- Email: [email protected]
Multilinear Cesàro maximal operators
J. Math. Anal. Appl. 397 (2013) 191–204
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
Multilinear Cesàro maximal operators✩ A.L. Bernardis a , R. Crescimbeni b , F.J. Martín-Reyes c,∗ a
IMAL-CONICET, Güemes 3450, (3000) Santa Fe, Argentina
b
Departamento de Matemática, Universidad Nacional del Comahue, Buenos Aires 1400, 8300 Neuquén, Argentina
c
Departamento de Análisis Matemático, Universidad de Málaga, 29071 Málaga, Spain
article
abstract
info
The study of the almost everywhere convergence of the product of m Cesàro-α averages leads to the characterization of the boundedness of the associated multi(sub)linear maximal operator. We characterize weighted weak type and strong type inequalities for this operator, extending results by Lerner et al. [A. Lerner, S. Ombrosi, C. Pérez, R. Torres, R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math. 220 (2009) 1222–1264.]. We also study the restricted weak type inequalities which are of particular interest in our case (they were not considered by Lerner et al.). © 2012 Elsevier Inc. All rights reserved.
Article history: Received 24 June 2011 Available online 22 July 2012 Submitted by R.H. Torres Keywords: Multi(sub)linear maximal operators Cesàro operators Weighted norm inequalities
1. Introduction Let us consider the dilated functions ϕR (x) = R1n ϕ( Rx ), R > 0, of a nonnegative integrable function ϕ defined on Rn such that ϕ = 1. It is well known that the study of the a.e. convergence and the convergence in the Lp norm of the averages PR f = f ∗ ϕR to f as R → 0 is related to the behavior of the maximal operator Mϕ f (x) = supR>0 |f | ∗ ϕR (x). When ϕ(x) = ϕ α (x) = Cn,α (1 − |x|∞ )α χQ (0,1) (x), where x = (x1 , . . . , xn ), |x|∞ = max1≤i≤n |xi |, −1 < α ≤ 0, and Cn,α is such that ϕ α = 1, we have PR f (x) = PRα f (x) =
2n+α Cn,α
|Q (x, R)|1+α/n
Q (x,R)
f (y)d(y, ∂ Q (x, R))α dy,
where Q (x, R) = [x1 − R, x1 + R] × · · · × [xn − R, xn + R], d(y, ∂ Q (x, R)) is the distance in the infinity norm from y to the boundary of Q (x, R) and |E | is the Lebesgue measure of the set E. These averages are called Cesàro-α averages. The maximal operator associated to these averages is (essentially) Mα f (x) = sup c
R >0
1
|Q (x, R)|1+α/n
Q (x,R)
|f (y)|d(y, ∂ Q (x, R))α dy.
If α = 0, Mαc is the centered Hardy–Littlewood maximal operator which is simply denoted by M c . The non-centered Hardy–Littlewood maximal operator is Mf (x) = supx∈Q |Q1 | Q |f (y)| dy, where the supremum is taken over all the cubes Q
✩ This research has been supported by grants MTM2008-06621-C02-01, MTM2008-06621-C02-02 and MTM2011-28149-C02-02 of the Ministerio de Economía y Competitividad (Spain) and grants FQM-354 and FQM-01509 of the Junta de Andalucía. First author was supported by CAI+D-UNL, CONICET (Argentina). Second author was supported by FaEA-Universidad Nacional del Comahue (Argentina). ∗ Corresponding author. E-mail addresses: [email protected] (A.L. Bernardis), [email protected] (R. Crescimbeni), [email protected] (F.J. Martín-Reyes).
0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.07.037
192
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
such that x ∈ Q (throughout the paper, a cube will be a cube with sides parallel to the axis). The non-centered version of the Cesàro-α maximal operator is given by Mα f (x) = sup x∈Q
1
|Q |1+α/n
|f (y)|d(y, ∂ Q )α dy. Q
The operators M and M c are pointwise equivalent. Classical results assert that M and M c are of weak type (1, 1) and of strong type (p, p), 1 < p ≤ +∞, with respect to the Lebesgue measure. The characterizations of the boundedness of M c (M) in weighted spaces are well-known (see [1] and [2]). 1 1 , 1+α ) and, consequently, it is of It follows from the results in [3] that Mαc , −1 < α ≤ 0, is of restricted weak type ( 1+α 1 strong type (p, p) for p > 1+α with respect to the Lebesgue measure. It is not clear if Mα and Mαc are pointwise equivalent for α ̸= 0. Therefore, the boundedness of Mα cannot be obtained from the corresponding result for Mαc . However, the weighted inequalities for Mα are equivalent to the corresponding one for Mαc (see [4]). We shall state below the results obtained in [4]. In order to state them, we introduce definitions and notations. A non-negative measurable function (a weight) w satisfies Ap,α , −1 < α ≤ 0, 1 < p < +∞, and we write w ∈ Ap,α , if there exists C > 0 such that
1/p
w(y) dy
Ap,α : Q
w
1−p′
(y)d(y, ∂ Q )
α p′
1/p′ dy
α
≤ C |Q |1+ n ,
Q
′
for every cube Q , where p is the conjugate exponent of p. Observe that Ap,0 = Ap is the Muckenhoupt Ap class. We recall that w satisfies A1 if there exists C > 0 such that M w(x) ≤ C w(x) for a.e. x. By definition, A1,0 is A1 . It is clear that Ap,α ⊂ Ap . ′
′
Consequently, if w ∈ Ap,α and w is not the function zero a.e. then w and w 1−p are locally integrable and 0 < w, w 1−p < ∞ a.e. Other property is in the following theorem (Corollary 3.4 in [4]). Theorem 1 ([4]). Let −1 < α ≤ 0, 1 < p < +∞, and let w be a weight on Rn . If w satisfies Ap,α , then there exists s ∈ (1, p) such that w satisfies A p ,α . s
1
The weighted weak Lp -norm of a measurable function f is defined by ∥f ∥Lp,∞ (w) = supλ>0 λ[w({x ∈ Rn : |f (x)| > λ})] p
1
and Lp,∞ (w) = {f : ∥f ∥Lp,∞ (w) < ∞}, where w(E ) = E w . The weighted Lp -norm of f is ∥f ∥Lp (w) = |f |p w p and Rn p L (w) = {f : ∥f ∥Lp (w) < ∞}. The following result (see [4]) characterizes the weighted weak and strong type inequalities for Mαc and Mα .
Theorem 2 ([4]). Let w be a weight on Rn , 1 < p < ∞ and −1 < α ≤ 0. Let Sα be either Mα or Mαc . The following conditions are equivalent. (a) w ∈ Ap,α . (b) There is C > 0 such that ∥Sα f ∥Lp,∞ (w) ≤ C ∥f ∥Lp (w) for all f ∈ Lp (w). (c) There is C > 0 such that ∥Sα f ∥Lp (w) ≤ C ∥f ∥Lp (w) for all f ∈ Lp (w). From this result, it is easy to prove that for all f ∈ Lp (w) with w ∈ Ap,α , the averages PRα f converge to f a.e. and in the L (w)-norm as R → 0. The restricted weak type inequalities require other classes of weights. A weight w satisfies RAp,α , 1 ≤ p < +∞, if there exists C > 0 such that p
w
RAp,α : Q
1p α χE (y)d(y, ∂ Q )α dy ≤ C |Q |1+ n χE w
1p Q
(3)
Q
for all cubes Q and all measurable sets E. It was proved (see [5]) that RAp,0 = RAp characterizes the restricted weak type for M. Theorem 4 ([4]). Let w be a weight on Rn , 1 ≤ p < ∞ and −1 < α ≤ 0. Let Sα be either Mα or Mαc . The following conditions are equivalent. (a) w ∈ RAp,α .
1
(b) There is C > 0 such that ∥Sα χE ∥Lp,∞ (w) ≤ C ∥χE ∥Lp (w) = (w(E )) p for every measurable set E. As before, it is easy to prove that if w ∈ RAp,α then the averages PRα f converge to f almost everywhere for all f ∈ Lp,1 (w) ∞ as R → 0, where Lp,1 (w) = {f : ∥f ∥Lp,1 (w) = 1p 0 λ1/p w({x ∈ Rn : |f (x)| > λ}) λ1 dλ < ∞}. Remark 5. We point out that we can only have non-trivial restricted weak-type inequalities if p ≥ 1/(1 + α) (the proof is as the proof in Theorem 8, the multilinear case, in Section 3). It follows from Theorem 1 that if there is a non-trivial function in Ap,α , α ̸= 0, then p > 1/(1 + α).
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
193
Let us fix a natural number m > 1. Let us take real numbers pi and αi , i = 1, . . . , m, pi > 1 and −1 < αi ≤ 0. Let α
p p
i and in the Lp (ν)-norm, where ν = fi ∈ Lpi (wi ) with wi ∈ Api ,αi . Then, limR→0 i=1 PR i fi = and i=1 fi a.e. i =1 w i m m 1 1 satisfies the following (multilinear) weighted are sufficient to obtain that M . The conditions w ∈ A = α i p ,α i =1 i =1 p i i i p
m
i
Mαi (fi )Lp (ν) ≤ C
m
strong type inequality:
i =1
The maximal operator associated to m
Mαc⃗ (⃗f )(x) = sup
i=1
α 1+ ni
|Q (x, R)|
m
αi
i=1
m
∥fi ∥Lpi (wi ) .
PR fi is, up to a constant,
1
R >0 i =1
m
m
Q (x,R)
|fi (y)|(dy, ∂ Q (x, R))αi dy,
where ⃗ f = (f1 , . . . , fm ) and α ⃗ = (α1 , . . . , αm ). Since Mαc⃗ (⃗f ) ≤ i=1 Mαi (fi ) it is reasonable to think that weaker conditions on the weights would imply the boundedness of Mαc⃗ in the weighted space. In this way we reach to the aim of this article: to characterize the weights for which we have restricted weak type, weak type and strong type inequalities for Mαc⃗ . We
m
follow the ideas in [6] where the authors studied the case α ⃗ = 0⃗ = (0, . . . , 0), that is, the multi(sub)linear version of the Hardy–Littlewood maximal operator M c := M⃗c . This operator is used in [6] ‘‘to obtain a precise control on multilinear 0 singular integral operators of Calderón–Zygmund type’’. The results in [6] are used in [7] to study the convergence of ergodic multilinear averages. The study of Mαc⃗ in this paper will be useful to study the Cesàro-α ergodic multilinear averages, extending the study in [7]. 2. Statement of the main results p1 pm mWe pcollect in the next theorem some of the main results in [6]. The space L (w1 ) × · · · × L (wm ) will be denoted by i (w ). L i i=1
Theorem 6 (see [6]). For i = 1, . . . , m, let 1 ≤ pi < ∞ and let p be such that
νw⃗ =
i =1 w
m
p/pi . i
m
1
|Q | where
1 |Q |
=
m
1 i=1 pi .
Let wi be weights and
The following statements are equivalent.
(i) There is C such that ∥M c (⃗ f )∥Lp,∞ (νw⃗ ) ≤ C i=1 ∥fi ∥Lpi (wi ) (⃗ f ∈ (ii) w ⃗ ∈ Ap⃗ , that is, there exists C > 0 such that for all cubes Q
1 p
νw⃗
1/p m
Q
Q
i=1 1−p′i
wi
1/p′i
1
|Q |
1−p′i
wi
1/p′i
m
i =1
Lpi (wi )).
≤ C,
Q
is understood in the case pi = 1 as (ess inf x∈Q wi (x))−1 .
If pi > 1 for every i, then (i) and (ii) are equivalent to the following. m m (iii) There is C such that ∥M c (⃗ f )∥Lp (νw⃗ ) ≤ C i=1 ∥fi ∥Lpi (wi ) (⃗ f ∈ i=1 Lpi (wi )).
Ap⃗ is related to Muckenhoupt Ap conditions in the following way (see [6]):
w ⃗ ∈ Ap⃗ ⇔ νw⃗ ∈ Amp and 1−p′i
where wi
1−p′i
wi
∈ Amp′i ,
i = 1, . . . , m, 1/m
∈ Amp′i in the case pi = 1 is understood as wi
(7)
∈ A1 .
Our aim is to extend this theorem to the Cesàro maximal operator Mαc⃗ . This extension is not straightforward. We explain in the next few lines one of the difficulties that appear in the study of Mαc⃗ .
The results in Theorem 6 are stated for the noncentered multilinear Hardy–Littlewood maximal operator M (⃗ f )(x) = m supx∈Q i=1 |Q1 | Q |fi (y)| dy. Since M and M c are pointwise equivalent, it is clear that the boundedness in (i) and (iii) is equivalent to the corresponding ones for M . The pointwise equivalence of M and M c makes easier some parts of the proof. However that is not the situation when we work with the Cesàro maximal operators. Therefore, we have to work harder to obtain the condition on the weights from the boundedness of Mαc⃗ . For future reference, we define the non-centered multilinear version of the Cesàro maximal operator by
Mα⃗ (⃗f )(x) = sup
m
x∈Q i=1
1 α 1+ ni
|Q |
|fi (y)|(d(y, ∂ Q ))αi dy. Q
⃗ These inequalities were not studied in [6] for The restricted weak-type inequalities are relevant for the case α ⃗ ̸= 0. α⃗ = 0⃗ but the proof in this case is easy. Since the proof for arbitrary α⃗ exemplifies the difference indicated above, we start
our study of Mα⃗ and Mαc⃗ characterizing the restricted weak-type weighted inequalities. To state our first main result we introduce some notation: given m measurable sets Ei , we denote (χE1 , . . . , χEm ) by χ ⃗E⃗ .
194
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
Theorem 8. For each i = 1, . . . , m, let −1 < αi ≤ 0, α¯ = α1 + · · · + αm , 1 ≤ pi < ∞ and let p be such that p pi
1 p
=
m
1 i=1 pi .
wi . The following statements are equivalent. p (i) There is C > 0 such that ∥Mα⃗ (χ ⃗E⃗ )∥Lp,∞ (νw⃗ ) ≤ C m i=1 ∥χEi ∥L i (wi ) for all measurable sets Ei , i = 1, . . . , m. c p ∥ (ii) There is C > 0 such that ∥Mα⃗ (χ ⃗E⃗ )∥Lp,∞ (νw⃗ ) ≤ C m ∥χ Ei L i (wi ) for all measurable sets Ei , i = 1, . . . , m. i=1 (iii) w ⃗ ∈ RAp⃗,⃗α , that is, there is C > 0 such that 1p p1 m m i 1 αi νw⃗ dy ≤ C χ w χ ( y ) d ( y , ∂ Q ) Ei i Ei α 1+ ni Q Q Q |Q | i=1 i=1 Let wi be weights and νw⃗ =
m
i =1
for all cubes Q and all measurable sets Ei , i = 1, . . . , m. 1 If νw⃗ is not 0 a.e. and statement (iii) holds then pi ≥ 1+α for all i. i
⃗ However, we remark that it is not very difficult As we have already said, this theorem is new even in the case α ⃗ = 0. to see that (i) ⇔ (iii). The proof of (i) ⇒ (iii) is straightforward. To prove (iii) ⇒ (ii) we follow the ideas in [6] but we have to use that (iii) implies that νw⃗ satisfies the doubling condition (this is not necessary when α ⃗ = 0⃗ because in this case c c ⃗ it suffices to estimate M⃗ = M ). The stronger difficulty appears when α ⃗ ̸= 0 in the proof of (ii) ⇒ (iii) since we cannot 0 use that Mα⃗ and Mαc⃗ are pointwise equivalent. In particular, as in [4], we establish the implication proving that (ii) implies certain conditions of one-sided nature. Our next step is the characterization of the weak type inequalities. 1 Theorem 9. For i = 1, . . . , m, let −1 < αi ≤ 0, α¯ = α1 + · · · + αm , αi = α¯ − αi , 1+α ≤ pi < ∞. Let p be such that i p/pi m m 1 1 = i=1 p . Let wi be weights and νw⃗ = i=1 wi . The following statements are equivalent. p i
(i) There is C such that ∥Mα⃗ (⃗ f )∥Lp,∞ (νw⃗ ) ≤ C
m pi p i=1 ∥fi ∥L i (wi ) (fi ∈ L (wi )). m p ≤ C i=1 ∥fi ∥Lpi (wi ) (fi ∈ L i (wi )).
