Characterization of two-weighted inequalities for multilinear fractional maximal operator

Linear Algebra Applications Nonlinear Analysis and 130 its (2016) 214–228 466 (2015) 102–116

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Linear Algebra and its Applications Nonlinear Analysis www.elsevier.com/locate/laa www.elsevier.com/locate/na

Inverse eigenvalue problem of Jacobi matrix Characterization of two-weighted inequalities for multilinear with mixed data fractional maximal operator 1 ∗ Ying Wei Mingming Cao, Qingying Xue School of MathematicalDepartment Sciences, Beijing Normal University, LaboratoryofofAeronautics Mathematics and Complex of Mathematics, Nanjing University and Astronautics, Systems, Ministry of Education, BeijingPR 100875, Nanjing 210016, ChinaPeople’s Republic of China

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Article history: Article history: In this the inverse eigenvalue problem of reconstructing In this paper, we restudied thepaper, two-weight problem of multilinear fractional maximal 2014 Received 9 March 2015 Received 16 January operator a Jacobi matrix from its of eigenvalues, leading principal Mα . First, we gave a characterization two-weight its inequalities for Mα Accepted 20 September 2014 Accepted 3 October 2015 submatrix part of two-weight the eigenvalues of its submatrix related to a multilinear analogueand of Sawyer’s condition S(⃗p,q) , which Communicated by EnzoAvailable Mitidierionline 22 October 2014 is extended considered. The necessary sufficient conditions for essentially improved and some known resultsand before. This was done mainly Submitted by Y. Wei existence and atomic uniqueness of the solution derived. by using the techniquethe of the well known decomposition of tent are space. Then Keywords: algorithm and some numerical we obtained the strongFurthermore, boundedness aof numerical Mα associated with multiple weight A(⃗p,q) MSC: Multilinear fractional maximal examples are given. class, which removed the power bump condition assumed in the known results 15A18 operator © 2014 Elsevier Inc. before. Finally, a new two-weight B(⃗p,q) condition was Published introducedbyand the twoTwo-weight inequalities 15A57 weight inequality of Mα with B(⃗p,q) condition was established. Tent space Keywords: © 2015 Elsevier Ltd. All rights reserved. S(⃗ p,q) condition Jacobi matrix B(⃗ p,q) condition Eigenvalue Inverse problem Submatrix

1. Introduction The two-weight problem for linear operators originated in the works of Muckenhoupt and Wheeden [14–16] in the 1970s. The general question is to find a necessary and sufficient condition for a pair of unrelated weights w and v for which the following estimate holds     T (f ) q ≤ Af  p , L (v) L (w) for a finite constant A independently of f . In 1982, Saywer [21] showed that the fractional maximal operator  1 Mα f (x) := sup |f (y)|dy 1−α/n Q∋x |Q| Q E-mail [email protected]. is bounded from Lp (w) to Lqaddress: (v) if and only if (w, v) satisfies that 1 Tel.: +86 13914485239.  1/q q M (σχ ) vdx α Q http://dx.doi.org/10.1016/j.laa.2014.09.031 Q v]SPublished := sup < ∞, 0024-3795/©[w, 2014 by Elsevier Inc. (p,q) σ(Q)1/p Q∈Q ∗ Corresponding author. E-mail addresses: [email protected] (M. Cao), [email protected] (Q. Xue).

http://dx.doi.org/10.1016/j.na.2015.10.004 0362-546X/© 2015 Elsevier Ltd. All rights reserved.

M. Cao, Q. Xue / Nonlinear Analysis 130 (2016) 214–228

215

where σ = w1−p , 0 ≤ α < n, 1 < p < n/α, Q is the family of all cubes in Rn with sides parallel to the axes. In 1984, Saywer [22] gave a sufficient and necessary condition for two-weight weak type inequality of the fractional integral operator Iα , which was defined by  f (y) dy. Iα f (x) = |x − y|n−α n R ′

They proved that the following inequality   A  q v {Iα f (x) > λ} ≤ q f Lp (w) , λ holds if and only if  

Iα (χQ v)(x)

 p′

w1−p dx ≤ B ′

q′ /p′ vdx < ∞,



for all cubes Q ∈ Rn .

Q

Q

In 1988, as for the fractional integral Iα and a kind of more general convolution operator with radial kernel decreasing in |x|, Saywer [23] obtained a characterization for two-weight strong type inequality. Inspired by the above results, the theory of two-weighted inequalities developed rapidly. Among such achievements are the nice works of Lacey and Li [7], Perez and Rela [19], Sawyer and Wheeden [24], and Wheeden [25]. Much later, the interests were focused on determining the sharp dependence of the Lp (w) → Lq (v) operator norm in terms of the relevant constant involving the weights, see [6,8,12] for more details. Moreover, in 2009, Moen [12] extended Sawyer’s result by proving that   Mα  p ≍ [w, v]S(p,q) . (1.1) L (w)→Lq (v) However, Saywer’s condition is often difficult to verify in practice, since it involves the maximal operator. Thus it is necessary to look for other simple sufficient conditions. The first attempt was made by Neugebauer [17] in 1983. He gave a sufficient condition closely in spirit to the classical Ap condition: if (w, v) satisfies  sup Q∈Q

1 |Q|



1   pr

pr

v dx Q

1 |Q|

 w

−p′ r

 dx

1 p′ r

< ∞,

for some r > 1,

(1.2)

