( y−x )

 CMB+ f (x) 1

1 B1−1 (2n−i−2 )

,

(3.8)

where in the last inequality we have used that y − x  2i+2 and that B1−1 is nondecreasing. Putting together (3.7) and (3.8) we obtain ∞

1 2  1 III  CMB+ f (x) 1 (B1−1 (2n−i−2 ))2 n=i+3 ∞

1 2  1  CM + f (x) = CM + f (x). (3.9) B1 B1 (n − i − 2)2 n=i+3

Let us estimate IV. For n ∈ Z, set  x+s y+s  1 1   f2 − f 2 . βn = sup   s s∈Jn  s x

y

Then, if βn = 0, we have that there exists sn ∈ Jn such that  x+s y+s  n  1  n 1   1 f2 − f2  > βn .   sn  2 sn x

y

If n  i + 1, then y + sn  y + 2n+1  x + 2i+2 + 2n+1  x + 2i+3 . Therefore in IV we may assume n  i + 2. Using again generalized Hölder’s inequality, we get that, for n  i + 2, 1 βn  C sn

y+s  n

x+sn

1 |f2 |  C sn

x+2  n+2

|f |χ[x+sn ,y+sn ) x

2n+2 + C M f (x)χ[x+sn ,y+sn ) B1 ,(x,x+2n+2 ) B1 sn

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M. Lorente et al. / J. Math. Anal. Appl. 336 (2007) 577–592

= CMB+ f (x) 1

1 n+2 B1−1 ( 2y−x )

 CMB+ f (x) 1

1 B1−1 (2n−i )

.

(3.10)

Then, IV  CM + f (x) B1

∞ 

1 2

1

(B1−1 (2n−i ))2 n=i+2

= CM + f (x). B1

(3.11)

Collecting inequalities (3.2)–(3.6), (3.9) and (3.11), we obtain (3.1). On the other hand, we have + + 2 that MB+ f = ML(1+log + L) f is pointwise equivalent to (M ) f (see [22]). As a consequence, 1 (3.1) gives  2   Mδ+,# O+ f (x)  C M + f (x) a.e. x ∈ R. To finish the proof of Theorem 1.1 we only have to observe that since w ∈ A+ ∞ , there exists + r > 1, such that w ∈ A+ r . Then, for δ small enough, we get that r < p/δ and thus, w ∈ Ap/δ . Therefore, by Theorem 4 in [18], we get     + p  +  + p  +  + δ  p δ w O f  w  Mδ O f M O f w= R

R



C

R

 +,#  + p Mδ O f wC

R



 + 2 p M f w,

(3.12)

R

whenever the left-hand side is finite. 4. Commutators The commutators of singular integrals with BMO functions have been extensively studied (see [2,6,24,25,20,21,10–13]). Since S + can be considered as a singular integral whose kernel satisfies a weaker condition (see [14]), it is interesting to know if the results about commutators of singular integrals can be extended to S + . In [14] we have proved that the classical results about boundedness with weights can be extended to S + and, furthermore, can be improved allowing a wider class of weights, since S + is a one-sided operator. The results in [20] and [21] have been improved in [11] for one-sided singular integrals. Observe that for standard Calderón–Zygmund singular integrals (satisfying the usual Lipschitz condition) one obtains Mf instead of M 2 f in Theorem 1.1. Therefore we cannot expect to obtain the same results for the commutator of S + as we obtained in [11] for one-sided singular integrals. However, we can give estimates of the same kind, increasing in one the iterations of M + . Concretely, for the kth order commutator of S + and O+ we have: Theorem 4.1. Let b ∈ BMO and k = 0, 1, 2, . . . . Let us define the kth order commutator of S + and O+ by      k +,k  Sb f (x) =  b(x) − b(y) H (x − y)f (y) dy  2 R

and



     k  . Ob+,k f (x) =  b(x) − b(y) K(x − y)f (y) dy   R

E

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585

(Observe that for k = 0 we obtain S + and O+ .) Then, for 0 < p < ∞ and w ∈ A+ ∞ , there exists C > 0 such that,     +,k p  +,k p  + k+2 p Sb f w  Ob f w  C M f w, f ∈ L∞ c , R

R

R

whenever the left-hand side is finite. Remark 4.2. In [10], the LA,k -Hörmander condition was introduced. If we just use that H , 1 the vector valued kernel of S + , satisfies the LA,k -Hörmander condition for A(t) = e 1+k+ , then Theorem 3.3 in [10] gives (M + )k+3 instead of (M + )k+2 in the previous inequality for Sb+,k . +,k Remark 4.3. In particular, we have that for 1 < p < ∞ and w ∈ A+ and Sb+,k are bounded p , Ob

in Lp (w), which was proved for Sb+,k using a different approach in [12].