(ii) There is C such that ∥Mαc⃗ (⃗ f )∥Lp,∞ (νw⃗ ) (iii) w ⃗ ∈ Ap⃗,⃗α , that is, there exists C > 0 such that for all cubes Q
1
|Q |
νw⃗
1/p m i=1
αi p′i n
(essinf x∈Q wi (x))−1 .
Q
|Q |1+
1/p′i
1
Q
where (|Q |−1−
αi p′i n
wi (y)
1−p′i
αi p′i
d(y, ∂ Q )
wi (y)1−pi d(y, ∂ Q )αi pi dy)1/pi in the case pi ′
≤ C,
dy
Q ′
′
= 1 (consequently, αi
= 0) is understood as
(iv) The following conditions hold. (a) νw⃗ ∈ Amp, α¯ , 1−p′i
(b) wi
m
∈ Amp′ , αi , for all i = 1, . . . , m and i m
1 ri
(c) wi ∈ A mpi , αi , for all i = 1, . . . , m, where ri = (m − 1)pi + 1. ri
m
1 for all i such that αi ̸= 0. If νw⃗ is not 0 a.e. and (iii) holds then pi > 1+α i
To prove the theorem we follow the ideas in [6], except when we work with the centered multilinear maximal operator. To prove (ii) ⇒ (iii) we have to show (ii) ⇒ (iv) ⇒ (iii). As in the first theorem, for the implication (ii) ⇒ (iv) we prove that (ii) implies conditions of one-sided nature. ′ When α ⃗ = 0⃗ then conditions (b) and (c) in statement (iv) are equivalent because, for 1 < q < ∞, u ∈ Aq ⇔ u1−q ∈ Aq′ . Therefore, when α ⃗ = 0⃗ the equivalence between (iii) and (iv) is nothing but the equivalence (7). ′ ⃗ = p, w If m = 1 (p ⃗ = w and α⃗ = α ), the conditions in (iv) yield νw⃗ = w ∈ Ap,α , w1−p ∈ Ap′ ,0 = Ap′ and w ∈ Ap,α . Since Ap,α ⊂ Ap , they are equivalent to νw⃗ = w ∈ Ap,α (statement (iv) is statement (a) in Theorem 2). The following theorem shows that Ap⃗,⃗α also characterizes the weighted strong type inequalities for the multilinear maximal Cesàro operators. 1 < pi < ∞. Let p, wi and νw⃗ be as in Theorem 9. The following Theorem 10. For i = 1, . . . , m, let −1 < αi ≤ 0 and 1+α i statements are equivalent.
(i) There is C such that ∥Mα⃗ (⃗ f )∥Lp (νw⃗ ) ≤ C
(ii) There is C such that ∥Mαc⃗ (⃗ f )∥Lp (νw⃗ ) (iii) w ⃗ ∈ Ap⃗,⃗α .
m pi p i=1 ∥fi ∥L i (wi ) (fi ∈ L (wi )). m p ≤ C i=1 ∥fi ∥Lpi (wi ) (fi ∈ L i (wi )).
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
195
The main tool to prove this theorem is the characterization of the weak type and the following openness property of the condition Ap⃗,⃗α . Theorem 11. For i = 1, . . . , m, let αi , pi , p, wi and νw⃗ be as in Theorem 10. If w ⃗ ∈ Ap⃗,⃗α then there exists r > 1 such that pi r
> 1, βi = αi 1+r (mm−1) ∈ (−1, 0] for all i = 1, . . . , m, and w ⃗ ∈ A p⃗ ,β⃗ . r
⃗ = and qi are the coordinates of q⃗ then qi < pi for all i. If m > 1 then βi < αi and we have that Notice that if q w ⃗ ∈ Ap⃗,⃗α ⇒ w ⃗ ∈ Aq⃗,β⃗ for some q⃗ and some β⃗ such that qi < pi and βi < αi ; this is an improvement of the result in [6] in ⃗ where this implication was proved with β⃗ = 0. ⃗ We point out that our argument in the proof of Theorem 11 the case α ⃗ = 0, is not the same as the corresponding result in [6]. When m = 1, the above theorem gives the result proved in [4]: if w ∈ Ap,α then there exists r > 1 such that w ∈ A p ,α . r Throughout the paper, if 1 < p < ∞ then p′ is its conjugate exponent and the letter C is used to denote positive constants ⃗ p r
whose values may change from line to line. 3. Proof of Theorem 8 First of all we give a result about the weights wi and νw⃗ . Proposition 12. For each i = 1, . . . , m, let pi , p, wi and νw⃗ be as in Theorem 8. Then there exists C > 0 (we can take C = 2m ) such that for each measurable set E of finite measure we can choose measurable subsets G1 , . . . , Gm ⊂ E such that C |Gi | ≥ |E |, i = 1, . . . , m, and
m i=1
wi
Gi
p1
i
≤C
E
νw⃗
1p
.
Proof. For any measurable set F of finite measure and any λ > 0, let Aλi,F := {x ∈ F : wi (x) ≤ λνw⃗ (x)}. Notice that there
exists λ such that |Aλi,F | >
1 2
|F |.
Let us fix any measurable set E of finite measure. Let F1 = E and λ1 = inf{λ : |Aλ1,F1 | >
|F1 |}. It is clear that λ1 > 0. Let µ1 be such that 2 < µ1 < λ1 . Then ≥ |F1 | and ≤ |F1 |. Let F2 = F1 \ Clearly |F2 | ≥ 12 |F1 | = 21 |E |. λ λ 1 Let λ2 = inf{λ : |Aλ2,F2 | > 2 |F2 |} and let µ2 be any number such that 22 < µ2 < λ2 ; then |A2,2F2 | ≥ 12 |F2 | ≥ 41 |E | and µ |A2,2F2 | ≤ 12 |F2 |. We continue in this way choosing sets Fi , numbers λi and µi , i = 1, . . . , m − 1, such that |Fi | ≥ 2i1−1 |E |, λ |A1,1F1 |
λ1
λ
|Ai,iFi | ≥ Gi = have
1 2i
λ Ai,iFi
µ
|E |, |Ai,Fi i | ≤ 12 |Fi | and
λi 2
µ |A1,1F1 |
1 2
µ A1,1F1 .
1 2
µ
< µi < λi . Let Fm = Fm−1 \ Amm−−11,Fm−1 . Notice that |Fm | ≥ 12 |Fm−1 | ≥
for i = 1, . . . , m − 1 and Gm = Fm . Clearly |Gi | ≥
wi (x) ≤ λi νw⃗ (x),
1 2
1 2m
1 2m
|E |. Let us take
|E | for all i = 1, . . . , m. By the definition of the sets Gi we
x ∈ Gi , i = 1, . . . , m − 1.
(13)
For Gm we have the following inequality: m−1
wm (x) <
(µi )
− ppm i
νw⃗ (x),
x ∈ Gm .
(14)
i=1
In fact, since Fm =
m−1 i=1
p
p p
(µi ) pi νw⃗ 1 m i =1
m−1 i=1
p m−1
+···+ p
wi
we have that wi (x) > µi νw⃗ (x), for x ∈ Fm and i ̸= m. Consequently, νw⃗ =
m
i=1
p p
wi i >
p
wmpm and (14) follows. Putting together (13) and (14) we obtain
p1
i
µ
E \ Ai,Fi i
1
m λi pi
m−1
≤C
Gi
i=1 1
1
≤ 2 p − pm
µi
i=1
νw⃗
1p
νw⃗
1p
Gi
≤ 2m−1
E
νw⃗
1p
.
E
To prove (ii) ⇒ (iii) in Theorem 8, we introduce some classes of weights. Definition 15. For each i = 1, . . . , m, let αi , α¯ , pi , p, wi and νw⃗ be as in Theorem 8. It is said that w ⃗ satisfies RA− ⃗,⃗ p α ,k , k = 1, . . . , n, if there exists C > 0 such that RA− ⃗,⃗ p α ,k :
νw⃗ Uk
1p m i=1
d(y, ∂ Q )αi dy Vk ∩Ei
α¯
≤ C |Q |m+ n
m i =1
wi Vk ∩Ei
p1
i
(16)
196
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
for all cubes Q = Q (z , R), z = (z1 , . . . , zm ), and all measurable sets Ei , i = 1, . . . , m, where Uk = Kk (z , R) ∩ {y : yk ≥ zk },
Vk = Kk (z , R) ∩ {y : yk ≤ zk }
(17)
and Kk (z , R) = {y ∈ Q (z , R) : |yj − zj | ≤ |yk − zk |, j = 1, . . . , n}. The class RA+ ⃗,⃗ p α ,k is defined analogously changing the roles of Uk and Vk . Notice that if m = 1 then RAp⃗,0⃗ coincides with the class RAp := RAp,0 introduced in (3). If m = 1 then the classes RA± ⃗ ⃗ ,0 ,k p
will be simply denoted by RA± p,k . We notice that it follows from the results in [4] that RAp =
n − (RA+ p,k ∩ RAp,k ),
p ≥ 1.
(18)
k=1
Taking into account these notations, we have the following result.
p⃗,⃗α := Proposition 19. Let pi , p, wi , and νw⃗ be as in Definition 15. Let RA
n
k=1
+ p⃗,⃗α then νw⃗ ∈ RApm . ⃗ ∈ RA (RA− ⃗,⃗ α ,k ). If w p α ,k ∩ RAp⃗,⃗
p⃗,⃗α . Clearly w ⃗ ⃗ . Therefore, for Proof. Let Q be any cube and let E be any measurable set. Assume that w ⃗ ∈ RA ⃗ ∈ RA p,0 k = 1, . . . , n, and for each family of measurable sets Ei , i = 1, . . . , m, we have that
1p m
νw⃗
Uk
|Ei ∩ Vk | ≤ C |Q |m
i=1
νw⃗
1p m
Vk
m
i=1
p1
i
,
(20)
.
(21)
Vk ∩Ei
i =1
|Ei ∩ Uk | ≤ C |Q |m
wi
m
wi
p1
i
Uk ∩Ei
i =1
By Proposition 12 with E = E ∩ Vk and taking Ei = Gi in (20) we have that
νw⃗
1 mp
νw⃗
|E ∩ Vk | ≤ C |Q |
1 mp
,
Vk ∩E
Uk
for k = 1, . . . , m. In a similar way, Proposition 12 with E = E ∩ Uk and (21) give
νw⃗
1 mp
νw⃗
|E ∩ Uk | ≤ C |Q |
1 mp
.
Uk ∩E
Vk
Therefore, νw⃗ ∈
n
k=1
− (RA+ ⃗ ∈ RAmp . mp,k ∩ RAmp,k ) and then, by (18), νw
Remark 22. It is clear that if νw⃗ ∈ RAmp then νw⃗ is an A∞ weight, so it is doubling. As a consequence, or taking E = Uk and E = Vk in the definition of RAmp , we get that νw⃗ (Q ) ≤ C νw⃗ (Uk ) and νw⃗ (Q ) ≤ C νw⃗ (Vk ). − The following lemma shows the relationship between the one-sided conditions RA+ ⃗,⃗ p α ,k and RAp⃗,⃗ α ,k with the general condition RAp⃗,⃗α (see (iii) in Theorem 8). This lemma is a key result in the proof of Theorem 8.
p⃗,⃗α = RAp⃗,⃗α . Lemma 23. Let pi , p, wi , and νw⃗ be as in Definition 15. Then RA p⃗,⃗α because the sets Uk , Vk ⊂ Q . Let w p⃗,⃗α and let us see that w Proof. It is obvious that RAp⃗,⃗α ⊂ RA ⃗ ∈ RA ⃗ ∈ RAp⃗,⃗α . Assume first that α ⃗ is equal to (0, . . . , αj , . . . , 0), that is, all the numbers αi are zero, except possibly the one in the place j. It follows from Proposition 19 and Remark 22 that there exists C > 0 such that for all cubes Q and every k = 1, . . . , n, we have νw⃗ (Q ) ≤ C νw⃗ (Uk ) and νw⃗ (Q ) ≤ C νw⃗ (Vk ), where Uk and Vk are the sets associated to the cube Q in Definition 15. We have to prove that w ⃗ ∈ RAp⃗,⃗α , that is, there exists a constant C > 0 such that for all cubes Q and all measurable sets Ei , i = 1, . . . , m,
νw⃗
p1 m m αj i wi . χEj (y)d(y, ∂ Q )αj dy |Ei ∩ Q | ≤ C |Q |m+ n
1p
i=1,i̸=j
Q
Q
i=1
Ei
We estimate the left hand side in the following way:
νw⃗
1p m n m χEj (y)d(y, ∂ Q )αj dy |Ei ∩ Q | = νw⃗ χEj (y)d(y, ∂ Q )αj dy |Ei ∩ Q |
1p
Q
i=1,i̸=j
Q n
+
k=1
νw⃗ Q
1p
k=1
m
χEj (y)d(y, ∂ Q )αj dy Vk
i=1,i̸=j
Q
|Ei ∩ Q | = I + II .
Uk
i=1,i̸=j
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
197
We shall only estimate I since II is estimated in a similar way. It will suffice to obtain the estimate for each term in the sum of I. We shall do it for k = 1 and without loss of generality we assume j = 1. Let Q be a cube such that for all y ∈ U1 , 1 , where U1 is the set U1 associated to d(y, ∂ Q ) = d(y, ∂ Q ), |Q | is comparable to | Q | and Q ⊂ U Q (see the figure below).
Then, since d(y, ∂ Q ) = d(y, ∂ Q ), for all y ∈ U1 ,
νw⃗
1p m m |Ei ∩ Q | ≤ νw⃗ |Ei ∩ Q |. χE1 (y)d(y, ∂ Q )α1 dy χE1 ∩Q (y)d(y, ∂ Q )α1 dy
1p
Q
U1
Q
i =2
1 U
i =2
If V1 is the set V1 associated to Q we have νw⃗ ( Q ) ≤ C νw⃗ (V1 ) by the doubling property of νw⃗ . This inequality and the p⃗,⃗α give assumption w ⃗ ∈ RA
νw⃗
1p
Q
1p m m χE1 ∩Q (y)d(y, ∂ Q )α1 dy |Ei ∩ Q ∩ U1 | χE1 (y)d(y, ∂ Q )α1 dy νw⃗ |Ei ∩ Q | ≤ C
U1
≤ C | Q |m+
α1 n
i =1
χEi ∩Q wi
p1
i
1 U
V1
i =2
m
α1
≤ C |Q |m+
n
1 U
m i =1
χEi wi
p1
i
i =2
.
Q
This proves the lemma in the particular case α ⃗ = (0, . . . , αj , . . . , 0). We prove now the general case. Let us fix a cube Q and sets E1 , . . . , Em . By Proposition 12 applied to the set E = Q there exist subsets G1 , . . . , Gm such that i = 1 , . . . , m,
C |Gi | ≥ |Q |,
m
and
wi
p1
i
Gi
i=1
νw⃗
≤C
1p
.
(24)
Q
p⃗,⃗α then w p⃗,⃗α , where α⃗ j = (0, . . . , αj , . . . , 0) for all j. By what we have already proved we have that Since w ⃗ ∈ RA ⃗ ∈ RA j w ⃗ ∈ RAp⃗,⃗αj . For fixed j, we apply that w ⃗ ∈ RAp⃗,⃗αj in the cube Q and with the sets Eij = Gi if i ̸= j and Ejj = Ej . Then we obtain νw⃗ (Q )
1 p
m
αj
αj m+ n
χEj (y)d(y, ∂ Q ) dy
|Gi ∩ Q | ≤ C |Q |
i=1,i̸=j
Q
p1 wj
m
j
i=1,i̸=j
Ej
wi
p1
i
,
Gi
for some constant independent of j and the sets. Multiplying these inequalities on j = 1, . . . , m we have
νw⃗ (Q )
m p
m j =1
αj
m
χEj (y)d(y, ∂ Q ) dy Q
m−1 |Gi |
¯ m2 + α n
≤ C |Q |
j =1
i =1
Then, the lemma follows from (24) and the last inequality.
p1
m
wj Ej
j
m i=1
wi
mp−1 i
.
Gi
Now we are ready to prove the characterization of the restricted weak type weighted inequalities for Mαc⃗ .
p⃗,⃗α = Proof of Theorem 8. The ⃗ ∈ RA implication (i) ⇒ (ii) is obvious. To prove (ii) ⇒ (iii) it will suffice to see that w n
k=1
+ RA− ⃗,⃗ p α ,k ∩ RAp⃗,⃗ α ,k .