Q

    then M f Lp (vp ) . f Lp (wp ) . Later, in 1995, P´erez [18] improved condition (1.2) by  sup Q∈Q

1 |Q|

 p1   1′  p r ′ 1 v p dx w−p r dx < ∞. |Q| Q Q



The two-weight problem for multilinear operators has also been studied recently. Let us first recall some related known results. In 2009, the multilinear fractional type maximal operator  m  1 |fi (yi )| dyi , for 0 ≤ α < mn, Mα (f⃗)(x) = sup α Q∋x |Q|1− mn Q i=1 Q∈Q

was first introduced and studied by Chen and Xue in [4], and also simultaneously defined and studied by Moen in [13]. Specially, when α = 0, M0 is coincide with the multilinear maximal function and will be denoted by M, which was introduced by Lerner, Ombrosi, P´erez, Torres and Trujillo-Gonz´alez in [9]. To study two-weight inequality, Moen [13] introduced the following two-weight condition [w, ⃗ v]A(⃗p,q) := sup |Q| Q∈Q

α 1 1 n+q−p



1 |Q|

 q1   1′  m  p 1 i 1−p′i vdx wi < ∞. dx |Q| Q Q i=1



(1.3)

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He showed that Mα is bounded from Lp1 (w1 ) × · · · × Lpm (wm ) to Lq,∞ (v) if and only if [w, ⃗ v]A(⃗p,q) < ∞. However, for strong boundedness of Mα , (w, ⃗ v) needs to satisfy a certain power bump variant of the multilinear A(⃗p,q) condition. That is, if for some r ∈ (1, ∞),   q1   1′   m  rp α 1 1 1 1 i r(1−p′i ) < ∞, (1.4) sup |Q| n + q − p vdx wi dx |Q| Q |Q| Q∈Q Q i=1 then Mα is bounded from Lp1 (w1 ) × · · · × Lpm (wm ) to Lq (v). Still more recently, for the two-weight case, Chen and Dami´an [2] gave another sufficient condition for the two-weight inequality to hold for the operator M. At the same time, Li and Sun [10] studied the problem of two-weight inequality for multilinear fractional maximal operator Mα . They gave a multilinear analogue of Sawyer’s two-weight test condition. More precisely, they proved that Mα was bounded from Lp1 (w1 ) × · · · Lpm (wm ) to Lq (v) if and only if [w, ⃗ v]S(⃗p,q) < ∞, where −1  1/q  m σi (Q)1/pi , [w, ⃗ v]S(⃗p,q) := sup Mα (σ1 χQ , . . . , σm χQ )q vdx Q∈Q

1−p′i

and σi = wi

Q

i=1

, i = 1, . . . , m. Moreover, they also pointed out that   Mα  p ≍ [w, ⃗ v]Sp⃗,q . L 1 (w )×···Lpm (w )→Lq (v) 1

m

It should be noted that the exponents were restricted strictly, by the reason that they require q ≥ maxi {pi } and 1q = p1 − α n . Thus α needs to satisfy α ≥ n(1/p − 1/ max{pi }). Moreover, since the interpolation theorem was used, their method is not suitable for the case 0 ≤ α < n(1/p − 1/ max{pi }). Motivated by the works above, in this paper, our first main purpose is to apply a new technique to improve Li and Sun’s result [10] by enlarging the ranges of exponents to 0 ≤ α < n(1/p − 1/ max{pi }). We first obtain a characterization of two-weight inequality with respect to a multilinear analogue of Sawyer’s two-weight test condition, S(⃗p,q) condition. However, it is worth mentioning that we do not adopt traditional methods as in [2]. Atomic decomposition of tent space, which is a powerful and convenient tool to tackle weighted inequalities, will be applied to establish two-weight inequalities as for different multiple weights conditions. In addition, a mixed A(⃗p,q) -Wp⃗∞ estimate for Mα will be given as well, which avoids the power bump condition used by Moen [13]. Finally, we consider a new quantity B(⃗p,q) condition and present a sufficient condition for the strong boundedness of Mα . The organization of this article is as follows. Definitions and main results will be listed in Section 2. In Section 3, we first introduce some definitions about tent space. Then by means of atomic decomposition of tent space, we present the proof of Theorem 2.1. Using the similar technique, we prove Theorems 2.2 and 2.5 in Sections 4 and 5 successively. In Section 6, the relationship among several kinds of weight conditions will be given. 2. Definitions and main results First we state the notation that we will follow in the sequel related to some constants involved in the theory of multiple weights. 1−p′i

Definition 2.1 ([2]). Let wi , v be nonnegative and locally integrable functions on Rn , σi = wi 1, . . . , m) and p1 = p11 + · · · + p1m with 1 < p1 , . . . , pm < ∞. We say that (1) w ⃗ satisfies the multiple Reverse H¨ older condition or RHp⃗ condition if p/pi   −1 m  m  p/p [w] ⃗ RHp⃗ := sup σi dx σi i dx < ∞; Q∈Q i=1