In [26] it was proved that S + is of weak type (1, 1) with respect to w, iff w ∈ A+ 1 . The commutator is more singular than the operator. A fact that is not apparent in the Lp (w) norm but it makes a difference near L1 (w). Theorem 4.4. Let b ∈ BMO, w ∈ A+ ∞ and k = 0, 1, 2 . . . . Then, there exists C > 0 such that   w x ∈ R: Sb+,k f (x) > λ    w x ∈ R: Ob+,k f (x) > λ    |f (x)| k+1 − |f (x)| log+ 1 + M w(x) dx, C λ λ

f ∈ L∞ c ,

R

whenever the left-hand side is finite. − Remark 4.5. If w ∈ A+ 1 , we can put w instead of M w in the right-hand side.

The following lemma will allow us to use induction in the proof of Theorem 4.1. Lemma 4.6. Let 0 < δ < γ < 1, b ∈ BMO and k ∈ N ∪ {0}. Then there exists C > 0 such that for any locally integrable f , k−1    +,j   + Mδ+,# Ob+,k f (x)  C Mγ+ Ob f (x) + CML(1+log + L)1+k f (x) j =0

C

k−1 

 +,j  k+2  Mγ+ Ob f (x) + C M + f (x)

a.e.

j =0

Proof. The case k = 0 follows from inequality (3.1) in the proof of Theorem 1.1. Let us prove the case k  1. Let λ be an arbitrary constant. Then, b(x) − b(y) = (b(x) − λ) − (b(y) − λ) and

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 Ob+,k f (x) =  

   k b(x) − b(y) K(x − y)f (y) dy  

R

E

 k      j  k−j   = Cj,k b(x) − λ K(x − y)f (y) dy  b(y) − λ   j =0

E

R

     k  b(y) − λ K(x − y)f (y) dy    R

E

 k      j  k−j   b(y) − λ + Cj,k b(x) − λ K(x − y)f (y) dy    j =1

R

E

  = O+ (b − λ)k f (x)   k k−j     s+j  k−j −s   b(x) − b(y) Cj,k,s b(x) − λ K(x − y)f (y) dy  +   j =1 s=0

R

k−1  k−m +,m     O+ (b − λ)k f (x) + Ck,m b(x) − λ Ob f (x),

E

(4.1)

m=0

where Cj,k (respectively Cj,k,s ) are absolute constants depending only on j and k (respectively j , k and s). Let x ∈ R and h > 0. Let i ∈ Z be such that 2i  h < 2i+1 and set J = [x, x + 2i+3 ). Then, write f = f1 + f2 , where f1 = f χJ and set λ = bJ . Then, for any a ∈ R we have

1 h

x+2h 

 +,k    O f (y) δ − |a|δ  dy

1 δ

b

x



1 h

x+2h 

  +,k   O f (y) − a δ dy

1 δ

b

x

 k−1 x+2h

1 δ  1   (k−m)δ  +,m δ b(y) − bJ  Ob f (y) dy C h m=0

+

1 h

x

x+2h 

 +   O (b − bJ )k f1 (y)δ dy

1 δ

x

+

1 h

x+2h 

  +  O (b − bJ )k f2 (y) − a δ dy

1  δ

x

= (I ) + (II) + (III).

(4.2)

Let us estimate (I ). Since 0 < δ < γ < 1, we can choose q such that 1 < q < γδ . Then, using Hölder’s and John–Nirenberg’s inequalities, we get

M. Lorente et al. / J. Math. Anal. Appl. 336 (2007) 577–592

587

1  x+2h

1 x+2h   k−1 δq δq  δq  (k−m)δq   +,m 1 1  dy b(y) − bJ  O (I )  C dy × f (y) b h h m=0

x

x

 1 k−m  x+8h  k−1 δq (k−m)   (k−m)δq  1 b(y) − bJ  C dy h m=0

×

x x+2h 

δq  +,m O f (y) dy

1 h

1

δq

b

x

C

k−1 

k−1    +  +,m  Ob f (x)  C Mδq Mγ+ Ob+,m f (x).

m=0

(4.3)

m=0

Kolmogorov’s inequality plus the fact that O+ is of weak type (1, 1) with respect to the Lebesgue measure imply 1 (II)  C h

x+2  i+3

    b(y) − bJ k f (y) dy.

x

Using now the generalized Hölder’s inequality with Bk+1 (t) = et t (1 + log+ t)k+1 we get,   (II)  C |b − bJ |k B ,J f B k+1 ,J .

1/(k+1)

− 1 and B k+1 (t) =

k+1

It follows from John–Nirenberg’s inequality that k+1 + (II)  Cb − bJ k+1 B1 ,J f B k+1 ,J  CbBMO MB k+1 f (x)  k+2  C M+ f (x).