Assume that (ii) holds and consider for each k = 1, . . . , n, the following non-centered multi(sub)linear maximal operators
Nα⃗−,k ⃗f (x) =
sup
m
x∈Uk (z ,R) i=1
1
|Q (z , R)|
α 1+ ni
Vk (z ,R)
|fi (y)|d(y, ∂ Q )αi dy
(25)
|fi (y)|d(y, ∂ Q )αi dy,
(26)
and
Nα⃗+,k ⃗f (x) =
sup
m
x∈Vk (z ,R) i=1
1 α 1+ ni
|Q (z , R)|
Uk (z ,R)
198
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
where Uk = Uk (z , R) and Vk = Vk (z , R) are the sets considered in Definition 15 associated to the cube Q = Q (z , R). It is easy f (x) ≤ C Mαc⃗ ⃗ f (x), for all k = 1, . . . , n and all measurable function f (see the proof f (x) ≤ C Mαc⃗ ⃗ f (x) and Nα⃗+,k ⃗ to see that Nα⃗−,k ⃗
p⃗,⃗α , ⃗ ∈ RA of Proposition 2.3 in [4]). Then we get the inequality in (ii) for the operators Nα⃗−,k and Nα⃗+,k . In order to see that w let Q = Q (z , R) be a fixed cube, and let Uk = Uk (z , R) and Vk = Vk (z , R), k = 1, . . . , n, its associated sets considered in Definition 15. Let E1 , . . . , Em be measurable sets. For each k we consider the multilinear function χ ⃗E⃗k = (χE k , . . . , χE k )
with Eik = Ei ∩ Vk . For x ∈ Uk we have Nα⃗−,k χ ⃗E⃗k (x) ≥
m
i =1
1 αi |Q |1+ n
1
Vk
m
χE k (y)d(y, ∂ Q )αi dy = λ. Then, using that Nα⃗−,k i
satisfies (ii), m C νw⃗ (Uk ) ≤ νw⃗ {x : Nα⃗−,k χ⃗E⃗k (x) ≥ λ} ≤ p ∥χ k ∥p . λ i=1 Ei pi ,wi
Therefore
νw⃗
1p m
Uk
¯ m+ α n
χE k (y)d(y, ∂ Q ) dy ≤ C |Q | i
Vk
i =1
αi
m
χEi wi
p1
i
.
Vk
i=1
+ p⃗,⃗α as Then we get that w ⃗ ∈ RA− ⃗ ∈ RA+ ⃗ ∈ RA ⃗,⃗ ⃗,⃗ p α ,k . In a similar way, using that Nα⃗ ,k satisfies (ii) we obtain that w p α ,k . Thus w we wished to prove. Finally, we prove (iii) ⇒ (i). Using the assumption (iii) we get, for every cube Q , that m i=1
1
|Q |
=
α 1+ ni
m i =1
αi
χEi (y)d(y, ∂ Q ) dy ≤ C Q
m i=1
1
νw⃗ (Q )
χEi Q
wi νw⃗ νw⃗
1/pi ≤
m
χEi wi
1/pi
1
Q
Q
Mνw⃗ (χEi wi νw⃗ −1 )(x)
νw⃗
1/p
1/pi
,
i =1
where Mνw⃗ is the Hardy–Littlewood maximal operator associated to the measure νw⃗ (x)dx defined as Mνw⃗ f (x) = supx∈Q 1 |f (y)|νw⃗ (y)dy. ν (Q ) Q w ⃗
By Proposition 19, Lemma 23 and Remark 22 we have that νw⃗ is a doubling weight. Then Mνw⃗ is of weak type (1, 1) with respect to νw⃗ . It follows from the last inequality, Hölder’s inequality for weak-spaces [8, p.15] and the weak-type (1, 1) inequality for Mνw⃗ that
∥Mα⃗ (χ⃗E⃗ )∥Lp,∞ (νw⃗ ) ≤ C
m 1/pi Mνw⃗ (χEi wi νw⃗ −1 ) p ,∞ L i
i=1
=C
m pi Mν (χE wi νw⃗ −1 )11/,∞ i w ⃗ L (ν i=1
w ⃗)
(νw⃗ )
≤C
m
∥χEi ∥Lpi (wi ) ,
i=1
and we are done. Finally, we are going to prove that if w ⃗ is not the function zero a.e. and w ⃗ ∈ RAp⃗,⃗α then pi ≥ 1/(1 + αi ). We shall prove it for i = 1. Since w ⃗ ∈ RAp⃗,⃗α implies that w ⃗ ∈ RAp⃗,β⃗ where β = (α1 , 0, . . . , 0), we may assume that αi = 0 for all i ≥ 2. There exists N such that the sets Fi,N = {x : wi (x) < N } have positive measure for all i. Let z be a Lebesgue point of the locally integrable functions νw⃗ , wi χFi,N and χFi,N , i = 1, . . . , m, such that νw⃗ (z ) ̸= 0. Without loss of generality, we may assume that z = 0. Let Q = Q (0, R) and Eλ = Q (0, (1 − λ)R) \ Q (0, (1 − 2λ)R), 0 < λ < 1/4. Notice that if x ∈ Eλ then |E | d(x, ∂ Q )α1 ≥ (2λR)α1 = λα1 |Q |α1 /n and C1 λ ≤ |Qλ| ≤ C2 λ (C1 and C2 depend on n but they are independent of λ). Now, keeping in mind these inequalities, we apply that w ⃗ ∈ RAp⃗,⃗α with the cube Q and the sets E1 = Eλ ∩ F1,N and Ei = Fi,N , i ≥ 2. Then we obtain
1
|Q |
νw⃗
1p
Q 1
≤ C (λ|Q |) p1
m |Eλ ∩ F1,N | |Fi,N ∩ Q | |Eλ | |Q | i=2 p1 p1 m 1 i 1 1 1 − 1 χF w1 χFi,N wi |Q | p p1 . |Eλ | Eλ 1,N | Q | Q i=2 1
λ1+α1 |Q | p
Notice that, for fixed λ, the family Eλ = Eλ (R) is a regular family which shrinks nicely to 0. If we let R tend to 0 we obtain
λ
1+α1 − p1
1
≤ C . Since λ can be chosen so small as we wish, we have 1 + α1 −
1 p1
≥ 0.
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
199
4. Proof of Theorem 9 If αi = 0 for all i then our theorem is Theorem 6 (proved in [6]). From now on, we assume αi ̸= 0 for some i which implies pi > 1 and pm > 1. We study only the case m > 1 since the case m = 1 is contained in Theorem 2. To prove (ii) ⇒ (iii), we work with the one-sided multi(sub)linear maximal operators Nα⃗−,k and Nα⃗+,k defined in (25) and (26). We also need to introduce the following multilinear classes of weights: for k = 1, . . . , n, it is said that w ⃗ = (w1 , . . . , wm ) satisfies A− ⃗,⃗ p α ,k if there is C > 0 such that
A− ⃗,⃗ p α ,k :
1
|Q |
νw⃗
1/p m
Uk
i=1
1/p′i
1 αi p′ 1+ n i
|Q |
1−p′i
wi (y)
d(y, ∂ Q )
αi p′i
dy
≤C
(27)
Vk
for all cubes Q = Q (z , R), where Uk and Vk are the sets defined in (17) (if pi = 1 then (|Q |−1−
αi p′i n
wi (y)1−pi d(y, ∂ Q )αi pi ′
Vk
′
dy)1/pi is understood as (essinfVk wi )−1 ). The class A+ ⃗,⃗ p α ,k is defined changing the roles of Uk and Vk . The following theorem contains the proof of (iii) ⇔ (iv) and shows the relationship between the multilinear classes of weights with a family of suitable linear conditions. ′
Theorem 28. For i = 1, . . . , m, αi , pi , wi and νw⃗ be as in Theorem 9 and assume that αi ̸= 0 for some i. Let let αi , α¯ , p⃗,⃗α = nk=1 (A− ∩ A+ ). Then the following statements are equivalent. w ⃗ = (w1 , . . . , wm ). Let A ⃗,⃗ ⃗,⃗ p α ,k p α ,k
p⃗,⃗α . (i) w ⃗ ∈A (ii) The following conditions hold. (a) νw⃗ ∈ Amp, α¯ , 1−p′i
(b) wi
m
1/m
∈ Amp′ , αi , for all i = 1, . . . , m where in the case pi = 1 is understood as wi
∈ A1 and
i m
1 ri
(c) wi ∈ A mpi , αi , for all i = 1, . . . , m, with ri = (m − 1)pi + 1. (iii) w ⃗ ∈ Ap⃗,⃗α .
ri
m
Proof of Theorem 28. In this proof we shall use the following lemma (see [4]). From now on, when m = 1 the classes A− ⃗,⃗ p α ,k
− + and A+ ⃗,⃗ p α ,k are denoted by Ap,α,k and Ap,α,k , respectively.
Lemma 29([4]). Let w be a weight on Rn and let −1 < α < 0. If 1 < p < ∞ then w satisfies Ap,α if and only if + w ∈ nk=1 A− Ap,α,k . p,α,k In the proof we consider the sets J = {i : pi = 1} and H = {i : pi > 1}. (i) ⇒ (ii). Let us prove (a). Let k ∈ {1, . . . , n}. It follows from the assumptions that pm > 1 since pi > 1 for some i. By p (mp−1) Hölder’s inequality with exponents si = pi (p −1) if i ∈ H and si = +∞ if i ∈ J, we have that i m p p α¯ ′ ′ − νw⃗ (y)1−(mp) d(y, ∂ Q ) m (mp) dy = wi (y) pi (mp−1) d(y, ∂ Q )αi mp−1 dy Vk i=1
Vk
≤C
1−p′i
wi (y)
d(y, ∂ Q )
Vk
i∈H
αi p′i
dy
p(pi −1) pi (mp−1)
− mpp−1
ess infy∈Vk wi (y)
.
i∈J
p⃗,⃗α ⊂ A− , we get that νw⃗ ∈ A− α¯ . Changing the role of the sets Uk and Vk , we also obtain that νw⃗ ∈ A+ α¯ Since w ⃗ ∈A ⃗,⃗ p α ,k mp, m ,k mp, m ,k for all k. By Lemma 29 we get (a). Since Ap,α ⊂ Ap for any p and α , then (a) implies that νw⃗ is doubling. Assume pi > 1. To prove (b), it suffices to show that for all k
1−p′i
wi Ek
1 mp′ i
1− 1 ′ 1 mp αi αi ′ − ′ ′ ′ i wi (y)1−pi mpi −1 d(y, ∂ Q ) m (mpi ) dy ≤ C |Q |1+ mn ,
(30)
Fk 1−p′i
with Ek = Uk , Vk and Fk = Uk , Vk . These inequalities imply that wi 1−p′i
by Lemma 29 we have wi
∈ A−
α mp′i , mi ,k
1−p′i
and wi
∈ A+
α mp′i , mi ,k
for all k; consequently,
∈ Amp′ , αi . We shall prove (30) with (Ek , Fk ) = (Uk , Uk ), being similar the proof of the case i m
(Ek , Fk ) = (Vk , Vk ). Once these inequalities have been proved, we observe that (30) with (Ek , Fk ) = (Uk , Uk ) implies the same 1−p′ 1−p′ 1−p′ 1−p′ inequality for αi = 0 which is equivalent to wi i ∈ Amp′i . Therefore, wi i is doubling. In particular, Uk wi i ≤ C Vk wi i
200
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204 1−p′i
1−p′i
≤ C Uk wi . These inequalities together with (30) in the cases (Ek , Fk ) = (Uk , Uk ) and (Ek , Fk ) = (Vk , Vk ) and V wi k imply (30) with (Ek , Fk ) = (Uk , Vk ) and (Ek , Fk ) = (Vk , Uk ). Now we prove (30) in the case (Ek , Fk ) = (Uk , Uk ). Observe that
1−p′i
wi
1 mp′ i
α
− nmi
≤ C |Q |
Uk
1−p′i
wi (y)
αi p′i
d(y, ∂ Q )
dy
1 mp′ i
.
(31)
Uk
On the other hand, with ri = (m − 1)pi + 1, we write
Ii =
−
1−p′i
wi (y)
1 mp′ −1 i
αi
′ ′ m (mpi )
d(y, ∂ Q )
1 mp′ i
1− dy
Uk
wi (y)
=
m
1 ri
wj (y)
m
p
− p ri
wj (y)
j i
j=1,j̸=i
Uk
Ii ≤
wi (y)
m
1 ri
d(y, ∂ Q )
mpri
αj p i ri
i
dy
.
(32)
j=1,j̸=i pr
pri pi
By Hölder’s inequality with si =
and sj = (p −j 1i)p for j ∈ H, j ̸= i, and sj = +∞ for j ∈ J, we get that j i
wj (y)
pi pj ri
ri s mp
si
i
i
dy
j=1,j̸=i
Uk
×
pi pj ri
j∈H ,j̸=i
p
− p ri
wj (y)
d(y, ∂ Q )
j i
αj
pi ri
ri s mp j i
sj
dy
Uk
− m1
ess infy∈Uk wj (y)
.
− m1
j∈J
Then, since νw⃗ is a doubling weight, we have that
νw⃗
Ii ≤ C
1 mp m
j∈H ,j̸=i
Vk
1−p′j
wj
αj p′j
(y)d(y, ∂ Q )
dy
1 mp′ j
Uk
ess infy∈Uk wj (y)
.
(33)
j∈J
Putting together (31), (33) and using (i) we get (30) with (Ek , Fk ) = (Uk , Uk ). To prove (c), we argue as before and, consequently, we only have to prove
mpri i
1 ri
wi
wi (y)
1 ri
− mp1i
d(y, ∂ Q )
ri −1
αi mpi ′ ( ) m
ri
dy
1
mp ( r i )′ i
αi
≤ C |Q |1+ nm .
Uk
Uk
To prove it we use (33) (see the definition of Ii in (32)) and we obtain 1 ri
wi
mpri
i
≤ |Q |
α
− nmi
1 ri
wi (y) d(y, ∂ Q )
αi pi ri
mpri dy
αi
i
= |Q |− nm Ii
Uk
Uk αi
≤ C |Q |− nm
νw⃗
1 mp m
j∈H ,j̸=i
Vk
×
ess infy∈Uk wj (y)
− m1
1′ mp ′ ′ j wj (y)1−pj d(y, ∂ Q )αj pj dy Uk
.
j∈J
p⃗,⃗α ⊂ A+ , we obtain (34) as follows: Then, using that w ⃗ ∈A ⃗,⃗ p α ,k 1 ri
wi
mpri i
Uk
− mp1i
ri −1
αi mpi ′ ( )
d(y, ∂ Q ) m
ri
dy
1 mp
( r i )′ i
Uk α
− nmi
≤ C |Q |
νw⃗ Uk
×
wi (y)
1 ri
1 mp
j∈H
− m1
ess infy∈Uk wj (y)
j∈J
wj (y)
1−p′j
αj p′j
d(y, ∂ Q )
dy
1 mp′ j
Uk αi
α¯
αi
≤ C |Q |− nm |Q |1+ nm = C |Q |1+ nm .
(34)
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204 1/m
Now we prove (b) and (c) for i ∈ J, that is wi
1
|Q |
1/m
wi
1/m
≤ C ess infUk wi
201
∈ A1 . It suffices to prove that for all cubes Q and some k
.
(35)
Uk 1/m
Inequality (35) implies that wi ∈ A1 as follows: as in the proof of Lemma 23, given a cube Q there exists a cube Q such k , where Uk is the set Uk associated to that |Q | is comparable to |Q | and Q ⊂ U Q ; by (35)
1
|Q |
1/m
wi
C
≤
Q
1/m
1/m
wi
k U
| Q|
≤ C ess infUk wi
1/m
≤ C ess infQ wi
.
Let us prove (35). By Hölder’s inequality with exponent pm, we have
1/m i
w
p
wi
≤ Uk
Uk
p pj
wj
1 mp
pmmp−1
p − p (pm −1)
wj
.
j
(36)
Uk j∈H
j∈H
Let sj = (m − 1/p)p′j , j ∈ H. Assume first that sj > 1 for all j ∈ H. Applying Hölder’s inequality with exponents sj and the inequality d(y, ∂ Q )
w
1/m i
−αj p′j
≤ |Q |−
w
≤
p i
Uk
Uk
αj p′j n
p pj
, we obtain 1 mp
wj
j∈H
1−p′j
wj
1′ αj mp ′ j (y)d(y, ∂ Q )αj pj dy |Q |− mn .