Q

Q i=1

(i =

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217

(2) w ⃗ satisfies the Wp⃗∞ condition if [w] ⃗ Wp⃗∞ := sup

  m

Q∈Q

  −1 m p/p < ∞. M (wi χQ )p/pi dx wi i dx

Q i=1

Q i=1

When m = 1, the linear Wp⃗∞ condition first appeared in the work of Wilson [11]. Here we formulate the main results of this paper as follows. Theorem 2.1. Let 0 ≤ α < mn, p1 = p11 + · · · + p1m with 1 < p1 , . . . , pm < ∞, and 0 < p ≤ q < ∞. Suppose that w1 , . . . , wm , v are weights and w ⃗ ∈ RHp⃗ . Then Mα : Lp1 (w1 ) × · · · × Lpm (wm ) → Lq (v) if and only if [w, ⃗ v]S(⃗p,q) < ∞. Moreover,   1/p ⃗ v]S(⃗p,q) [w] ⃗ RHp⃗ . [w, ⃗ v]S(⃗p,q) . Mα Lp1 (w1 )×···×Lpm (wm )→Lq (v) . [w, Remark 2.1. When m = 1, the Reverse H¨ older condition always holds and we recover the linear result of m Moen (1.1). In one-weight A(⃗p,q) case (that is, taking v = i=1 ωiq and 1q = p1 − α n in (1.3), this type of weights was defined and studied by [4,13], more properties of this weights can be found in [3]), the Reverse H¨ older condition always holds as well, which will be proved in the following Proposition 2.3. Thus, the above theorem are sharp for one-weight A(⃗p,q) case. Moreover, as for the multilinear case, we enlarge the ranges of exponents in the result of Li and Sun [10] from n(1/p − 1/ max{pi }) ≤ α < mn to 0 ≤ α < mn. 1 p

Theorem 2.2. Let 0 ≤ α < mn,

1 p1

=

+ ··· +

1 pm

with 1 < p1 , . . . , pm < ∞, and 0 < p ≤ q < ∞.

1−p′ wi i

Suppose that w1 , . . . , wm , v are weights and σi = (i = 1, . . . , m). Assume that ⃗σ ∈ Wp⃗∞ . Then ⃗ v) ∈ A(⃗p,q) . Moreover, Mα : Lp1 (w1 ) × · · · × Lpm (wm ) → Lq (v) if and only if (w, m      1/p fi  p Mα (f⃗) q . [w, . ⃗ v] [⃗ σ ] ∞ A(⃗ p,q) W L i (wi ) L (v) p ⃗

i=1

Remark 2.2. As for the linear case M , we give an alternative proof of Theorem 1.7 [6]. Here we avoid the power bump condition (1.4). However, the two-weight A(⃗p,q) condition is not strong enough to obtain the boundedness of Mα from Lp1 (w1 ) × · · · × Lpm (wm ) to Lq (v), see Remark 7.3 [13]. Thus, we further assumed the condition ⃗σ ∈ Wp⃗∞ . It should be noted that as for the one-weight A(⃗p,q) case, the condition ⃗σ ∈ Wp⃗∞ always holds (see Proposition 2.3 below). Thus, our assumption is pretty much natural in the above theorems. Proposition 2.3. Let α, p⃗, q, w, ⃗ v and ⃗σ be the same as in Theorem 2.2. Then one-weight condition w ⃗ ∈ A(⃗p,q) implies that [w] ⃗ RHp⃗ ≤

m 

−p′i p/pi ]A∞ ,

[wi

[⃗σ ]Wp⃗∞ .

i=1

m 

−p′i 2p/pi ]A∞

[wi

< ∞,

i=1

where  [w] ⃗ A(⃗p,q) = sup

Q∈Q

1 |Q|

  m Q i=1

 q1 wiq dx

 1′  m   p 1 i −p′ wi i dx < ∞. |Q| Q i=1

As we mentioned in the above, Sawyer’s condition is difficult to check. Therefore, it is necessary to consider another sufficient condition. To establish the strong boundedness of Mα , we introduce a different new quantity as follows.

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Definition 2.2. Let 0 ≤ α < mn, 0 < q < ∞ and (w, ⃗ v) satisfies the B(⃗p,q) condition if 1

1 p

Q∈Q

1 p1

+ ··· +

1 pm

with 1 < p1 , . . . , pm < ∞. We say that

 q1     m 1 1 vdx wi dx |Q| Q |Q| Q i=1   m  − p1  1 log × exp wi i dx < ∞. |Q| Q i=1 1

[w, ⃗ v]B(⃗p,q) := sup |Q| n + q − p α

=



When m = 1, α = 0 and p = q, the B(⃗p,q) was first introduced by Hyt¨onen and P´erez [6] to study the two-weight boundedness of Hardy–Littlewood operator. As for the case α = 0 and p = q, the multiple B(⃗p,q) condition was investigated by Chen and Dami´ an in [2]. A(⃗p,q) , B(⃗p,q) and S(⃗p,q) have the following relationship, Proposition 2.4. Let α, p⃗, q, w, ⃗ v and ⃗σ be the same as in Theorem 2.2. Then (i) When α = 0, p = q, m = 1 and v = w, we have 1

[⃗σ , v]B(⃗p,q) ≤ [w]Ap−1 ; p (ii) For the general case, we have [w, ⃗ v]A(⃗p,q) ≤ [w, ⃗ v]S(⃗p,q) ≤ cn,⃗p,q [⃗σ , v]B(⃗p,q) . For the smallest weights class B(⃗p,q) , we can obtain more better form as follows, Theorem 2.5. Let 0 ≤ α < mn,