(4.4)

For (III) we take a = O+ ((b − bJ )k f2 )(x). Then, by Jensen’s inequality, 1 (III)  C i 2

x+2  i+3

 +     O (b − bJ )k f2 (y) − O+ (b − bJ )k f2 (x) dy

x

1 C i 2

x+2  i+3

      V (b − bJ )k f2 (y) − V (b − bJ )k f2 (x) dy. E

(4.5)

x

For j  3, let Ij = [x + 2j , x + 2j +1 ) and I˜j = [x, x + 2j +1 ). As in inequality (3.6) we have       V (b − bJ )k f2 (y) − V (b − bJ )k f2 (x) E n n   

y+2 x+2     1 1   k k  (b − b ) f − (b − b ) f (s) χ  J 2 J 2 J n   2n 2n y

x

n∈Z, s∈R E

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M. Lorente et al. / J. Math. Anal. Appl. 336 (2007) 577–592

 x+s 

y+s  1 1  k k + (b − bJ ) f2 − (b − bJ ) f2 χJn (s)  s s x

y

    

n∈Z, s∈R E

= (III n ) + (III s ).

(4.6)

For (III n ), we proceed as in the estimate of (III) in Theorem 1.1. Since y ∈ (x, x + 2i+2 ) and f2 has support in (x + 2i+3 , ∞), it follows that, if n  i + 2, then x + 2n  x + 2i+2 and y + 2n  x + 2i+2 + 2n  x + 2i+2 + 2i+2 = x + 2i+3 . As a consequence, we only have to take into account n > i + 2. Therefore 2 1 ∞  y+2n  2   1  k (III n ) = f (b − bJ )   n 2  n=i+3

C

x+2n

n n  2 1 2 1 ∞  y+2   ∞   2  1 y+2  2   1   k k f (b − bIn )  +C f (bIn − bJ )   n  n 2  2 

n=i+3

=C

x+2n

∞    (IV n )2

n=i+3



1 2

∞    (Vn )2

+C

n=i+3

1

x+2n

2

(4.7)

.

n=i+3

Using the generalized Hölder’s inequality (2.1) with A = B1 , B = B k+1 and C = B k , followed by John–Nirenberg’s inequality we get √   k  2  (IV n )  C n b(t) − bIn  f (t)χ(x+2n ,y+2n ) (t) dt 2 In

   C (b − bIn )k B

k ,I˜n

f χ(x+2n ,y+2n ) B k ,I˜n

 CbkBMO f B k+1 ,I˜n χ(x+2n ,y+2n ) B1 ,I˜n  CM +

B k+1

1

f (x)

B1−1 (2n−i−2 )

(4.8)

.

For (Vn ) again the generalized Hölder’s inequality is used to obtain (Vn )  C(n − i − 1)k f B k+1 ,I˜n χ(x+2n ,y+2n ) Bk+1 ,I˜n  C(n − i − 1)k M +

B k+1

f (x)

1 −1 Bk+1 (2n−i−2 )

(4.9)

.

Putting together inequalities (4.8) and (4.9) we get  

(III n )  CM + f (x) B k+1 + CM +

B k+1

 CM +

B k+1

1

1

2

(B1−1 (2n−i−2 ))2 ni+3

  f (x) (n − i − 1)2k ni+3

f (x)  C(M + )k+2 f (x).

1

1 2

−1 (Bk+1 (2n−i−2 ))2

(4.10)

M. Lorente et al. / J. Math. Anal. Appl. 336 (2007) 577–592

589

Let us estimate (III s ). As in Theorem 1.1, for n ∈ Z, set  x+s  y+s 1  1  k k  (b − bJ ) f2 − (b − bJ ) f2 . βn = sup   s s∈Jn  s x

y

Then, if βn = 0 there exists sn ∈ Jn , such that   x+s y+s  n  1 1  n 1  k k  (b − bJ ) f2 − (b − bJ ) f2  > βn .   2  sn sn x

y

If n  i + 1, then y + sn  y + 2n+1  x + 2i+2 + 2n+1  x + 2i+3 . Therefore we only have to consider n  i + 2 in the estimate of III s . Then 1 βn  C sn

y+s  n

  (b − bJ )k f2 

x+sn

x+2  n+2    2n+2 1  b(t) − bJ k f (t)χ[x+s ,y+s ) (t) dt C n n n+2 sn 2 x    k 1  b(t) − bI  C n+2 f (t)χ[x+sn ,y+sn ) (t) dt n+1 2 I˜n+1

+C

1 2n+2



 (bI

n+1

 − bJ )k f (t)χ[x+sn ,y+sn ) (t) dt.