(37)
Uk
j∈H
If sj0 = 1 for some j0 then pi = 1 for all i ̸= j0 . Consequently, H = {j0 } and inequality (37) follows from (36) since αj p′j
−α p′
d(y, ∂ Q ) j j ≤ |Q |− n (in this case we do not need to use Hölder’s inequality). By (i) and using that νw⃗ is doubling, we obtain (35) as follows:
w
1/m i
w
≤C
p i
Uk
Uk
wj
1 mp
wjp
j∈I ,j̸=i
Uk
= C ess infUk w
νw⃗
1 − mp (ess infUk wj )1/m |Q |
Uk
j∈H
wip
≤C
p pj
1/m i
j∈I
1 mp
p pj
wj
νw⃗
1 − mp
(ess infUk wi )1/m |Q |
Uk
j∈H
|Q |.
(ii) ⇒ (iii). We consider again the sets J and H. Let #(H ) be the number of elements of H. Notice that 1 α¯ #(H ) α¯ #(H ) − 1 +1 =C d(y, ∂ Q ) m2 νw⃗ (y) m2 p νw⃗ (y) m2 p dy |Q | nm2 Q α¯
d(y, ∂ Q ) m2 d(y, ∂ Q )
=C
α( ¯ #(H )−1) m2
−
νw⃗ (y)
1 m2 p
Q
−
1 m2 p
α¯
d(y, ∂ Q ) m2
Q
|Q |
nm2
+1
1 m2 pi
wi
αi
(y)d(y, ∂ Q ) m2
(y) dy
i∈H
By Hölder’s inequality with s0 = α¯ #(H )
1 m2 pi
wi
i =1
νw⃗ (y)
=C
m
≤C
m2 p mp−1
−
νw⃗ (y)
and si =
1 m2 p
s0
1 2
wim (y) dy.
i∈J
m2 pi ri
d(y, ∂ Q )
, i = 1, . . . , m, we get
s1
α¯ s0
0
m2
dy
Q
×
i∈H
wi (y)
si pi m2
d(y, ∂ Q )
si αi m2
s1 i
Q
i∈J α¯ p
1
s1 dy
i
Q
mp2−1
νw⃗ (y)− mp−1 d(y, ∂ Q ) mp−1 dy
=C
wi (y)
dy
si m2
m p
Q
×
i∈H
1 ri
wi (y) d(y, ∂ Q ) Q
αi pi ri
dy
ri m2 pi
m1 1 wi (y) m dy .
i∈J
Q
202
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
Raising to the mth power,
1
1 ≤ C
α¯ p 1+ n(mp−1)
|Q | ×
νw⃗ (y)
d(y, ∂ Q )
mpmp−1
α¯ p mp−1
dy
Q
1 αi pi 1+ nr
|Q |
i∈H
− mp1−1
1 ri
wi (y) d(y, ∂ Q )
αi pi ri
mpri
i
1 1 wim . |Q | Q i∈J
dy
Q
i
mpi ri
By Hölder’s inequality with si =
(38)
we get
mpri 1 m 1 − p 1−1 mp′i i 1 1 ri i w w 1≤ , |Q | Q i |Q | Q i i=1 where
1 |Q |
− p 1−1
Q
wi
i
1 mp′ i
(39)
is understood as ess infQ wi
−1/m
if pi = 1. Multiplying inequalities (38) and (39) and the result
by
1
|Q |
νw⃗
1 mp m
1
Q
i =1
|Q |
αi p′i
wi (y)1−pi d(y, ∂ Q )αi pi dy ′
′
1 mp′ i
,
(40)
Q
1+ n
and using (ii) we obtain that (40) is smaller than a constant C , as we wished to prove. Finally, the implication (iii) ⇒ (i) is obvious since Uk , Vk ⊂ Q . Proof of Theorem 9. (i) ⇒ (ii) is obvious. To prove (ii) ⇒ (iii) we consider the operators Nα⃗−,k and Nα⃗+,k . Since Nα⃗−,k ⃗ f ( x) ≤ C Mαc⃗ ⃗ f (x) and Nα⃗+,k ⃗ f (x) ≤ C Mαc⃗ ⃗ f (x), we get that the inequality in (ii) holds for Nα⃗−,k and Nα⃗+,k . Proceeding as in Lemma 2.4
p⃗,⃗α which is equivalent to w in [4] we get that w ⃗ ∈A ⃗ ∈ Ap⃗,⃗α by Theorem 28. To prove (iii) implies (i), notice that by Hölder’s inequality we get if pi > 1
αi
fi (y)d(y, ∂ Q ) dy ≤
Q
pi
fi wi
1/pi
Q
and if pi = 1 (αi = 0) then m
1
|Q |
i=1
α 1+ ni
f Q i
− p 1−1
wi (y)
i
αi p′i
d(y, ∂ Q )
1/p′i dy
,
Q
(y)d(y, ∂ Q )αi dy ≤
fi (y)d(y, ∂ Q )αi dy ≤ C Q
m
wi
f Q i p
fi i wi
ess infQ wi
1/pi
νw⃗
. Now, by using the hypothesis we get
−1/p
.
Q
Q
i =1
−1
1/pi
p
It follows that Mα⃗ (⃗ f )(x) ≤ C i=1 Mνw⃗ (fi i wi νw⃗ −1 (x)) , where Mνw⃗ is the Hardy–Littlewood maximal operator associated to the doubling measure νw⃗ (x)dx (by Theorem 28, νw⃗ ∈ Amp, α¯ ⊂ Amp ). Using Hölder’s inequality for weakm spaces [8, p.15] and the weak-type (1, 1) inequality for Mνw⃗ we get
m
∥Mα⃗ (⃗f )∥Lp,∞ (νw⃗ ) ≤ C
m 1/pi p Mνw⃗ (fi i wi νw⃗ −1 ) p ,∞ L i
i=1
≤C
m pi Mν (f pi wi νw⃗ −1 )11/,∞ w ⃗ i L (ν
w ⃗)
i=1
1 r
It follows from Remark 5 and wi i ∈ A mpi , αi that ri
m
mpi ri
>
(νw⃗ ) m
≤C
∥fi ∥Lpi (wi ) .
i =1 1 , αi ̸= 0, which is equivalent to pi > 1+α . i
1 α 1+ mi
5. Proof of Theorems 11 and 10 Proof of Theorem 11. Let w ⃗ ∈ Ap⃗,⃗α . Using Theorem 28 and Theorem 1 we get that there exists r0 > 1 such that 1 r
wi i ∈ A mpi , αi . We also have w ⃗ ∈ Ap⃗ . By Hölder’s inequality with si = r0 ri
m
r0 m
|Q |
=
m i=1
1 mpi
wi Q
1 − mp
wi
i
r0 ≤
m i =1
1 ri
wi Q
0 ri rmp i
Q
mpi ri
− p 1−1
wi
i
and s′i = r0 mp′ i
.
mpi , pi −1
(41)
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
203
1 ri
Since wi ∈ A mpi , αi , we have r0 ri
m
m
1 ri
wi
0 ri rmp i
wi (y)
0 ri
i
d(y, ∂ Q )
0 ri 1− rmp
αi pi mpi −r0 ri
i
dy
≤C
Q
Q
i =1
r
− mp −0r
m
αi
α¯
|Q |1+ nm = C |Q |m+ nm .
(42)
i=1
Using (41), (42) and w ⃗ ∈ Ap⃗ we obtain r0 m
νw⃗
|Q |
r0 mp m
Q
and βi = αi
Taking r = can be written as
νw⃗
r /p
m Q
i =1
Q
mr r0
where we have used
d(y, ∂ Q )
wi (y)
i
1+r (m−1) m
we get that
0 ri
αi pi mpi −r0 ri
0 ri 1− rmp i
dy
α¯
≤ C |Q |r0 +m+ nm .
Q
i=1
r0 m−(m−1)r0
r
− mp −0r
−r0 pi m−r0 ri
= 1 − ( pri )′ and
αi pi mpi −ri r0
= βi ( pri )′ . Then the above inequality
rri mr pi ′ pi ′ β¯ r −p wi (y)1−( r ) d(y, ∂ Q )βi ( r ) dy 0 i ≤ C |Q |m+ n ,
¯ α¯ m − r0 (m − 1) + mn = m + βn . Since
mr r0
rri pi
−
= [( pri )′ ]−1 , we are done.
Proof of Theorem 10. Since (i) ⇒ (ii) is trivial and by Theorem 9 we get that (ii) ⇒ (iii), we shall only prove (iii) ⇒ (i). As in the proof of Theorem 3.7 in [6] it is enough to prove that there exists δ < 1 such that
Mα⃗ (⃗f )(x) ≤ C
m
−δ Mνw⃗ |fi |δ pi wiδ νw (x) ⃗
δ1p
.
i
(43)
i=1
In fact, by Hölder’s inequality with exponents νw⃗ we get
|Mα⃗ (⃗f )| νw⃗ p
1p ≤C
and the strong type ( 1δ , 1δ ) inequality of Mνw⃗ with respect to the measure
pi p
m
p −δ δ pi Mνw⃗ |fi |pi δ wiδ νw νw⃗ ⃗
1p
i =1 m
≤C
δ pi
Mνw⃗ (|fi |
1
δ wiδ νw−δ ⃗ )
νw⃗
p1
i
≤C
i=1
|fi (y)|d(y, ∂ Q ) dy ≤
pi δ
|fi |
Q
Q
= Q
Let λ =
1+r (m−1) . mr
p1
i
.
|fi |pi δ wiδ νw1⃗−δ
δ1p
δ − p δ− 1
i
wi (y)
i
< δ < 1, where r is the number in Theorem 11. By
νw⃗ (y)
δ−1 pi δ−1
d(y, ∂ Q )
αi pi δ pi δ−1
pipδ−δ 1 dy
i
Q
δ1p
i
Ii .
It is clear that λ ∈ (0, 1) and δ − p δ− 1
wi (y)
Ii =
wiδ νw1⃗−δ
1 pi (1− 1r )+1
Hölder’s inequality with exponent pi δ , αi
|fi | wi pi
i=1
Now, we shall prove (43). Let δ be such that maxi=1,...,m
m
i
νw⃗ (y)
δ−1 pi δ−1
d(y, ∂ Q )
αi λpi δ pi δ−1
d(y, ∂ Q )
αi (1−λ)δ pi δ pi −1
δpδip−1 dy
i
.
Q r (δ p −1)
Let si = δ(p i−r ) . We have that si > 1 and, for later computations, i we get that
wi (y)
Ii ≤
− p r−r i
βi pi
ppi −rr
d(y, ∂ Q ) pi −r dy
i
1 s′i
=
δ pi (r −1)+r (δ−1) . By Hölder inequality with exponent si r (pi δ−1)
IIi ,
Q
where IIi =
ti =
Q
νw⃗ (y)
(δ−1)s′i δ pi −1
d(y, ∂ Q )
αi (1−λ)pi δ s′i pi δ−1
pi δ−′1 pi δ s i
dy
and βi = αi
δ pi − 1 αi (1 − λ)pi δ s′i ti′ > 1 and > −1, (1 − δ)s′i (mp − 1) pi δ − 1
1+r (m−1) . m
Now, choosing δ close to 1 such that
204
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
and applying Hölder’s inequality with ti we get
− mp1−1
IIi ≤ Q
νw⃗
(1−δ)(δpmp−1) i
αi (1−λ) n
|Q |
(For subsequent computations, notice that
νw⃗ ∈ Amp we get that m
|fi (y)|d(y, ∂ Q )αi dy ≤ C
1 s′i ti′
m
. 1 s′i
=
(1−δ)(mp−1) .) pi δ−1
Putting together the above estimates and using that
δ1p i |fi (y)|δpi wi (y)δ νw⃗ (y)1−δ dy
1
−
νw⃗ (Q ) Q ppi −rr 1p m m (1−δ)mp αi (1−λ) pi δ−1 βi pi + n + i δp − p r−r pi δ s′ t ′ p − r i i . × wi (y) i d(y, ∂ Q ) i νw⃗ |Q | i
Q
i=1
pi δ−1 pi δ s′ t ′ i i
+
i=1
i =1
Q
Q
i=1
Then, using Theorem 11 we get 1
|Q |
¯ m+ α n
m i=1
αi
|fi (y)|d(y, ∂ Q ) dy ≤ C
m
Q
−m− αn¯
|Q |
δ pi
|fi |
νw⃗ (Q )
i=1
×
1
Q
|Q |
β¯ m r + nr
wiδ νw1⃗−δ
|Q |
δ1p
(1−δ)m δ
i
|Q |
α( ¯ 1−λ) n
m
|Q |
pi δ−1 pi δ s′ t ′ i i
.
i =1
Since the last factor is 1 we are done.
6. Remarks and examples We start establishing some properties of the linear classes Ap,α . Assume that p(1 + α) > 1. Applying Hölder’s inequality, it is easy to see that Aq ⊂ Ap,α for 1 < q < p(1 + α). Using that w ∈ Ap ⇒ Ap−ε we have Ap(1+α) ⊂ Ap,α . Other interesting relation is that w ∈ A1 ⇒ w ∈ RA 1 ,α . We can also see that a power weight 1+α
w(x) = |x|β ∈ Ap,α ⇔ −n < β < p(n + α) − n ⇔ |x|β ∈ Ap(1+ αn ) .
(44)
These properties in the one-sided case can be found in [9].
⃗, α⃗ satisfy the assumptions on Theorem 9. Let Example 45. We work in R (n = 1) with m = 2. We assume that p w1 (x) = |x|γ1 and w2 (x) = |x|γ2 . We are going to determine the conditions on γ1 and γ2 to have w ⃗ ∈ Ap⃗,⃗α or, equivalently, 1−p′i
(a) νw⃗ ∈ A2p, α¯ , (b) wi 2
1 r
∈ A2p′ , αi , for i = 1, 2 and (c) wi i ∈ A 2pi , αi , for i = 1, 2, where ri = pi + 1. Using (44), w ⃗ ∈ Ap⃗,⃗α i
2
ri
2
if and only if the following relations hold: −1 < p γ1 + p γ2 < 2p + p(α1 + α2 ) − 1, −p1 − 1 − p1 α2 < γ1 < p1 − 1, 1 2 −p2 − 1 − p2 α1 < γ2 < p2 − 1, −p1 − 1 < γ1 < p1 + p1 α1 − 1 and −p2 − 1 < γ2 < p2 + p2 α2 − 1. Let p1 = 2, p2 = 4, p = 4/3, α1 = −1/3, α2 = −1/2. p
p
(1) If γ1 = −1, γ2 = 0 then w ⃗ ∈ Ap⃗,⃗α but w1 ̸∈ Ap1 ,α1 . Therefore, Mα⃗ applies Lp1 (w1 ) × Lp2 (w2 ) into Lp (νw⃗ ) but Mα1 is not bounded in Lp1 (w1 ). (2) If γ1 = −1 = γ2 then (b) and (c) are satisfied but (a) is not satisfied. (3) If γ1 = 1/3, γ2 = −11/3 + ε , ε small, then (a) and (b) are satisfied but (c) is not satisfied (the second condition in (c) is satisfied). Notice that in this example (a) and (c) imply (b). References [1] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972) 207–226. [2] J. García-Cuerva, J.L. Rubio de Francia, Weighted norm inequalities and related topics, in: North-Holland Mathematics Studies, 116, in: Notas de Matemática [Mathematical Notes], vol. 104, North-Holland Publishing Co, Amsterdam, 1985. [3] W. Jurkat, J. Troutman, Maximal inequalities related to generalized a.e. continuity, Trans. Amer. Math. Soc. 252 (1979) 49–64. [4] A.L. Bernardis, F.J. Martín-Reyes, The Cesàro maximal operator in dimension greater than one, J. Math. Anal. Appl. 228 (2003) 69–77. [5] R.A. Kerman, A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math. 71 (1981/82) 277–284. [6] A. Lerner, S. Ombrosi, C. Pérez, R. Torres, R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math. 220 (2009) 1222–1264. [7] F.J. Martín-Reyes, A dominated ergodic theorem for some bilinear averages, J. Math. Anal. Appl. 368 (2010) 469–481. [8] L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, 2004. [9] F.J. Martín-Reyes, A. de la Torre, Some weighted inequalities for general one-sided maximal operators, Studia Math. 122 (1997) 1–14.