1 p

=

1 p1

+ ··· + 1−p′i

that w1 , . . . , wm , v are weights and σi = wi

1 pm

with 1 < p1 , . . . , pm < ∞, and 0 < p ≤ q < ∞. Suppose

(i = 1, . . . , m). If (⃗σ , v) ∈ B(⃗p,q) , then

m      fi  p Mα (f⃗) q . [⃗σ , v]B . (⃗ p ,q) L (v) L i (wi ) i=1

3. Proof of Theorem 2.1 Our next argument of the proof follows a scheme similar to the one used by Rakotondratsimba in [20]. It was shown how the atomic decomposition of tent spaces can be used to get a characterization of the weight functions w, v for which the fractional maximal operators Mα sends the weighted Lebesgue spaces Lp (w) into Lq (v). As a matter of convenience, we first are going to introduce the notions of tent spaces needed. Our definitions will be slightly different from those introduced by Coifman, Meyer, and Stein in [5], since we use the dyadic versions of the spaces they defined. Let X be the cone [0, ∞)n minus the set of dyadics points, i.e. X = [0, ∞)n \ {(2−ℓ k1 , . . . , 2−ℓ kn ); ℓ ∈ Z, ki ∈ N}. The upper half-space is defined by  = X × {2−ℓ ; ℓ ∈ Z}. X  there is a unique (open) dyadic cube Q = Qyw containing y and having the side For each couple (y, w) ∈ X, length w = 2−ℓ . We write  (x) (y, w) ∈ Γ

if and only if x ∈ Qyw .

(3.1)

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Define  = Ω

c

 

 (x) Γ

,

for any Ω ⊂ [0, ∞)n .

x∈Ω c

Thus  (y, w) ∈ Ω

if and only if Qyw ⊂ Ω .

 by Finally, define the functional A∞ acting on each measurable function f⃗˜ on X A∞ (f⃗˜)(x) =

m  sup |fi (y, w)|. (x) i=1 (y,w)∈Γ

(3.2)

The following lemma is essential to the proof of our theorems. Lemma 3.1 (Atomic Decomposition). Let 0 < p < ∞ and ν be a nonnegative locally integrable function  on Rn . There is C > 0 such that for all functions fi (y, w = 2−ℓ ) with a support contained in Q[0, γ] ⃗ and ∥A∞ (f˜)∥Lp (ν) < ∞, one can find nonnegative scalars {λj }∞ , dyadic cubes {Qj }∞ and functions j=1

j=1

{ aj (y, w)}∞ j=1 which satisfy the following conditions: 1

Q supp  aj are disjoint and | aj (y, w)| ≤ ν(Qkj )− p χ  (y, w); j

m 

fi (y, w) =

i=1



λj  aj (y, w) a.e.;

(3.3) (3.4)

j



λpj

 p1

→ −   ≤ C A∞ ( f )Lp (ν) ;

(3.5)

j



r λrj ν(Qj )− p χQj (x) ≤ CA∞ (f⃗˜)(x)r (x),

for any r ≥ 1.

(3.6)

j

 Here χ Q (·, ·) is the characteristic function of tent Q. Proof. The essential idea of the proof can be tracked back to the works of [5,20] (where they treated the case for m = 1). Let Ωk = {A∞ (f⃗˜) > 2k }, k ∈ Z. Since A∞ (f⃗˜) is supported by (0, γ)n , there exist dyadic  cubes {Qkj }∞ j=1 with disjoint interiors such that Ωk = j Qkj . Then  k =  kj , and Ω k+1 ⊂ Ω k . Ω Q j

We define scalars {λj } and functions { akj (y, w)} as follows 1

λkj = 2k+1 ν(Qkj ) p , 1

 akj (y, w) = 2−(k+1) χ E (y, w)ν(Qkj )− p

m 

kj

fi (y, w),

i=1

kj = Q  kj \ Ω k+1 is the support of  where E akj and disjoint. It is obvious that (3.3) holds. The equality (3.4) can be obtained as follows m m    fi (y, w) = fi (y, w) χΩ  \Ω  (y, w) k

i=1

=

k i=1 m 

k+1

fi (y, w) χE (y, w) =



kj

k,j i=1

k,j

λkj  akj (y, w).

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M. Cao, Q. Xue / Nonlinear Analysis 130 (2016) 214–228

The inequality (3.5) follows from 

λpkj =



k,j

2(k+1)p

ν(Qkj )

j

k

.





 p 2kp ν(Ωk ) ≤ A∞ (f⃗˜)Lp (ν) .

k

Now it remains to estimate the inequality (3.6). Let r ≥ 1, then     r 2(k+1)r χQkj = 2(k+1)r χΩk (x) λrkj ν(Qkj )− p χQkj (x) = j

k

k,j

∞ 

.

ℓ=0 ∞ 

≤

2−ℓ



k

2(k+ℓ)r χ{2k+ℓ
⃗ ˜

∞ (f )≤2

k+ℓ+1 }

(x)

k

2−ℓ A∞ (f⃗˜)(x)r . A∞ (f⃗˜)(x)r .