I˜n+1

By the generalized Hölder’s inequality (2.1) with the Young functions used in (4.8) and (4.9), we get   βn  C (b − bIn+1 )k B ,I˜ f B k+1 ,I˜n+1 χ(x+sn ,y+sn ) B1 ,I˜n+1 k

n+1

+ C(n − i)k f B k+1 ,I˜n+1 χ(x+sn ,y+sn ) Bk+1 ,I˜n+1   1 1 k  CM + f (x) + (n − i) n+2 B k+1 −1 2n+2 B1−1 ( 2y−x ) Bk+1 ( y−x )   k 1 (n − i)  CM + f (x) + −1 . −1 n−i B k+1 B1 (2 ) Bk+1 (2n−i ) Then,  (III s )  CM + f (x) B k+1  CM +

B k+1

∞   n=i+2

1 B1−1 (2n−i )

k+2  f (x)  C M + f (x).



2 12 +

2 12  ∞   (n − i)k n=i+2

−1 Bk+1 (2n−i )

(4.11)

Collecting now inequalities (4.2)–(4.6), (4.10) and (4.11) we finish the proof of Lemma 4.6.

2

590

M. Lorente et al. / J. Math. Anal. Appl. 336 (2007) 577–592

Proof of Theorem 4.1. Let us observe that from the definition of  · E , it follows that Sb+,k f  Ob+,k f , therefore the first inequality in Theorem 4.1 holds trivially. For the second one, we will proceed by induction on k. The case k = 0 is Theorem 1.1. Let now k ∈ N and suppose that Theorem 4.1 holds for j = 1, . . . , k − 1. In order to prove the case j = k we proceed as + in (3.12): since w ∈ A+ ∞ , there exists r > 1, such that w ∈ Ar . Then, for δ and γ small enough, + 0 < δ < γ < 1, we get that r < p/γ < p/δ and thus, w ∈ A+ p/γ ⊂ Ap/δ . Then by Theorem 4 in [18] and Lemma 4.6 we have    +,k      O f  p  M + O+,k f  p  C M +,# O+,k f  p b

δ

L (w)

C

b

 +  +,j  M O f  γ

j =0

Since w

∈ A+ p/γ b

b

Lp (w)

b

L (w)

 k+2  + C M + f Lp (w) .

(4.12)

we obtain

 +  +,j  M O f  γ

δ

L (w)

k−1 

 Lp (w)

=

   +  +,j γ  p γ w  C O +,j f  M Ob f . b Lp (w)

R

Then, by recurrence, we can continue the chain of inequalities in (4.12) by C

k−1   + j +2   M f j =0

Lp (w)

  k+2  k+2  + C M + f Lp (w)  C  M + f Lp (w) .

2

Proof of Theorem 4.4. To prove this theorem we shall use the following results: (i) For any weight w, we have that k+2    f (x) > λ  C w x ∈ R: M +

 R

  |f | k+1 − |f | + log 1 + M w. λ λ

(ii) Let 1 < p0 < ∞ and F be a family of couples of nonnegative functions such that, for w ∈ A+ ∞,   f (x)p0 w(x) dx  C g(x)p0 w(x) dx, (4.13) R

R

for all (f, g) ∈ F . Let φ ∈ Δ2 and such that there exist some exponents 0 < r0 < s0 < ∞, such that φ(t r0 )s0 is quasi convex. Then, for all w ∈ A+ ∞,     sup φ(λ)w x ∈ R: f (x) > λ  C sup φ(λ)w x ∈ R: g(x) > λ λ>0

λ>0

for all (f, g) ∈ F , such that the left-hand side is finite. Result (i) is a direct consequence of Theorem 3 in [22], since the pair (w, M − w) ∈ A+ 1. The proof of (ii) follows exactly as in Theorem 3.1 in [7], then we omit it. The Coifman type estimate in Theorem 4.1 gives inequality (4.13) for the family of func1 , where B k+1 (t) = tions (Ob+,k f, (M + )k+2 f ) (k  0). Also observe that φ(t) = B k+1 (1/t)

t (1 + log+ t)k+1 , belongs to Δ2 and φ(t r ), is quasi convex for r > 1 large enough.

M. Lorente et al. / J. Math. Anal. Appl. 336 (2007) 577–592

591

In order to prove Theorem 4.4, it suffices to consider λ = 1 (the general case follows by applying the result to the function f/λ). We may also assume bBMO = 1. Then by (ii) and (i),     w x ∈ R: Ob+,k f (x) > 1  sup φ(t)w x ∈ R: Ob+,k f (x) > t t>0

k+2     C sup φ(t)w x ∈ R: M + f (x) > t t>0    |f | M −w  C sup φ(t) B k+1 t t>0 R     1  C sup φ(t)B k+1 B k+1 |f | M − w t t>0 R    − 2  C B k+1 |f | M w. R

Acknowledgment We want to thank the referee for his helpful comments and suggestions.

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