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
Multilinear Cesàro maximal operators✩ A.L. Bernardis a , R. Crescimbeni b , F.J. Martín-Reyes c,∗ a
IMAL-CONICET, Güemes 3450, (3000) Santa Fe, Argentina
b
Departamento de Matemática, Universidad Nacional del Comahue, Buenos Aires 1400, 8300 Neuquén, Argentina
c
Departamento de Análisis Matemático, Universidad de Málaga, 29071 Málaga, Spain
article
abstract
info
The study of the almost everywhere convergence of the product of m Cesàro-α averages leads to the characterization of the boundedness of the associated multi(sub)linear maximal operator. We characterize weighted weak type and strong type inequalities for this operator, extending results by Lerner et al. [A. Lerner, S. Ombrosi, C. Pérez, R. Torres, R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math. 220 (2009) 1222–1264.]. We also study the restricted weak type inequalities which are of particular interest in our case (they were not considered by Lerner et al.). © 2012 Elsevier Inc. All rights reserved.
Article history: Received 24 June 2011 Available online 22 July 2012 Submitted by R.H. Torres Keywords: Multi(sub)linear maximal operators Cesàro operators Weighted norm inequalities
1. Introduction Let us consider the dilated functions ϕR (x) = R1n ϕ( Rx ), R > 0, of a nonnegative integrable function ϕ defined on Rn such that ϕ = 1. It is well known that the study of the a.e. convergence and the convergence in the Lp norm of the averages PR f = f ∗ ϕR to f as R → 0 is related to the behavior of the maximal operator Mϕ f (x) = supR>0 |f | ∗ ϕR (x). When ϕ(x) = ϕ α (x) = Cn,α (1 − |x|∞ )α χQ (0,1) (x), where x = (x1 , . . . , xn ), |x|∞ = max1≤i≤n |xi |, −1 < α ≤ 0, and Cn,α is such that ϕ α = 1, we have PR f (x) = PRα f (x) =
2n+α Cn,α
|Q (x, R)|1+α/n
Q (x,R)
f (y)d(y, ∂ Q (x, R))α dy,
where Q (x, R) = [x1 − R, x1 + R] × · · · × [xn − R, xn + R], d(y, ∂ Q (x, R)) is the distance in the infinity norm from y to the boundary of Q (x, R) and |E | is the Lebesgue measure of the set E. These averages are called Cesàro-α averages. The maximal operator associated to these averages is (essentially) Mα f (x) = sup c
R >0
1
|Q (x, R)|1+α/n
Q (x,R)
|f (y)|d(y, ∂ Q (x, R))α dy.
If α = 0, Mαc is the centered Hardy–Littlewood maximal operator which is simply denoted by M c . The non-centered Hardy–Littlewood maximal operator is Mf (x) = supx∈Q |Q1 | Q |f (y)| dy, where the supremum is taken over all the cubes Q
✩ This research has been supported by grants MTM2008-06621-C02-01, MTM2008-06621-C02-02 and MTM2011-28149-C02-02 of the Ministerio de Economía y Competitividad (Spain) and grants FQM-354 and FQM-01509 of the Junta de Andalucía. First author was supported by CAI+D-UNL, CONICET (Argentina). Second author was supported by FaEA-Universidad Nacional del Comahue (Argentina). ∗ Corresponding author. E-mail addresses: [email protected] (A.L. Bernardis), [email protected] (R. Crescimbeni), [email protected] (F.J. Martín-Reyes).
0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.07.037
192
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
such that x ∈ Q (throughout the paper, a cube will be a cube with sides parallel to the axis). The non-centered version of the Cesàro-α maximal operator is given by Mα f (x) = sup x∈Q
1
|Q |1+α/n
|f (y)|d(y, ∂ Q )α dy. Q
The operators M and M c are pointwise equivalent. Classical results assert that M and M c are of weak type (1, 1) and of strong type (p, p), 1 < p ≤ +∞, with respect to the Lebesgue measure. The characterizations of the boundedness of M c (M) in weighted spaces are well-known (see [1] and [2]). 1 1 , 1+α ) and, consequently, it is of It follows from the results in [3] that Mαc , −1 < α ≤ 0, is of restricted weak type ( 1+α 1 strong type (p, p) for p > 1+α with respect to the Lebesgue measure. It is not clear if Mα and Mαc are pointwise equivalent for α ̸= 0. Therefore, the boundedness of Mα cannot be obtained from the corresponding result for Mαc . However, the weighted inequalities for Mα are equivalent to the corresponding one for Mαc (see [4]). We shall state below the results obtained in [4]. In order to state them, we introduce definitions and notations. A non-negative measurable function (a weight) w satisfies Ap,α , −1 < α ≤ 0, 1 < p < +∞, and we write w ∈ Ap,α , if there exists C > 0 such that
1/p
w(y) dy
Ap,α : Q
w
1−p′
(y)d(y, ∂ Q )
α p′
1/p′ dy
α
≤ C |Q |1+ n ,
Q
′
for every cube Q , where p is the conjugate exponent of p. Observe that Ap,0 = Ap is the Muckenhoupt Ap class. We recall that w satisfies A1 if there exists C > 0 such that M w(x) ≤ C w(x) for a.e. x. By definition, A1,0 is A1 . It is clear that Ap,α ⊂ Ap . ′
′
Consequently, if w ∈ Ap,α and w is not the function zero a.e. then w and w 1−p are locally integrable and 0 < w, w 1−p < ∞ a.e. Other property is in the following theorem (Corollary 3.4 in [4]). Theorem 1 ([4]). Let −1 < α ≤ 0, 1 < p < +∞, and let w be a weight on Rn . If w satisfies Ap,α , then there exists s ∈ (1, p) such that w satisfies A p ,α . s
1
The weighted weak Lp -norm of a measurable function f is defined by ∥f ∥Lp,∞ (w) = supλ>0 λ[w({x ∈ Rn : |f (x)| > λ})] p
1
and Lp,∞ (w) = {f : ∥f ∥Lp,∞ (w) < ∞}, where w(E ) = E w . The weighted Lp -norm of f is ∥f ∥Lp (w) = |f |p w p and Rn p L (w) = {f : ∥f ∥Lp (w) < ∞}. The following result (see [4]) characterizes the weighted weak and strong type inequalities for Mαc and Mα .
Theorem 2 ([4]). Let w be a weight on Rn , 1 < p < ∞ and −1 < α ≤ 0. Let Sα be either Mα or Mαc . The following conditions are equivalent. (a) w ∈ Ap,α . (b) There is C > 0 such that ∥Sα f ∥Lp,∞ (w) ≤ C ∥f ∥Lp (w) for all f ∈ Lp (w). (c) There is C > 0 such that ∥Sα f ∥Lp (w) ≤ C ∥f ∥Lp (w) for all f ∈ Lp (w). From this result, it is easy to prove that for all f ∈ Lp (w) with w ∈ Ap,α , the averages PRα f converge to f a.e. and in the L (w)-norm as R → 0. The restricted weak type inequalities require other classes of weights. A weight w satisfies RAp,α , 1 ≤ p < +∞, if there exists C > 0 such that p
w
RAp,α : Q
1p α χE (y)d(y, ∂ Q )α dy ≤ C |Q |1+ n χE w
1p Q
(3)
Q
for all cubes Q and all measurable sets E. It was proved (see [5]) that RAp,0 = RAp characterizes the restricted weak type for M. Theorem 4 ([4]). Let w be a weight on Rn , 1 ≤ p < ∞ and −1 < α ≤ 0. Let Sα be either Mα or Mαc . The following conditions are equivalent. (a) w ∈ RAp,α .
1
(b) There is C > 0 such that ∥Sα χE ∥Lp,∞ (w) ≤ C ∥χE ∥Lp (w) = (w(E )) p for every measurable set E. As before, it is easy to prove that if w ∈ RAp,α then the averages PRα f converge to f almost everywhere for all f ∈ Lp,1 (w) ∞ as R → 0, where Lp,1 (w) = {f : ∥f ∥Lp,1 (w) = 1p 0 λ1/p w({x ∈ Rn : |f (x)| > λ}) λ1 dλ < ∞}. Remark 5. We point out that we can only have non-trivial restricted weak-type inequalities if p ≥ 1/(1 + α) (the proof is as the proof in Theorem 8, the multilinear case, in Section 3). It follows from Theorem 1 that if there is a non-trivial function in Ap,α , α ̸= 0, then p > 1/(1 + α).
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
193
Let us fix a natural number m > 1. Let us take real numbers pi and αi , i = 1, . . . , m, pi > 1 and −1 < αi ≤ 0. Let α
p p
i and in the Lp (ν)-norm, where ν = fi ∈ Lpi (wi ) with wi ∈ Api ,αi . Then, limR→0 i=1 PR i fi = and i=1 fi a.e. i =1 w i m m 1 1 satisfies the following (multilinear) weighted are sufficient to obtain that M . The conditions w ∈ A = α i p ,α i =1 i =1 p i i i p
m
i
Mαi (fi )Lp (ν) ≤ C
m
strong type inequality:
i =1
The maximal operator associated to m
Mαc⃗ (⃗f )(x) = sup
i=1
α 1+ ni
|Q (x, R)|
m
αi
i=1
m
∥fi ∥Lpi (wi ) .
PR fi is, up to a constant,
1
R >0 i =1
m
m
Q (x,R)
|fi (y)|(dy, ∂ Q (x, R))αi dy,
where ⃗ f = (f1 , . . . , fm ) and α ⃗ = (α1 , . . . , αm ). Since Mαc⃗ (⃗f ) ≤ i=1 Mαi (fi ) it is reasonable to think that weaker conditions on the weights would imply the boundedness of Mαc⃗ in the weighted space. In this way we reach to the aim of this article: to characterize the weights for which we have restricted weak type, weak type and strong type inequalities for Mαc⃗ . We
m
follow the ideas in [6] where the authors studied the case α ⃗ = 0⃗ = (0, . . . , 0), that is, the multi(sub)linear version of the Hardy–Littlewood maximal operator M c := M⃗c . This operator is used in [6] ‘‘to obtain a precise control on multilinear 0 singular integral operators of Calderón–Zygmund type’’. The results in [6] are used in [7] to study the convergence of ergodic multilinear averages. The study of Mαc⃗ in this paper will be useful to study the Cesàro-α ergodic multilinear averages, extending the study in [7]. 2. Statement of the main results p1 pm mWe pcollect in the next theorem some of the main results in [6]. The space L (w1 ) × · · · × L (wm ) will be denoted by i (w ). L i i=1
Theorem 6 (see [6]). For i = 1, . . . , m, let 1 ≤ pi < ∞ and let p be such that
νw⃗ =
i =1 w
m
p/pi . i
m
1
|Q | where
1 |Q |
=
m
1 i=1 pi .
Let wi be weights and
The following statements are equivalent.
(i) There is C such that ∥M c (⃗ f )∥Lp,∞ (νw⃗ ) ≤ C i=1 ∥fi ∥Lpi (wi ) (⃗ f ∈ (ii) w ⃗ ∈ Ap⃗ , that is, there exists C > 0 such that for all cubes Q
1 p
νw⃗
1/p m
Q
Q
i=1 1−p′i
wi
1/p′i
1
|Q |
1−p′i
wi
1/p′i
m
i =1
Lpi (wi )).
≤ C,
Q
is understood in the case pi = 1 as (ess inf x∈Q wi (x))−1 .
If pi > 1 for every i, then (i) and (ii) are equivalent to the following. m m (iii) There is C such that ∥M c (⃗ f )∥Lp (νw⃗ ) ≤ C i=1 ∥fi ∥Lpi (wi ) (⃗ f ∈ i=1 Lpi (wi )).
Ap⃗ is related to Muckenhoupt Ap conditions in the following way (see [6]):
w ⃗ ∈ Ap⃗ ⇔ νw⃗ ∈ Amp and 1−p′i
where wi
1−p′i
wi
∈ Amp′i ,
i = 1, . . . , m, 1/m
∈ Amp′i in the case pi = 1 is understood as wi
(7)
∈ A1 .
Our aim is to extend this theorem to the Cesàro maximal operator Mαc⃗ . This extension is not straightforward. We explain in the next few lines one of the difficulties that appear in the study of Mαc⃗ .
The results in Theorem 6 are stated for the noncentered multilinear Hardy–Littlewood maximal operator M (⃗ f )(x) = m supx∈Q i=1 |Q1 | Q |fi (y)| dy. Since M and M c are pointwise equivalent, it is clear that the boundedness in (i) and (iii) is equivalent to the corresponding ones for M . The pointwise equivalence of M and M c makes easier some parts of the proof. However that is not the situation when we work with the Cesàro maximal operators. Therefore, we have to work harder to obtain the condition on the weights from the boundedness of Mαc⃗ . For future reference, we define the non-centered multilinear version of the Cesàro maximal operator by
Mα⃗ (⃗f )(x) = sup
m
x∈Q i=1
1 α 1+ ni
|Q |
|fi (y)|(d(y, ∂ Q ))αi dy. Q
⃗ These inequalities were not studied in [6] for The restricted weak-type inequalities are relevant for the case α ⃗ ̸= 0. α⃗ = 0⃗ but the proof in this case is easy. Since the proof for arbitrary α⃗ exemplifies the difference indicated above, we start
our study of Mα⃗ and Mαc⃗ characterizing the restricted weak-type weighted inequalities. To state our first main result we introduce some notation: given m measurable sets Ei , we denote (χE1 , . . . , χEm ) by χ ⃗E⃗ .
194
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
Theorem 8. For each i = 1, . . . , m, let −1 < αi ≤ 0, α¯ = α1 + · · · + αm , 1 ≤ pi < ∞ and let p be such that p pi
1 p
=
m
1 i=1 pi .
wi . The following statements are equivalent. p (i) There is C > 0 such that ∥Mα⃗ (χ ⃗E⃗ )∥Lp,∞ (νw⃗ ) ≤ C m i=1 ∥χEi ∥L i (wi ) for all measurable sets Ei , i = 1, . . . , m. c p ∥ (ii) There is C > 0 such that ∥Mα⃗ (χ ⃗E⃗ )∥Lp,∞ (νw⃗ ) ≤ C m ∥χ Ei L i (wi ) for all measurable sets Ei , i = 1, . . . , m. i=1 (iii) w ⃗ ∈ RAp⃗,⃗α , that is, there is C > 0 such that 1p p1 m m i 1 αi νw⃗ dy ≤ C χ w χ ( y ) d ( y , ∂ Q ) Ei i Ei α 1+ ni Q Q Q |Q | i=1 i=1 Let wi be weights and νw⃗ =
m
i =1
for all cubes Q and all measurable sets Ei , i = 1, . . . , m. 1 If νw⃗ is not 0 a.e. and statement (iii) holds then pi ≥ 1+α for all i. i
⃗ However, we remark that it is not very difficult As we have already said, this theorem is new even in the case α ⃗ = 0. to see that (i) ⇔ (iii). The proof of (i) ⇒ (iii) is straightforward. To prove (iii) ⇒ (ii) we follow the ideas in [6] but we have to use that (iii) implies that νw⃗ satisfies the doubling condition (this is not necessary when α ⃗ = 0⃗ because in this case c c ⃗ it suffices to estimate M⃗ = M ). The stronger difficulty appears when α ⃗ ̸= 0 in the proof of (ii) ⇒ (iii) since we cannot 0 use that Mα⃗ and Mαc⃗ are pointwise equivalent. In particular, as in [4], we establish the implication proving that (ii) implies certain conditions of one-sided nature. Our next step is the characterization of the weak type inequalities. 1 Theorem 9. For i = 1, . . . , m, let −1 < αi ≤ 0, α¯ = α1 + · · · + αm , αi = α¯ − αi , 1+α ≤ pi < ∞. Let p be such that i p/pi m m 1 1 = i=1 p . Let wi be weights and νw⃗ = i=1 wi . The following statements are equivalent. p i
(i) There is C such that ∥Mα⃗ (⃗ f )∥Lp,∞ (νw⃗ ) ≤ C
m pi p i=1 ∥fi ∥L i (wi ) (fi ∈ L (wi )). m p ≤ C i=1 ∥fi ∥Lpi (wi ) (fi ∈ L i (wi )).
(ii) There is C such that ∥Mαc⃗ (⃗ f )∥Lp,∞ (νw⃗ ) (iii) w ⃗ ∈ Ap⃗,⃗α , that is, there exists C > 0 such that for all cubes Q
1
|Q |
νw⃗
1/p m i=1
αi p′i n
(essinf x∈Q wi (x))−1 .