ℓ=0



So, we complete the proof of Lemma 3.1. Lemma 3.2. Let

1 p

1 p1

=

1−p′i

1 pm

+ ··· +

with 1 < p1 , . . . , pm < ∞. Suppose that w1 , . . . , wm are weights

and σi = wi (i = 1, . . . , m), and ⃗σ ∈ RHp⃗ . Then there is a constant c > 0 such that for all p1 ∞ ⃗ f ∈ L (w1 ) × · · · × Lpm (wm ), one can find nonnegative scalars {λj }∞ j=1 and dyadic cubes {Qj }j=1 satisfying 

λpj

 p1

≤C

m    fi  p ; L i (w )

m      − 1 λj σi (Qj ) pi χQj   j

(3.7)

i

i=1

j

Lp (ν⃗ σ)

i=1

r/p

≤C

r ⃗ Md,γ ⃗ RHp⃗ α (f )(x) χ(0,∞)n (x) ≤ [w]

m    fi  p ; L i (wi )

(3.8)

i=1



λrj

j

m 

− pr

σi (Qj )

i

Mα (σ1 χQj , . . . , σm χQj )(x)r χQj (x),

(3.9)

i=1

⃗ for all most everywhere x ∈ (0, ∞)n and all r > 0, where the truncated version maximal operator Md,γ α (f ) is defined as follows  m  1 ⃗)(x) = sup Md,γ ( f |fi (yi )|dyi , x ∈ Rn . α α 1− mn Q∋x,Q∈D |Q| Q n i=1 Q⊂(0,γ)

Proof. Set ν = ν⃗σ =

m

i=1

p/pi

σi

in Lemma 3.1. Write  1 |fi /σi |σi dx, fi (y, w) = σi (Qyw ) Qyw

i = 1, . . . , m.

According to (3.1) and (3.2), we have −−→ A∞ (f⃗˜)(x) ≤ M⃗σd (f /σ)(x),

x ∈ Rn ,

−−→ where f /σ = (f1 /σ1 , . . . , fm /σm ) and the dyadic version multilinear weighted maximal operator M⃗σd is well defined as follows  m  1 M⃗σd (f⃗)(x) = sup |fi (yi )|σi dyi . (3.10) σ (Q) Q Q∋x i=1 i Q∈D

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M. Cao, Q. Xue / Nonlinear Analysis 130 (2016) 214–228

Using the H¨ older’s inequality and the boundedness of Mσdi on Lpi (σi ), we obtain   A∞ (f⃗˜) p L (ν

⃗ σ)

 −−→  ≤ M⃗σd (f /σ)Lp (ν ) ⃗ σ m     ≤  Mσdi (fi /σi ) p

L (ν⃗ σ)

i=1

≤

m   d  Mσ (fi /σi ) p i L i (σi ) i=1

m m       fi /σi  p fi  p . = . L i (σi ) L i (wi ) i=1

(3.11)

i=1

Then, (3.5) and (3.11) imply the inequality (3.7). Similarly, the inequality (3.8) follows from (3.6) with r = 1 and (3.11). Next we turn our attention to the proof of (3.9). A fundamental observation is that ⃗ Md,γ α (f )(x) =

sup Qyw ∋x,Q∈D Qyw ⊂(0,γ)n

m  σi (Qyw )  α fi (y, w). 1− mn |Q yw | i=1

Let r > 0 and Qyw ∋ x. Then by (3.3) and (3.4), we have  r   r m m σi (Qyw )  σi (Qyw ) r r f (y, w) = λ  a (y, w) α α i j j |Qyw |1− mn |Qyw |1− mn i=1 i=1 j ≤



λrj



m 

p pi

σi

− pr  m

σi (Qyw )   (y, w) α χ j |Qyw |1− mn Q i=1

Qj i=1

j r/p

≤ [w] ⃗ RHp⃗



λrj

j r/p

≤ [w] ⃗ RHp⃗

 j

m 

σi (Qj )

− pr

i

i=1

λrj

m 

σi (Qj )

− pr

i

r

 m

r σi (Qyw ) χ  (y, w) α j |Qyw |1− mn Q i=1

Mα (σ1 χQj , . . . , χQj )(x)r χQj (x).

i=1

Taking the supremum on cubes Qyw ∋ x with Qyw ⊂ (0, γ)n , we get (3.9). 

So far, we have proved Lemma 3.2.

We also need the following key lemma, which connects the maximal operator with the corresponding dyadic version of the maximal operator. Lemma 3.3 ([1]). Let x, t ∈ Rn , 0 ≤ α < mn. For any locally integral functions f⃗, k ≥ 0, we define the following truncated version maximal operator,  m  1 ⃗)(x) := |fi (yi )|dyi . M(k) ( f sup α α Q∋x,Q∈Q, |Q|1− mn Q i=1 every side length of Q≤2k

Then we have, q ⃗ M(k) α (f )(x)

Cn,α ≤ |Bk |



 q τ−t ◦ Mdα ◦ ⃗τt (f⃗)(x) dt,

Bk

where Bk = [−2k+2 , 2k+2 ]n , τt g(x) = g(x − t), ⃗τt f⃗ = (τt f1 , . . . , τt fm ). Now let us see how Lemmas 3.2 and 3.3 imply Theorem 2.1.

for any k ≥ 0,

(3.12)

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Proof of Theorem 2.1. The necessity is obvious. Indeed, from the boundedness of Mα , it follows that   [w, ⃗ v]S(⃗p,q) ≤ supMα Lp1 (w

p 1 )×···L m (wm

Q

m m  −1      σ χ σi (Q)1/pi i Q Lpi (w ) )→Lq (v) i

i=1

i=1

  = Mα Lp1 (w1 )×···Lpm (wm )→Lq (v) . To prove the sufficiency, we only need to estimate m      1/p Mα (f⃗) q . [w, fi  p ⃗ v] [ w] ⃗ . S(⃗ p,q) RHp L (v) L i (w ) ⃗ i

i=1

We first prove the boundedness for the dyadic version operator Mdα , then give the proof for Mα • Estimate for Mdα . We need to show, m   d    1/p Mα (f⃗) q . [w, fi  p . ⃗ v] [ w] ⃗ S (⃗ p,q) RHp L (v) L i (wi ) ⃗

(3.13)

i=1

As done in [21] by applying translations and reflections of the cone [0, ∞)n , it is sufficient to show m      d,γ 1/p fi  p Mα (f⃗)χ(0,γ)n  q . [w, ⃗ v] . [ w] ⃗ S (⃗ p,q) RHp L (v) L i (wi ) ⃗

(3.14)

i=1

Note that p ≤ q and Minkowski’s inequality. By the inequalities (3.7), (3.9) and S(⃗p,q) condition, we obtain  d,γ    ⃗p  q Mα (f⃗)χ(0,γ)n p q = Md,γ α (f ) χ(0,γ)n L (v) p L (v)

m      −p σi (Qj ) pi Mdα (σ1 χQj , . . . , σm χQj )p χQj  λpj ≤ [w] ⃗ RHp⃗ 

≤ [w] ⃗ RHp⃗

 j

λpj

m 

σi (Qj )

− pp



i

 pq Mdα (σ1 χQj , . . . , σm χQj )q vdx

Qj

i=1

≤ [w, ⃗ v]pS(⃗p,q) [w] ⃗ RHp⃗

q

L p (v)

i=1

j



λpj

j

. [w, ⃗ v]pS(⃗p,q) [w] ⃗ RHp⃗

m   p fi  p . L i (w ) i

i=1

Therefore, the inequality (3.14) is proved. • Estimate for Mα . From Lemma 3.3 and Fubini’s theorem, it follows that   q   (k) q 1    d ⃗ Mα (f⃗) q . τ ◦ M ◦ ⃗ τ ( f ) dt 1  −t t α L (v) |Bk | Bk L (v)    1 τ−t ◦ Mdα ◦ ⃗τt (f⃗)q q dt ≤ L (v) |Bk | Bk   d  1 Mα⃗τt (f⃗)q q ≤ dt. L (τt v) |Bk | Bk

(3.15)

M. Cao, Q. Xue / Nonlinear Analysis 130 (2016) 214–228

223

Since (w, ⃗ v) satisfies S(⃗p,q) condition and RHp⃗ condition, we can further verify (⃗τt w, ⃗ τt ) also satisfies S(⃗p,q) condition and RHp⃗ condition independently of t. Therefore, combining (3.13) with (3.15) we deduce that   m  (k) q   1 q/p q Mα (f⃗) q . [⃗τt w, τt fi q p ⃗ τ v] [⃗ τ w] ⃗ dt t S RH L (v) L i (τt wi ) p ⃗ (⃗ p,q) |Bk | Bk i=1 =

q/p [w, ⃗ v]qS(⃗p,q) [w] ⃗ RHp⃗

q/p

= [w, ⃗ v]qS(⃗p,q) [w] ⃗ RHp⃗

1 |Bk |



m   q fi  p dt L i (wi )

Bk i=1

m   q fi  p . L i (wi ) i=1

Finally, letting k tend to infinity, we finish the proof.



4. Proof of Theorem 2.2 In this section, we shall prove Theorem 2.2. Before doing that, we first present a key lemma, which is similar to Lemma 3.2 as follows. Lemma 4.1. Let p1 = p11 + · · · + p1m with 1 < p1 , . . . , pm < ∞. Suppose that wi (i = 1, . . . , m) is nonnegative locally integral function on Rn . Then there exists a constant c > 0 such that for all ∞ f⃗ ∈ Lp1 (w1 ) × · · · × Lpm (wm ), we can find nonnegative scalars {λj }∞ j=1 and dyadic cubes {Qj }j=1 satisfying supp  aj are disjoint and | aj (y, w)| ≤

m 

− p1

σi (Qj )

i

m 

fi (y, w) =



i=1



χ Q  (y, w);

(4.1)

j

i=1

λj  aj (y, w) a.e.;

(4.2)

j

λpj

 p1

≤ c[⃗σ ]Wp⃗∞

m    fi  p ; L i (w )

(4.3)

i

i=1

j r ⃗ Md,γ α (f )(x) ≤



m  1 r  α ′ λrj |Qj | n −m σi (Qj ) pi χQj (x),

j

(4.4)

i=1

for all most everywhere x ∈ (0, ∞)n and all r > 0. k , E kj be as the same in Lemma 3.1. We define scalars {λj } and functions Proof. Let the notations Ωk , Qkj , Ω { akj (y, w)} as follows λkj = 2k+1

m 

1

σi (Qkj ) pi ,

i=1

 akj (y, w) = 2−(k+1) χ E (y, w)

m 

kj

− p1

σi (Qkj )

i

fi (y, w),

i=1

kj = Q  kj \ Ω k+1 is the support of  where E akj and disjoint. It is easy to verify that the inequalities (4.1) and (4.2) hold. In addition, applying the standard argument as before, we will get the inequality (4.4) as well.