Q
|Q |1+
1/p′i
1
Q
where (|Q |−1−
αi p′i n
wi (y)
1−p′i
αi p′i
d(y, ∂ Q )
wi (y)1−pi d(y, ∂ Q )αi pi dy)1/pi in the case pi ′
≤ C,
dy
Q ′
′
= 1 (consequently, αi
= 0) is understood as
(iv) The following conditions hold. (a) νw⃗ ∈ Amp, α¯ , 1−p′i
(b) wi
m
∈ Amp′ , αi , for all i = 1, . . . , m and i m
1 ri
(c) wi ∈ A mpi , αi , for all i = 1, . . . , m, where ri = (m − 1)pi + 1. ri
m
1 for all i such that αi ̸= 0. If νw⃗ is not 0 a.e. and (iii) holds then pi > 1+α i
To prove the theorem we follow the ideas in [6], except when we work with the centered multilinear maximal operator. To prove (ii) ⇒ (iii) we have to show (ii) ⇒ (iv) ⇒ (iii). As in the first theorem, for the implication (ii) ⇒ (iv) we prove that (ii) implies conditions of one-sided nature. ′ When α ⃗ = 0⃗ then conditions (b) and (c) in statement (iv) are equivalent because, for 1 < q < ∞, u ∈ Aq ⇔ u1−q ∈ Aq′ . Therefore, when α ⃗ = 0⃗ the equivalence between (iii) and (iv) is nothing but the equivalence (7). ′ ⃗ = p, w If m = 1 (p ⃗ = w and α⃗ = α ), the conditions in (iv) yield νw⃗ = w ∈ Ap,α , w1−p ∈ Ap′ ,0 = Ap′ and w ∈ Ap,α . Since Ap,α ⊂ Ap , they are equivalent to νw⃗ = w ∈ Ap,α (statement (iv) is statement (a) in Theorem 2). The following theorem shows that Ap⃗,⃗α also characterizes the weighted strong type inequalities for the multilinear maximal Cesàro operators. 1 < pi < ∞. Let p, wi and νw⃗ be as in Theorem 9. The following Theorem 10. For i = 1, . . . , m, let −1 < αi ≤ 0 and 1+α i statements are equivalent.
(i) There is C such that ∥Mα⃗ (⃗ f )∥Lp (νw⃗ ) ≤ C
(ii) There is C such that ∥Mαc⃗ (⃗ f )∥Lp (νw⃗ ) (iii) w ⃗ ∈ Ap⃗,⃗α .
m pi p i=1 ∥fi ∥L i (wi ) (fi ∈ L (wi )). m p ≤ C i=1 ∥fi ∥Lpi (wi ) (fi ∈ L i (wi )).
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
195
The main tool to prove this theorem is the characterization of the weak type and the following openness property of the condition Ap⃗,⃗α . Theorem 11. For i = 1, . . . , m, let αi , pi , p, wi and νw⃗ be as in Theorem 10. If w ⃗ ∈ Ap⃗,⃗α then there exists r > 1 such that pi r
> 1, βi = αi 1+r (mm−1) ∈ (−1, 0] for all i = 1, . . . , m, and w ⃗ ∈ A p⃗ ,β⃗ . r
⃗ = and qi are the coordinates of q⃗ then qi < pi for all i. If m > 1 then βi < αi and we have that Notice that if q w ⃗ ∈ Ap⃗,⃗α ⇒ w ⃗ ∈ Aq⃗,β⃗ for some q⃗ and some β⃗ such that qi < pi and βi < αi ; this is an improvement of the result in [6] in ⃗ where this implication was proved with β⃗ = 0. ⃗ We point out that our argument in the proof of Theorem 11 the case α ⃗ = 0, is not the same as the corresponding result in [6]. When m = 1, the above theorem gives the result proved in [4]: if w ∈ Ap,α then there exists r > 1 such that w ∈ A p ,α . r Throughout the paper, if 1 < p < ∞ then p′ is its conjugate exponent and the letter C is used to denote positive constants ⃗ p r
whose values may change from line to line. 3. Proof of Theorem 8 First of all we give a result about the weights wi and νw⃗ . Proposition 12. For each i = 1, . . . , m, let pi , p, wi and νw⃗ be as in Theorem 8. Then there exists C > 0 (we can take C = 2m ) such that for each measurable set E of finite measure we can choose measurable subsets G1 , . . . , Gm ⊂ E such that C |Gi | ≥ |E |, i = 1, . . . , m, and
m i=1
wi
Gi
p1
i
≤C
E
νw⃗
1p
.
Proof. For any measurable set F of finite measure and any λ > 0, let Aλi,F := {x ∈ F : wi (x) ≤ λνw⃗ (x)}. Notice that there
exists λ such that |Aλi,F | >
1 2
|F |.
Let us fix any measurable set E of finite measure. Let F1 = E and λ1 = inf{λ : |Aλ1,F1 | >
|F1 |}. It is clear that λ1 > 0. Let µ1 be such that 2 < µ1 < λ1 . Then ≥ |F1 | and ≤ |F1 |. Let F2 = F1 \ Clearly |F2 | ≥ 12 |F1 | = 21 |E |. λ λ 1 Let λ2 = inf{λ : |Aλ2,F2 | > 2 |F2 |} and let µ2 be any number such that 22 < µ2 < λ2 ; then |A2,2F2 | ≥ 12 |F2 | ≥ 41 |E | and µ |A2,2F2 | ≤ 12 |F2 |. We continue in this way choosing sets Fi , numbers λi and µi , i = 1, . . . , m − 1, such that |Fi | ≥ 2i1−1 |E |, λ |A1,1F1 |
λ1
λ
|Ai,iFi | ≥ Gi = have
1 2i
λ Ai,iFi
µ
|E |, |Ai,Fi i | ≤ 12 |Fi | and
λi 2
µ |A1,1F1 |
1 2
µ A1,1F1 .
1 2
µ
< µi < λi . Let Fm = Fm−1 \ Amm−−11,Fm−1 . Notice that |Fm | ≥ 12 |Fm−1 | ≥
for i = 1, . . . , m − 1 and Gm = Fm . Clearly |Gi | ≥
wi (x) ≤ λi νw⃗ (x),
1 2
1 2m
1 2m
|E |. Let us take
|E | for all i = 1, . . . , m. By the definition of the sets Gi we
x ∈ Gi , i = 1, . . . , m − 1.
(13)
For Gm we have the following inequality: m−1
wm (x) <
(µi )
− ppm i
νw⃗ (x),
x ∈ Gm .
(14)
i=1
In fact, since Fm =
m−1 i=1
p
p p
(µi ) pi νw⃗ 1 m i =1
m−1 i=1
p m−1
+···+ p
wi
we have that wi (x) > µi νw⃗ (x), for x ∈ Fm and i ̸= m. Consequently, νw⃗ =
m
i=1
p p
wi i >
p
wmpm and (14) follows. Putting together (13) and (14) we obtain
p1
i
µ
E \ Ai,Fi i
1
m λi pi
m−1
≤C
Gi
i=1 1
1
≤ 2 p − pm
µi
i=1
νw⃗
1p
νw⃗
1p
Gi
≤ 2m−1
E
νw⃗
1p
.
E
To prove (ii) ⇒ (iii) in Theorem 8, we introduce some classes of weights. Definition 15. For each i = 1, . . . , m, let αi , α¯ , pi , p, wi and νw⃗ be as in Theorem 8. It is said that w ⃗ satisfies RA− ⃗,⃗ p α ,k , k = 1, . . . , n, if there exists C > 0 such that RA− ⃗,⃗ p α ,k :
νw⃗ Uk
1p m i=1
d(y, ∂ Q )αi dy Vk ∩Ei
α¯
≤ C |Q |m+ n
m i =1
wi Vk ∩Ei
p1
i
(16)
196
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
for all cubes Q = Q (z , R), z = (z1 , . . . , zm ), and all measurable sets Ei , i = 1, . . . , m, where Uk = Kk (z , R) ∩ {y : yk ≥ zk },
Vk = Kk (z , R) ∩ {y : yk ≤ zk }
(17)
and Kk (z , R) = {y ∈ Q (z , R) : |yj − zj | ≤ |yk − zk |, j = 1, . . . , n}. The class RA+ ⃗,⃗ p α ,k is defined analogously changing the roles of Uk and Vk . Notice that if m = 1 then RAp⃗,0⃗ coincides with the class RAp := RAp,0 introduced in (3). If m = 1 then the classes RA± ⃗ ⃗ ,0 ,k p
will be simply denoted by RA± p,k . We notice that it follows from the results in [4] that RAp =
n − (RA+ p,k ∩ RAp,k ),
p ≥ 1.
(18)
k=1
Taking into account these notations, we have the following result.
p⃗,⃗α := Proposition 19. Let pi , p, wi , and νw⃗ be as in Definition 15. Let RA
n
k=1
+ p⃗,⃗α then νw⃗ ∈ RApm . ⃗ ∈ RA (RA− ⃗,⃗ α ,k ). If w p α ,k ∩ RAp⃗,⃗
p⃗,⃗α . Clearly w ⃗ ⃗ . Therefore, for Proof. Let Q be any cube and let E be any measurable set. Assume that w ⃗ ∈ RA ⃗ ∈ RA p,0 k = 1, . . . , n, and for each family of measurable sets Ei , i = 1, . . . , m, we have that
1p m
νw⃗
Uk
|Ei ∩ Vk | ≤ C |Q |m
i=1
νw⃗
1p m
Vk
m
i=1
p1
i
,
(20)
.
(21)
Vk ∩Ei
i =1
|Ei ∩ Uk | ≤ C |Q |m
wi
m
wi
p1
i
Uk ∩Ei
i =1
By Proposition 12 with E = E ∩ Vk and taking Ei = Gi in (20) we have that
νw⃗
1 mp
νw⃗
|E ∩ Vk | ≤ C |Q |
1 mp
,
Vk ∩E
Uk
for k = 1, . . . , m. In a similar way, Proposition 12 with E = E ∩ Uk and (21) give
νw⃗
1 mp
νw⃗
|E ∩ Uk | ≤ C |Q |
1 mp
.
Uk ∩E
Vk
Therefore, νw⃗ ∈
n
k=1
− (RA+ ⃗ ∈ RAmp . mp,k ∩ RAmp,k ) and then, by (18), νw
Remark 22. It is clear that if νw⃗ ∈ RAmp then νw⃗ is an A∞ weight, so it is doubling. As a consequence, or taking E = Uk and E = Vk in the definition of RAmp , we get that νw⃗ (Q ) ≤ C νw⃗ (Uk ) and νw⃗ (Q ) ≤ C νw⃗ (Vk ). − The following lemma shows the relationship between the one-sided conditions RA+ ⃗,⃗ p α ,k and RAp⃗,⃗ α ,k with the general condition RAp⃗,⃗α (see (iii) in Theorem 8). This lemma is a key result in the proof of Theorem 8.
p⃗,⃗α = RAp⃗,⃗α . Lemma 23. Let pi , p, wi , and νw⃗ be as in Definition 15. Then RA p⃗,⃗α because the sets Uk , Vk ⊂ Q . Let w p⃗,⃗α and let us see that w Proof. It is obvious that RAp⃗,⃗α ⊂ RA ⃗ ∈ RA ⃗ ∈ RAp⃗,⃗α . Assume first that α ⃗ is equal to (0, . . . , αj , . . . , 0), that is, all the numbers αi are zero, except possibly the one in the place j. It follows from Proposition 19 and Remark 22 that there exists C > 0 such that for all cubes Q and every k = 1, . . . , n, we have νw⃗ (Q ) ≤ C νw⃗ (Uk ) and νw⃗ (Q ) ≤ C νw⃗ (Vk ), where Uk and Vk are the sets associated to the cube Q in Definition 15. We have to prove that w ⃗ ∈ RAp⃗,⃗α , that is, there exists a constant C > 0 such that for all cubes Q and all measurable sets Ei , i = 1, . . . , m,
νw⃗
p1 m m αj i wi . χEj (y)d(y, ∂ Q )αj dy |Ei ∩ Q | ≤ C |Q |m+ n
1p
i=1,i̸=j
Q
Q
i=1
Ei
We estimate the left hand side in the following way:
νw⃗
1p m n m χEj (y)d(y, ∂ Q )αj dy |Ei ∩ Q | = νw⃗ χEj (y)d(y, ∂ Q )αj dy |Ei ∩ Q |
1p
Q
i=1,i̸=j
Q n
+
k=1
νw⃗ Q
1p
k=1
m
χEj (y)d(y, ∂ Q )αj dy Vk
i=1,i̸=j
Q
|Ei ∩ Q | = I + II .
Uk
i=1,i̸=j
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
197
We shall only estimate I since II is estimated in a similar way. It will suffice to obtain the estimate for each term in the sum of I. We shall do it for k = 1 and without loss of generality we assume j = 1. Let Q be a cube such that for all y ∈ U1 , 1 , where U1 is the set U1 associated to d(y, ∂ Q ) = d(y, ∂ Q ), |Q | is comparable to | Q | and Q ⊂ U Q (see the figure below).
Then, since d(y, ∂ Q ) = d(y, ∂ Q ), for all y ∈ U1 ,
νw⃗
1p m m |Ei ∩ Q | ≤ νw⃗ |Ei ∩ Q |. χE1 (y)d(y, ∂ Q )α1 dy χE1 ∩Q (y)d(y, ∂ Q )α1 dy
1p
Q
U1
Q
i =2
1 U
i =2
If V1 is the set V1 associated to Q we have νw⃗ ( Q ) ≤ C νw⃗ (V1 ) by the doubling property of νw⃗ . This inequality and the p⃗,⃗α give assumption w ⃗ ∈ RA
νw⃗
1p
Q
1p m m χE1 ∩Q (y)d(y, ∂ Q )α1 dy |Ei ∩ Q ∩ U1 | χE1 (y)d(y, ∂ Q )α1 dy νw⃗ |Ei ∩ Q | ≤ C
U1
≤ C | Q |m+
α1 n
i =1
χEi ∩Q wi
p1
i
1 U
V1
i =2
m
α1
≤ C |Q |m+
n
1 U
m i =1
χEi wi
p1
i
i =2
.
Q
This proves the lemma in the particular case α ⃗ = (0, . . . , αj , . . . , 0). We prove now the general case. Let us fix a cube Q and sets E1 , . . . , Em . By Proposition 12 applied to the set E = Q there exist subsets G1 , . . . , Gm such that i = 1 , . . . , m,
C |Gi | ≥ |Q |,
m
and
wi
p1
i
Gi
i=1
νw⃗
≤C
1p
.
(24)
Q
p⃗,⃗α then w p⃗,⃗α , where α⃗ j = (0, . . . , αj , . . . , 0) for all j. By what we have already proved we have that Since w ⃗ ∈ RA ⃗ ∈ RA j w ⃗ ∈ RAp⃗,⃗αj . For fixed j, we apply that w ⃗ ∈ RAp⃗,⃗αj in the cube Q and with the sets Eij = Gi if i ̸= j and Ejj = Ej . Then we obtain νw⃗ (Q )
1 p
m
αj
αj m+ n
χEj (y)d(y, ∂ Q ) dy
|Gi ∩ Q | ≤ C |Q |
i=1,i̸=j
Q
p1 wj
m
j
i=1,i̸=j
Ej
wi
p1
i
,
Gi
for some constant independent of j and the sets. Multiplying these inequalities on j = 1, . . . , m we have
νw⃗ (Q )
m p
m j =1
αj
m
χEj (y)d(y, ∂ Q ) dy Q
m−1 |Gi |
¯ m2 + α n
≤ C |Q |
j =1
i =1
Then, the lemma follows from (24) and the last inequality.
p1
m
wj Ej
j
m i=1
wi
mp−1 i
.
Gi
Now we are ready to prove the characterization of the restricted weak type weighted inequalities for Mαc⃗ .
p⃗,⃗α = Proof of Theorem 8. The ⃗ ∈ RA implication (i) ⇒ (ii) is obvious. To prove (ii) ⇒ (iii) it will suffice to see that w n
k=1
+ RA− ⃗,⃗ p α ,k ∩ RAp⃗,⃗ α ,k .
Assume that (ii) holds and consider for each k = 1, . . . , n, the following non-centered multi(sub)linear maximal operators
Nα⃗−,k ⃗f (x) =
sup
m
x∈Uk (z ,R) i=1
1
|Q (z , R)|
α 1+ ni
Vk (z ,R)
|fi (y)|d(y, ∂ Q )αi dy
(25)
|fi (y)|d(y, ∂ Q )αi dy,
(26)
and
Nα⃗+,k ⃗f (x) =
sup
m
x∈Vk (z ,R) i=1
1 α 1+ ni
|Q (z , R)|
Uk (z ,R)
198
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
where Uk = Uk (z , R) and Vk = Vk (z , R) are the sets considered in Definition 15 associated to the cube Q = Q (z , R). It is easy f (x) ≤ C Mαc⃗ ⃗ f (x), for all k = 1, . . . , n and all measurable function f (see the proof f (x) ≤ C Mαc⃗ ⃗ f (x) and Nα⃗+,k ⃗ to see that Nα⃗−,k ⃗
p⃗,⃗α , ⃗ ∈ RA of Proposition 2.3 in [4]). Then we get the inequality in (ii) for the operators Nα⃗−,k and Nα⃗+,k . In order to see that w let Q = Q (z , R) be a fixed cube, and let Uk = Uk (z , R) and Vk = Vk (z , R), k = 1, . . . , n, its associated sets considered in Definition 15. Let E1 , . . . , Em be measurable sets. For each k we consider the multilinear function χ ⃗E⃗k = (χE k , . . . , χE k )
with Eik = Ei ∩ Vk . For x ∈ Uk we have Nα⃗−,k χ ⃗E⃗k (x) ≥
m
i =1
1 αi |Q |1+ n
1
Vk
m
χE k (y)d(y, ∂ Q )αi dy = λ. Then, using that Nα⃗−,k i
satisfies (ii), m C νw⃗ (Uk ) ≤ νw⃗ {x : Nα⃗−,k χ⃗E⃗k (x) ≥ λ} ≤ p ∥χ k ∥p . λ i=1 Ei pi ,wi
Therefore
νw⃗
1p m
Uk
¯ m+ α n
χE k (y)d(y, ∂ Q ) dy ≤ C |Q | i
Vk
i =1
αi
m
χEi wi
p1
i
.
Vk
i=1
+ p⃗,⃗α as Then we get that w ⃗ ∈ RA− ⃗ ∈ RA+ ⃗ ∈ RA ⃗,⃗ ⃗,⃗ p α ,k . In a similar way, using that Nα⃗ ,k satisfies (ii) we obtain that w p α ,k . Thus w we wished to prove. Finally, we prove (iii) ⇒ (i). Using the assumption (iii) we get, for every cube Q , that m i=1
1
|Q |
=
α 1+ ni
m i =1
αi
χEi (y)d(y, ∂ Q ) dy ≤ C Q
m i=1
1
νw⃗ (Q )
χEi Q
wi νw⃗ νw⃗
1/pi ≤
m
χEi wi
1/pi
1
Q
Q
Mνw⃗ (χEi wi νw⃗ −1 )(x)
νw⃗
1/p
1/pi
,
i =1
where Mνw⃗ is the Hardy–Littlewood maximal operator associated to the measure νw⃗ (x)dx defined as Mνw⃗ f (x) = supx∈Q 1 |f (y)|νw⃗ (y)dy. ν (Q ) Q w ⃗
By Proposition 19, Lemma 23 and Remark 22 we have that νw⃗ is a doubling weight. Then Mνw⃗ is of weak type (1, 1) with respect to νw⃗ . It follows from the last inequality, Hölder’s inequality for weak-spaces [8, p.15] and the weak-type (1, 1) inequality for Mνw⃗ that
∥Mα⃗ (χ⃗E⃗ )∥Lp,∞ (νw⃗ ) ≤ C
m 1/pi Mνw⃗ (χEi wi νw⃗ −1 ) p ,∞ L i
i=1
=C
m pi Mν (χE wi νw⃗ −1 )11/,∞ i w ⃗ L (ν i=1
w ⃗)
(νw⃗ )
≤C
m
∥χEi ∥Lpi (wi ) ,
i=1
and we are done. Finally, we are going to prove that if w ⃗ is not the function zero a.e. and w ⃗ ∈ RAp⃗,⃗α then pi ≥ 1/(1 + αi ). We shall prove it for i = 1. Since w ⃗ ∈ RAp⃗,⃗α implies that w ⃗ ∈ RAp⃗,β⃗ where β = (α1 , 0, . . . , 0), we may assume that αi = 0 for all i ≥ 2. There exists N such that the sets Fi,N = {x : wi (x) < N } have positive measure for all i. Let z be a Lebesgue point of the locally integrable functions νw⃗ , wi χFi,N and χFi,N , i = 1, . . . , m, such that νw⃗ (z ) ̸= 0. Without loss of generality, we may assume that z = 0. Let Q = Q (0, R) and Eλ = Q (0, (1 − λ)R) \ Q (0, (1 − 2λ)R), 0 < λ < 1/4. Notice that if x ∈ Eλ then |E | d(x, ∂ Q )α1 ≥ (2λR)α1 = λα1 |Q |α1 /n and C1 λ ≤ |Qλ| ≤ C2 λ (C1 and C2 depend on n but they are independent of λ). Now, keeping in mind these inequalities, we apply that w ⃗ ∈ RAp⃗,⃗α with the cube Q and the sets E1 = Eλ ∩ F1,N and Ei = Fi,N , i ≥ 2. Then we obtain
1
|Q |
νw⃗
1p
Q 1
≤ C (λ|Q |) p1
m |Eλ ∩ F1,N | |Fi,N ∩ Q | |Eλ | |Q | i=2 p1 p1 m 1 i 1 1 1 − 1 χF w1 χFi,N wi |Q | p p1 . |Eλ | Eλ 1,N | Q | Q i=2 1
λ1+α1 |Q | p
Notice that, for fixed λ, the family Eλ = Eλ (R) is a regular family which shrinks nicely to 0. If we let R tend to 0 we obtain
λ
1+α1 − p1
1
≤ C . Since λ can be chosen so small as we wish, we have 1 + α1 −
1 p1
≥ 0.
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
199
4. Proof of Theorem 9 If αi = 0 for all i then our theorem is Theorem 6 (proved in [6]). From now on, we assume αi ̸= 0 for some i which implies pi > 1 and pm > 1. We study only the case m > 1 since the case m = 1 is contained in Theorem 2. To prove (ii) ⇒ (iii), we work with the one-sided multi(sub)linear maximal operators Nα⃗−,k and Nα⃗+,k defined in (25) and (26). We also need to introduce the following multilinear classes of weights: for k = 1, . . . , n, it is said that w ⃗ = (w1 , . . . , wm ) satisfies A− ⃗,⃗ p α ,k if there is C > 0 such that
A− ⃗,⃗ p α ,k :
1
|Q |
νw⃗
1/p m
Uk
i=1
1/p′i
1 αi p′ 1+ n i
|Q |
1−p′i
wi (y)
d(y, ∂ Q )
αi p′i
dy
≤C
(27)
Vk
for all cubes Q = Q (z , R), where Uk and Vk are the sets defined in (17) (if pi = 1 then (|Q |−1−
αi p′i n
wi (y)1−pi d(y, ∂ Q )αi pi ′
Vk
′
dy)1/pi is understood as (essinfVk wi )−1 ). The class A+ ⃗,⃗ p α ,k is defined changing the roles of Uk and Vk . The following theorem contains the proof of (iii) ⇔ (iv) and shows the relationship between the multilinear classes of weights with a family of suitable linear conditions. ′
Theorem 28. For i = 1, . . . , m, αi , pi , wi and νw⃗ be as in Theorem 9 and assume that αi ̸= 0 for some i. Let let αi , α¯ , p⃗,⃗α = nk=1 (A− ∩ A+ ). Then the following statements are equivalent. w ⃗ = (w1 , . . . , wm ). Let A ⃗,⃗ ⃗,⃗ p α ,k p α ,k
p⃗,⃗α . (i) w ⃗ ∈A (ii) The following conditions hold. (a) νw⃗ ∈ Amp, α¯ , 1−p′i
(b) wi
m
1/m
∈ Amp′ , αi , for all i = 1, . . . , m where in the case pi = 1 is understood as wi
∈ A1 and
i m
1 ri
(c) wi ∈ A mpi , αi , for all i = 1, . . . , m, with ri = (m − 1)pi + 1. (iii) w ⃗ ∈ Ap⃗,⃗α .
ri
m
Proof of Theorem 28. In this proof we shall use the following lemma (see [4]). From now on, when m = 1 the classes A− ⃗,⃗ p α ,k
− + and A+ ⃗,⃗ p α ,k are denoted by Ap,α,k and Ap,α,k , respectively.
Lemma 29([4]). Let w be a weight on Rn and let −1 < α < 0. If 1 < p < ∞ then w satisfies Ap,α if and only if + w ∈ nk=1 A− Ap,α,k . p,α,k In the proof we consider the sets J = {i : pi = 1} and H = {i : pi > 1}. (i) ⇒ (ii). Let us prove (a). Let k ∈ {1, . . . , n}. It follows from the assumptions that pm > 1 since pi > 1 for some i. By p (mp−1) Hölder’s inequality with exponents si = pi (p −1) if i ∈ H and si = +∞ if i ∈ J, we have that i m p p α¯ ′ ′ − νw⃗ (y)1−(mp) d(y, ∂ Q ) m (mp) dy = wi (y) pi (mp−1) d(y, ∂ Q )αi mp−1 dy Vk i=1
Vk
≤C
1−p′i
wi (y)
d(y, ∂ Q )
Vk
i∈H
αi p′i
dy
p(pi −1) pi (mp−1)
− mpp−1
ess infy∈Vk wi (y)
.
i∈J
p⃗,⃗α ⊂ A− , we get that νw⃗ ∈ A− α¯ . Changing the role of the sets Uk and Vk , we also obtain that νw⃗ ∈ A+ α¯ Since w ⃗ ∈A ⃗,⃗ p α ,k mp, m ,k mp, m ,k for all k. By Lemma 29 we get (a). Since Ap,α ⊂ Ap for any p and α , then (a) implies that νw⃗ is doubling. Assume pi > 1. To prove (b), it suffices to show that for all k
1−p′i
wi Ek
1 mp′ i
1− 1 ′ 1 mp αi αi ′ − ′ ′ ′ i wi (y)1−pi mpi −1 d(y, ∂ Q ) m (mpi ) dy ≤ C |Q |1+ mn ,
(30)
Fk 1−p′i
with Ek = Uk , Vk and Fk = Uk , Vk . These inequalities imply that wi 1−p′i
by Lemma 29 we have wi
∈ A−
α mp′i , mi ,k
1−p′i
and wi
∈ A+
α mp′i , mi ,k
for all k; consequently,
∈ Amp′ , αi . We shall prove (30) with (Ek , Fk ) = (Uk , Uk ), being similar the proof of the case i m
(Ek , Fk ) = (Vk , Vk ). Once these inequalities have been proved, we observe that (30) with (Ek , Fk ) = (Uk , Uk ) implies the same 1−p′ 1−p′ 1−p′ 1−p′ inequality for αi = 0 which is equivalent to wi i ∈ Amp′i . Therefore, wi i is doubling. In particular, Uk wi i ≤ C Vk wi i
200
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204 1−p′i
1−p′i
≤ C Uk wi . These inequalities together with (30) in the cases (Ek , Fk ) = (Uk , Uk ) and (Ek , Fk ) = (Vk , Vk ) and V wi k imply (30) with (Ek , Fk ) = (Uk , Vk ) and (Ek , Fk ) = (Vk , Uk ). Now we prove (30) in the case (Ek , Fk ) = (Uk , Uk ). Observe that
1−p′i
wi
1 mp′ i
α
− nmi
≤ C |Q |
Uk
1−p′i
wi (y)
αi p′i
d(y, ∂ Q )
dy
1 mp′ i
.
(31)
Uk
On the other hand, with ri = (m − 1)pi + 1, we write
Ii =
−
1−p′i
wi (y)
1 mp′ −1 i
αi
′ ′ m (mpi )
d(y, ∂ Q )
1 mp′ i
1− dy
Uk
wi (y)
=
m
1 ri
wj (y)
m
p
− p ri
wj (y)
j i
j=1,j̸=i
Uk
Ii ≤
wi (y)
m
1 ri
d(y, ∂ Q )
mpri
αj p i ri
i
dy
.
(32)
j=1,j̸=i pr
pri pi
By Hölder’s inequality with si =
and sj = (p −j 1i)p for j ∈ H, j ̸= i, and sj = +∞ for j ∈ J, we get that j i
wj (y)
pi pj ri
ri s mp
si
i
i
dy
j=1,j̸=i
Uk
×
pi pj ri
j∈H ,j̸=i
p
− p ri
wj (y)
d(y, ∂ Q )
j i
αj
pi ri
ri s mp j i
sj
dy
Uk
− m1
ess infy∈Uk wj (y)
.
− m1
j∈J
Then, since νw⃗ is a doubling weight, we have that
νw⃗
Ii ≤ C
1 mp m
j∈H ,j̸=i
Vk
1−p′j
wj
αj p′j
(y)d(y, ∂ Q )
dy
1 mp′ j
Uk
ess infy∈Uk wj (y)
.
(33)
j∈J
Putting together (31), (33) and using (i) we get (30) with (Ek , Fk ) = (Uk , Uk ). To prove (c), we argue as before and, consequently, we only have to prove
mpri i
1 ri
wi
wi (y)
1 ri
− mp1i
d(y, ∂ Q )
ri −1
αi mpi ′ ( ) m
ri
dy
1
mp ( r i )′ i
αi
≤ C |Q |1+ nm .
Uk
Uk
To prove it we use (33) (see the definition of Ii in (32)) and we obtain 1 ri
wi
mpri
i
≤ |Q |
α
− nmi
1 ri
wi (y) d(y, ∂ Q )
αi pi ri
mpri dy
αi
i
= |Q |− nm Ii
Uk
Uk αi
≤ C |Q |− nm
νw⃗
1 mp m
j∈H ,j̸=i
Vk
×
ess infy∈Uk wj (y)
− m1
1′ mp ′ ′ j wj (y)1−pj d(y, ∂ Q )αj pj dy Uk
.
j∈J
p⃗,⃗α ⊂ A+ , we obtain (34) as follows: Then, using that w ⃗ ∈A ⃗,⃗ p α ,k 1 ri
wi
mpri i
Uk
− mp1i
ri −1
αi mpi ′ ( )
d(y, ∂ Q ) m
ri
dy
1 mp
( r i )′ i
Uk α
− nmi
≤ C |Q |
νw⃗ Uk
×
wi (y)
1 ri
1 mp
j∈H
− m1
ess infy∈Uk wj (y)
j∈J
wj (y)
1−p′j
αj p′j
d(y, ∂ Q )
dy
1 mp′ j
Uk αi
α¯
αi
≤ C |Q |− nm |Q |1+ nm = C |Q |1+ nm .
(34)
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204 1/m
Now we prove (b) and (c) for i ∈ J, that is wi
1
|Q |
1/m
wi
1/m
≤ C ess infUk wi
201
∈ A1 . It suffices to prove that for all cubes Q and some k
.
(35)
Uk 1/m
Inequality (35) implies that wi ∈ A1 as follows: as in the proof of Lemma 23, given a cube Q there exists a cube Q such k , where Uk is the set Uk associated to that |Q | is comparable to |Q | and Q ⊂ U Q ; by (35)
1
|Q |
1/m
wi
C
≤
Q
1/m
1/m
wi
k U
| Q|
≤ C ess infUk wi
1/m
≤ C ess infQ wi
.
Let us prove (35). By Hölder’s inequality with exponent pm, we have
1/m i
w
p
wi
≤ Uk
Uk
p pj
wj
1 mp
pmmp−1
p − p (pm −1)
wj
.
j
(36)
Uk j∈H
j∈H
Let sj = (m − 1/p)p′j , j ∈ H. Assume first that sj > 1 for all j ∈ H. Applying Hölder’s inequality with exponents sj and the inequality d(y, ∂ Q )
w
1/m i
−αj p′j
≤ |Q |−
w
≤
p i
Uk
Uk
αj p′j n
p pj
, we obtain 1 mp
wj
j∈H
1−p′j
wj
1′ αj mp ′ j (y)d(y, ∂ Q )αj pj dy |Q |− mn .