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Next we turn our attention to the proof of (4.3). 

λpkj =



=



k,j

2(k+1)p

m  j

k

2(k+1)p

i=1



|Qkj |

j

k

≤

p

σi (Qkj ) pi



2(k+1)p

i=1 m 



|Qkj | p

M (σi χQkj ) pi dx,

Qkj i=1

j

k

m  p  σi (Qkj )  pi

where M is the classical Hardy–Littlewood maximal function. By the Wp⃗∞ condition and the boundedness of A∞ which has been proved in (3.11), we get  p    λkj . [⃗σ ]Wp⃗∞ 2(k+1)p ν⃗σ (Qkj ) ≤ [⃗σ ]Wp⃗∞ 2(k+1)p ν⃗σ (Ωk ) k,j

j

k

= [⃗σ ]Wp⃗∞

∞   ℓ=0

. [⃗σ ]Wp⃗∞

∞ 

k

  2(k+1)p ν⃗σ {2k+ℓ < A∞ (f⃗˜) ≤ 2k+ℓ+1 }

k

2−ℓ

ℓ=0



  2(k+ℓ)p ν⃗σ {2k+ℓ < A∞ (f⃗˜) ≤ 2k+ℓ+1 }

k

 p . [⃗σ ]Wp⃗∞ A∞ (f⃗˜)Lp (ν

⃗ σ)

. [⃗σ ]Wp⃗∞

m   p fi  p .  L i (wi )

(4.5)

i=1

Proof of Theorem 2.2. By the similar argument as in Theorem 2.1, it suffices to estimate m     d,γ  1/p fi  p Mα (f⃗)χ(0,γ)n  q . [w, . ⃗ v] [⃗ σ ] ∞ A(⃗ p,q) W L i (wi ) L (v) p ⃗

i=1

Using the inequalities (4.3), (4.4) and A(⃗p,q) condition, we obtain  d,γ    ⃗p Mα (f⃗)χ(0,γ)n p q = Md,γ  q α (f ) χ(0,γ)n L (v) p L (v)

m    1 p  α ′   λpj |Qj | n −m ≤ σi (Qj ) pi χQj  j

≤



q

L p (v)

i=1

m  1 p  p α ′ σi (Qj ) pi λpj v(Qj ) q |Qj | n −m

j

i=1

  q1   1′ p   m   p p α 1 1 1 1 i 1−p′ = λj |Qj | n + q − p vdx wi i dx |Q | |Q | j j Qj Qj j i=1 ≤ [w, ⃗ v]pA(⃗p,q)



λpj . [w, ⃗ v]pA(⃗p,q) [⃗σ ]Wp⃗∞

j

where in the third step we used p ≤ q and Minkowski’s inequality.

m   p fi  p , L i (wi ) i=1



5. Proof of Theorem 2.5 To prove Theorem 2.5, we need the following lemma. Lemma 5.1. Let p1 = p11 + · · · + p1m with 1 < p1 , . . . , pm < ∞. Then there exists a constant c > 0 such that ∞ for all f⃗ ∈ Lp1 (w1 ) × · · · × Lpm (wm ), we can find nonnegative scalars {λj }∞ j=1 and dyadic cubes {Qj }j=1

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M. Cao, Q. Xue / Nonlinear Analysis 130 (2016) 214–228

satisfying supp  aj are disjoint and 1 −p

| aj (y, w)| ≤ |Qj | m 

fi (y, w) =



i=1



   m  − p1 1 i Q exp log σi dx χ j (y, w); |Qj | Qj i=1

(5.1)

λj  aj (y, w) a.e.;

(5.2)

j

λpj

 p1

≤c

j

m    fi  p ; L i (wi )

(5.3)

i=1

r ⃗ Md,γ α (f )(x)

≤

 j

λrj

 r   m m   − p1 1 1 σi (Qj )  −p i χQj (x), exp log |Qj | σi dx α |Qj | Qj |Qj |1− mn i=1 i=1

(5.4)

for all most everywhere x ∈ (0, ∞)n and all r > 0. k , E kj be as the same in Lemma 3.1. We define scalars {λj } and functions Proof. Let the notations Ωk , Qkj , Ω { akj (y, w)} as follows    m 1  1 1 p λkj = 2k+1 |Qkj | p exp log σi i dx , |Q| Q i=1    m m  − p1 1 1 σi i dx  akj (y, w) = 2−(k+1) χ E (y, w)|Qkj |− p exp log fi (y, w). kj |Q| Q i=1 i=1 Here we only show the inequality (5.3), since the others are trivial. Indeed,    m p  p    1 pi (k+1)p λkj = log σi dx 2 |Qkj | exp |Qkj | Qkj i=1 j k,j k      1 . 2(k+1)p exp log ν⃗σ dx dx |Qkj | Qkj Qkj j k   . 2(k+1)p M0 (ν⃗σ χΩk )dx, k