(37)
Uk
j∈H
If sj0 = 1 for some j0 then pi = 1 for all i ̸= j0 . Consequently, H = {j0 } and inequality (37) follows from (36) since αj p′j
−α p′
d(y, ∂ Q ) j j ≤ |Q |− n (in this case we do not need to use Hölder’s inequality). By (i) and using that νw⃗ is doubling, we obtain (35) as follows:
w
1/m i
w
≤C
p i
Uk
Uk
wj
1 mp
wjp
j∈I ,j̸=i
Uk
= C ess infUk w
νw⃗
1 − mp (ess infUk wj )1/m |Q |
Uk
j∈H
wip
≤C
p pj
1/m i
j∈I
1 mp
p pj
wj
νw⃗
1 − mp
(ess infUk wi )1/m |Q |
Uk
j∈H
|Q |.
(ii) ⇒ (iii). We consider again the sets J and H. Let #(H ) be the number of elements of H. Notice that 1 α¯ #(H ) α¯ #(H ) − 1 +1 =C d(y, ∂ Q ) m2 νw⃗ (y) m2 p νw⃗ (y) m2 p dy |Q | nm2 Q α¯
d(y, ∂ Q ) m2 d(y, ∂ Q )
=C
α( ¯ #(H )−1) m2
−
νw⃗ (y)
1 m2 p
Q
−
1 m2 p
α¯
d(y, ∂ Q ) m2
Q
|Q |
nm2
+1
1 m2 pi
wi
αi
(y)d(y, ∂ Q ) m2
(y) dy
i∈H
By Hölder’s inequality with s0 = α¯ #(H )
1 m2 pi
wi
i =1
νw⃗ (y)
=C
m
≤C
m2 p mp−1
−
νw⃗ (y)
and si =
1 m2 p
s0
1 2
wim (y) dy.
i∈J
m2 pi ri
d(y, ∂ Q )
, i = 1, . . . , m, we get
s1
α¯ s0
0
m2
dy
Q
×
i∈H
wi (y)
si pi m2
d(y, ∂ Q )
si αi m2
s1 i
Q
i∈J α¯ p
1
s1 dy
i
Q
mp2−1
νw⃗ (y)− mp−1 d(y, ∂ Q ) mp−1 dy
=C
wi (y)
dy
si m2
m p
Q
×
i∈H
1 ri
wi (y) d(y, ∂ Q ) Q
αi pi ri
dy
ri m2 pi
m1 1 wi (y) m dy .
i∈J
Q
202
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
Raising to the mth power,
1
1 ≤ C
α¯ p 1+ n(mp−1)
|Q | ×
νw⃗ (y)
d(y, ∂ Q )
mpmp−1
α¯ p mp−1
dy
Q
1 αi pi 1+ nr
|Q |
i∈H
− mp1−1
1 ri
wi (y) d(y, ∂ Q )
αi pi ri
mpri
i
1 1 wim . |Q | Q i∈J
dy
Q
i
mpi ri
By Hölder’s inequality with si =
(38)
we get
mpri 1 m 1 − p 1−1 mp′i i 1 1 ri i w w 1≤ , |Q | Q i |Q | Q i i=1 where
1 |Q |
− p 1−1
Q
wi
i
1 mp′ i
(39)
is understood as ess infQ wi
−1/m
if pi = 1. Multiplying inequalities (38) and (39) and the result
by
1
|Q |
νw⃗
1 mp m
1
Q
i =1
|Q |
αi p′i
wi (y)1−pi d(y, ∂ Q )αi pi dy ′
′
1 mp′ i
,
(40)
Q
1+ n
and using (ii) we obtain that (40) is smaller than a constant C , as we wished to prove. Finally, the implication (iii) ⇒ (i) is obvious since Uk , Vk ⊂ Q . Proof of Theorem 9. (i) ⇒ (ii) is obvious. To prove (ii) ⇒ (iii) we consider the operators Nα⃗−,k and Nα⃗+,k . Since Nα⃗−,k ⃗ f ( x) ≤ C Mαc⃗ ⃗ f (x) and Nα⃗+,k ⃗ f (x) ≤ C Mαc⃗ ⃗ f (x), we get that the inequality in (ii) holds for Nα⃗−,k and Nα⃗+,k . Proceeding as in Lemma 2.4
p⃗,⃗α which is equivalent to w in [4] we get that w ⃗ ∈A ⃗ ∈ Ap⃗,⃗α by Theorem 28. To prove (iii) implies (i), notice that by Hölder’s inequality we get if pi > 1
αi
fi (y)d(y, ∂ Q ) dy ≤
Q
pi
fi wi
1/pi
Q
and if pi = 1 (αi = 0) then m
1
|Q |
i=1
α 1+ ni
f Q i
− p 1−1
wi (y)
i
αi p′i
d(y, ∂ Q )
1/p′i dy
,
Q
(y)d(y, ∂ Q )αi dy ≤
fi (y)d(y, ∂ Q )αi dy ≤ C Q
m
wi
f Q i p
fi i wi
ess infQ wi
1/pi
νw⃗
. Now, by using the hypothesis we get
−1/p
.
Q
Q
i =1
−1
1/pi
p
It follows that Mα⃗ (⃗ f )(x) ≤ C i=1 Mνw⃗ (fi i wi νw⃗ −1 (x)) , where Mνw⃗ is the Hardy–Littlewood maximal operator associated to the doubling measure νw⃗ (x)dx (by Theorem 28, νw⃗ ∈ Amp, α¯ ⊂ Amp ). Using Hölder’s inequality for weakm spaces [8, p.15] and the weak-type (1, 1) inequality for Mνw⃗ we get
m
∥Mα⃗ (⃗f )∥Lp,∞ (νw⃗ ) ≤ C
m 1/pi p Mνw⃗ (fi i wi νw⃗ −1 ) p ,∞ L i
i=1
≤C
m pi Mν (f pi wi νw⃗ −1 )11/,∞ w ⃗ i L (ν
w ⃗)
i=1
1 r
It follows from Remark 5 and wi i ∈ A mpi , αi that ri
m
mpi ri
>
(νw⃗ ) m
≤C
∥fi ∥Lpi (wi ) .
i =1 1 , αi ̸= 0, which is equivalent to pi > 1+α . i
1 α 1+ mi
5. Proof of Theorems 11 and 10 Proof of Theorem 11. Let w ⃗ ∈ Ap⃗,⃗α . Using Theorem 28 and Theorem 1 we get that there exists r0 > 1 such that 1 r
wi i ∈ A mpi , αi . We also have w ⃗ ∈ Ap⃗ . By Hölder’s inequality with si = r0 ri
m
r0 m
|Q |
=
m i=1
1 mpi
wi Q
1 − mp
wi
i
r0 ≤
m i =1
1 ri
wi Q
0 ri rmp i
Q
mpi ri
− p 1−1
wi
i
and s′i = r0 mp′ i
.
mpi , pi −1
(41)
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
203
1 ri
Since wi ∈ A mpi , αi , we have r0 ri
m
m
1 ri
wi
0 ri rmp i
wi (y)
0 ri
i
d(y, ∂ Q )
0 ri 1− rmp
αi pi mpi −r0 ri
i
dy
≤C
Q
Q
i =1
r
− mp −0r
m
αi
α¯
|Q |1+ nm = C |Q |m+ nm .
(42)
i=1
Using (41), (42) and w ⃗ ∈ Ap⃗ we obtain r0 m
νw⃗
|Q |
r0 mp m
Q
and βi = αi
Taking r = can be written as
νw⃗
r /p
m Q
i =1
Q
mr r0
where we have used
d(y, ∂ Q )
wi (y)
i
1+r (m−1) m
we get that
0 ri
αi pi mpi −r0 ri
0 ri 1− rmp i
dy
α¯
≤ C |Q |r0 +m+ nm .
Q
i=1
r0 m−(m−1)r0
r
− mp −0r
−r0 pi m−r0 ri
= 1 − ( pri )′ and
αi pi mpi −ri r0
= βi ( pri )′ . Then the above inequality
rri mr pi ′ pi ′ β¯ r −p wi (y)1−( r ) d(y, ∂ Q )βi ( r ) dy 0 i ≤ C |Q |m+ n ,
¯ α¯ m − r0 (m − 1) + mn = m + βn . Since
mr r0
rri pi
−
= [( pri )′ ]−1 , we are done.
Proof of Theorem 10. Since (i) ⇒ (ii) is trivial and by Theorem 9 we get that (ii) ⇒ (iii), we shall only prove (iii) ⇒ (i). As in the proof of Theorem 3.7 in [6] it is enough to prove that there exists δ < 1 such that
Mα⃗ (⃗f )(x) ≤ C
m
−δ Mνw⃗ |fi |δ pi wiδ νw (x) ⃗
δ1p
.
i
(43)
i=1
In fact, by Hölder’s inequality with exponents νw⃗ we get
|Mα⃗ (⃗f )| νw⃗ p
1p ≤C
and the strong type ( 1δ , 1δ ) inequality of Mνw⃗ with respect to the measure
pi p
m
p −δ δ pi Mνw⃗ |fi |pi δ wiδ νw νw⃗ ⃗
1p
i =1 m
≤C
δ pi
Mνw⃗ (|fi |
1
δ wiδ νw−δ ⃗ )
νw⃗
p1
i
≤C
i=1
|fi (y)|d(y, ∂ Q ) dy ≤
pi δ
|fi |
Q
Q
= Q
Let λ =
1+r (m−1) . mr
p1
i
.
|fi |pi δ wiδ νw1⃗−δ
δ1p
δ − p δ− 1
i
wi (y)
i
< δ < 1, where r is the number in Theorem 11. By
νw⃗ (y)
δ−1 pi δ−1
d(y, ∂ Q )
αi pi δ pi δ−1
pipδ−δ 1 dy
i
Q
δ1p
i
Ii .
It is clear that λ ∈ (0, 1) and δ − p δ− 1
wi (y)
Ii =
wiδ νw1⃗−δ
1 pi (1− 1r )+1
Hölder’s inequality with exponent pi δ , αi
|fi | wi pi
i=1
Now, we shall prove (43). Let δ be such that maxi=1,...,m
m
i
νw⃗ (y)
δ−1 pi δ−1
d(y, ∂ Q )
αi λpi δ pi δ−1
d(y, ∂ Q )
αi (1−λ)δ pi δ pi −1
δpδip−1 dy
i
.
Q r (δ p −1)
Let si = δ(p i−r ) . We have that si > 1 and, for later computations, i we get that
wi (y)
Ii ≤
− p r−r i
βi pi
ppi −rr
d(y, ∂ Q ) pi −r dy
i
1 s′i
=
δ pi (r −1)+r (δ−1) . By Hölder inequality with exponent si r (pi δ−1)
IIi ,
Q
where IIi =
ti =
Q
νw⃗ (y)
(δ−1)s′i δ pi −1
d(y, ∂ Q )
αi (1−λ)pi δ s′i pi δ−1
pi δ−′1 pi δ s i
dy
and βi = αi
δ pi − 1 αi (1 − λ)pi δ s′i ti′ > 1 and > −1, (1 − δ)s′i (mp − 1) pi δ − 1
1+r (m−1) . m
Now, choosing δ close to 1 such that
204
A.L. Bernardis et al. / J. Math. Anal. Appl. 397 (2013) 191–204
and applying Hölder’s inequality with ti we get
− mp1−1
IIi ≤ Q
νw⃗
(1−δ)(δpmp−1) i
αi (1−λ) n
|Q |
(For subsequent computations, notice that
νw⃗ ∈ Amp we get that m
|fi (y)|d(y, ∂ Q )αi dy ≤ C
1 s′i ti′
m
. 1 s′i
=
(1−δ)(mp−1) .) pi δ−1
Putting together the above estimates and using that
δ1p i |fi (y)|δpi wi (y)δ νw⃗ (y)1−δ dy
1
−
νw⃗ (Q ) Q ppi −rr 1p m m (1−δ)mp αi (1−λ) pi δ−1 βi pi + n + i δp − p r−r pi δ s′ t ′ p − r i i . × wi (y) i d(y, ∂ Q ) i νw⃗ |Q | i
Q
i=1
pi δ−1 pi δ s′ t ′ i i
+
i=1
i =1
Q
Q
i=1
Then, using Theorem 11 we get 1
|Q |
¯ m+ α n
m i=1
αi
|fi (y)|d(y, ∂ Q ) dy ≤ C
m
Q
−m− αn¯
|Q |
δ pi
|fi |
νw⃗ (Q )
i=1
×
1
Q
|Q |
β¯ m r + nr
wiδ νw1⃗−δ
|Q |
δ1p
(1−δ)m δ
i
|Q |
α( ¯ 1−λ) n
m
|Q |
pi δ−1 pi δ s′ t ′ i i
.
i =1
Since the last factor is 1 we are done.
6. Remarks and examples We start establishing some properties of the linear classes Ap,α . Assume that p(1 + α) > 1. Applying Hölder’s inequality, it is easy to see that Aq ⊂ Ap,α for 1 < q < p(1 + α). Using that w ∈ Ap ⇒ Ap−ε we have Ap(1+α) ⊂ Ap,α . Other interesting relation is that w ∈ A1 ⇒ w ∈ RA 1 ,α . We can also see that a power weight 1+α
w(x) = |x|β ∈ Ap,α ⇔ −n < β < p(n + α) − n ⇔ |x|β ∈ Ap(1+ αn ) .
(44)
These properties in the one-sided case can be found in [9].
⃗, α⃗ satisfy the assumptions on Theorem 9. Let Example 45. We work in R (n = 1) with m = 2. We assume that p w1 (x) = |x|γ1 and w2 (x) = |x|γ2 . We are going to determine the conditions on γ1 and γ2 to have w ⃗ ∈ Ap⃗,⃗α or, equivalently, 1−p′i
(a) νw⃗ ∈ A2p, α¯ , (b) wi 2
1 r
∈ A2p′ , αi , for i = 1, 2 and (c) wi i ∈ A 2pi , αi , for i = 1, 2, where ri = pi + 1. Using (44), w ⃗ ∈ Ap⃗,⃗α i
2
ri
2
if and only if the following relations hold: −1 < p γ1 + p γ2 < 2p + p(α1 + α2 ) − 1, −p1 − 1 − p1 α2 < γ1 < p1 − 1, 1 2 −p2 − 1 − p2 α1 < γ2 < p2 − 1, −p1 − 1 < γ1 < p1 + p1 α1 − 1 and −p2 − 1 < γ2 < p2 + p2 α2 − 1. Let p1 = 2, p2 = 4, p = 4/3, α1 = −1/3, α2 = −1/2. p
p
(1) If γ1 = −1, γ2 = 0 then w ⃗ ∈ Ap⃗,⃗α but w1 ̸∈ Ap1 ,α1 . Therefore, Mα⃗ applies Lp1 (w1 ) × Lp2 (w2 ) into Lp (νw⃗ ) but Mα1 is not bounded in Lp1 (w1 ). (2) If γ1 = −1 = γ2 then (b) and (c) are satisfied but (a) is not satisfied. (3) If γ1 = 1/3, γ2 = −11/3 + ε , ε small, then (a) and (b) are satisfied but (c) is not satisfied (the second condition in (c) is satisfied). Notice that in this example (a) and (c) imply (b). References [1] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972) 207–226. [2] J. García-Cuerva, J.L. Rubio de Francia, Weighted norm inequalities and related topics, in: North-Holland Mathematics Studies, 116, in: Notas de Matemática [Mathematical Notes], vol. 104, North-Holland Publishing Co, Amsterdam, 1985. [3] W. Jurkat, J. Troutman, Maximal inequalities related to generalized a.e. continuity, Trans. Amer. Math. Soc. 252 (1979) 49–64. [4] A.L. Bernardis, F.J. Martín-Reyes, The Cesàro maximal operator in dimension greater than one, J. Math. Anal. Appl. 228 (2003) 69–77. [5] R.A. Kerman, A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math. 71 (1981/82) 277–284. [6] A. Lerner, S. Ombrosi, C. Pérez, R. Torres, R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math. 220 (2009) 1222–1264. [7] F.J. Martín-Reyes, A dominated ergodic theorem for some bilinear averages, J. Math. Anal. Appl. 368 (2010) 469–481. [8] L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, 2004. [9] F.J. Martín-Reyes, A. de la Torre, Some weighted inequalities for general one-sided maximal operators, Studia Math. 122 (1997) 1–14.