Rn

  1  log |f (x)|dx is the logarithmic maximal function described in [6, Lemma where M0 (f )(x) = supQ exp |Q| Q 2.1]. Notice that M0 is bounded from Ls (Rn ) (s > 0) into itself. Applying the same technique as in (4.5), we deduce that m  p    p  p fi  p λkj . 2(k+1)p ν⃗σ (Ωk ) ≤ A∞ (f⃗˜)Lp (ν ) . .  L i (w ) i

⃗ σ

k,j

i=1

k

Proof of Theorem 2.5. Proceeding as we did in the proof of Theorem 2.1, we only need to prove m   d,γ    Mα (f⃗)χ(0,γ)n  q . [⃗σ , v]B fi  p . (⃗ p,q) L (v) L i (w ) i

i=1

By the inequalities (5.3), (5.4) and Minkowski’s inequality, we obtain  d,γ    ⃗p Mα (f⃗)χ(0,γ)n p q = Md,γ  q α (f ) χ(0,γ)n L p (v) L (v)      p  m m    − p1 σi (Qj )  1 p −1 i  ≤ λ |Q | exp log σ dx χ α j Qj  q j i  1− |Qj | Qj |Qj | mn L p (v) j i=1 i=1

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p    m m  − p1 1 σi (Qj )  i dx exp log σ α i |Qj | Qj |Qj |1− mn j i=1 i=1 p  1 m   m  p  − p1 α 1 v(Qj ) q  σi (Qj ) 1 + q1 − p i n = λj |Qj | exp log σi dx |Qj | |Qj | |Qj | Qj j i=1 i=1

≤



p

λpj |Qj |−1 v(Qj ) q

≤ [⃗σ , v]pB(⃗p,q)



m   p fi  p , L i (wi )

λpj . [⃗σ , v]pB(⃗p,q)

j

i=1

where in the next to last inequality we have used the B(⃗p,q) condition.



6. Proof of Propositions 2.3–2.4 Before proving Proposition 2.3, we first give the following more generalized reverse H¨older’s inequality, which can be viewed as an extension of the reverse Jensen’s inequality in the theory of weighted inequalities. m Proposition 6.1 ([26]). Let 0 ≤ θi ≤ 1 and 0 ≤ i=1 θi ≤ 1. If wi ∈ A∞ , then for any cube Q, we have  θi   m m m  θi −1   wiθi dx. [wi ]θAi∞ wi dx ≤ |Q| i Q

i=1

Q i=1

i=1

We also need the following connection among different weights. Proposition 6.2 ([6]). For any weight w, define [w]′A∞ = sup Q

1 w(Q)

 M (wχQ )dx. Q

Then [w]′A∞ . [w]A∞ . p (1−p′i )

Proof of Proposition 2.3. In the one-weight case, σi = wi i that

−p′i

= wi

. From Theorem 2.2 [4], it follows

σi ∈ Amp′ ⊂ A∞ . Then by Proposition 6.1, we obtain m   Q

i=1

Therefore, [w] ⃗ RHp⃗ ≤

p/pi    m m p/p p/p σi dx ≤ [σi ]A∞i σi i dx. Q i=1

i=1

−p′i p/pi ]A∞ . i=1 [wi

m

In addition, for any Q ∈ Q, Proposition 6.2 and the inequality (6.1) imply that     −1 m m p/p M (σi χQ )p/pi dx σi i dx Q i=1 m  

≤

.

i=1 m  i=1

Q i=1

1 σi (Q) 2p/p



p/pi  p/pi   −1 m  m p/pi M (σi χQ )dx σi dx σi dx

Q

[σi ]A∞ i . 

i=1

Q

Q i=1

(6.1)

M. Cao, Q. Xue / Nonlinear Analysis 130 (2016) 214–228

227

Proof of Proposition 2.4. Indeed, (i) Let α = 0, p = q, m = 1 and v = w. From Jensen’s inequality, it follows that    q1       m m  − p1 1 α 1 1 1 + q1 − p i n vdx σi dx exp log |Q| σi dx |Q| Q |Q| Q |Q| Q i=1 i=1   p1        1 1 1 1 −p 1−p′ wdx w dx exp log σ dx = |Q| Q |Q| Q |Q| Q 1  1   p−1  p−1 −1   p(p−1)     1 1 1 1 1−p′ = wdx w dx wdx exp log wdx |Q| Q |Q| Q |Q| Q |Q| Q 1

≤ [w]Ap−1 . p (ii) In the general case,  q1   1′  m  p 1 i 1−p′i vdx wi dx |Q| |Q| Q Q i=1   − p1  m m 1/q  i σi (Q) q = σ dx vdx α i 1− mn Q i=1 |Q| Q i=1  1/q  −1 m 1/pi q σi (Q) ≤ Mα (σ1 χQ , . . . , σm χQ ) vdx 1 1 α n+q−p



1 |Q|



Q

i=1

≤ [w, ⃗ v]S(⃗p,q) . That is, [w, ⃗ v]A(⃗p,q) ≤ [w, ⃗ v]S(⃗p,q) . For the remaining inequality, it may be a little hard to prove it directly. However, by means of Theorems 2.1 and 2.5, we deduce that   [w, ⃗ v]S . [⃗σ , v]B . Mα  p . p q (⃗ p,q)

Thus, we have proved Proposition 2.4.

L

1 (w1 )×···×L m (wm )→L

(v)

(⃗ p,q)



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