New York Journal of Mathematics New York J. Math.

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New York Journal of Mathematics

New York J. Math. 24(2018) 980–1003.

Two-weight inequalities for multilinear commutators

Ishwari Kunwar and Yumeng Ou

Abstract. We prove Bloom type two-weight inequalities for commutators of multilinear singular integral operators that include Calder´on- Zygmund operators and their dyadic counterparts. Such estimates are further extended to a general higher order multilinear setting. The proof involves a pointwise sparse domination of multilinear commutators.

Contents

1. Introduction and statement of main results 980 1.1. Definitions of multilinear CZ and dyadic operators 987 2. Sparse domination of multilinear commutators 988 3. Bloom inequalities for multilinear commutators 992 3.1. First order multilinear commutators 992 3.2. Higher order multilinear commutators 995 3.3. Lower bound of multilinear commutators 998 Appendix A. Proof of Theorem 1 and 3 for dyadic operators 1000

Acknowledgments 1001

References 1002

1. Introduction and statement of main results

In this article, we study commutators of certain BMO functions and singular integral operators in the multilinear setting. Our basic object is

[b, T]_β(f1,· · · , fm) :=bT(f1,· · · , fm)−T(f1,· · · , bf_β,· · ·, fm), where T is a m-linear operator acting on functions onRⁿ, m≥1, and b is understood as the pointwise multiplication operator by functionb.

We prove two-weight inequalities, first of their kind in the multilinear setting, for commutators of this type and its full multilinear, higher order extensions. More precisely, our goal is to understand how the operator norm of the commutator acting on weighted L^p spaces is controlled by certain

Received January 9, 2018.

2010Mathematics Subject Classification. Primary: 42B20; Secondary: 42B25.

Key words and phrases. Multilinear Calder´on-Zygmund operators, multilinear dyadic operators, sparse domination, commutators, weighted BMO..

ISSN 1076-9803/2018

980

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BMO norms (determined by the weights) of the symbol function b, and we are particularly interested in the cases when T is a multilinear Calder´on- Zygmund operator or its dyadic counterparts.

The main results of the article are the following. We first obtain a two- weight estimate for the full first order multilinear commutators (Theorem 1), i.e. commutator of the form

(1) [bm,· · · ,[b2,[b1, T]1]2· · ·]m,

where in each component there is a function commuted once. We then develop an abstract scheme (Theorem 4) that bootstraps from there to the higher order case, where multiple different functions are commuted in some components. As the formulation is very technical, we defer the definition of the general higher order multilinear commutators. Note that such two- weight norm inequalities are new even in the first order case for multilinear commutators. Moreover, lower bound estimates for multilinear commutators are also studied (Theorem 7).

Commutator estimates in terms of BMO norms of the symbol functions have attracted a great amount of interest in the past decades, as it gives straightforward characterization of BMO spaces, implies weak factorization of Hardy spaces and Div-Curl estimates, and is closely connected to Han- kel operators in operator theory. We refer the reader to [4, 6, 24] and the references therein for more detailed discussion of these connections in the unweighted linear theory. In the linear setting (i.e. whenT is a linear operator), weighted and two-weight type inequalities have been recently studied in works such as [3, 9, 10, 11]. In the multilinear setting, such estimates have also been considered where the natural class of weights and structure of the commutators are much more complicated. Lerner et al. [17] proved one-weight estimates for certain first order multilinear commutators. In the work of P´erez et al. [25], one-weight estimates for the full first order multilinear commutators (1) are considered too. There is also another independent work of Tang [27] who treated first order multilinear commutators but only for a subclass of multilinear weights: products of classical weights. Our result seems to be the first of its kind that extends such estimates to a setting where challenges from multilinear, iterated higher order, and two-weight are overcome simultaneously.

To begin with, we introduce relevant classes of weights and BMO spaces.

A positive, locally integrable functionwis called a MuckenhouptAp weight if

[w]_A_p := sup

Q Q

w(x)dx

Q

w(x)^1−p⁰dx p−1

<∞, 1< p <∞, where the supreme is taken over all cubesQ⊂Rⁿ. When studying multilinear singular integrals, one usually works with weight vectorsw~ = (w1, . . . , wm) where eachwj is a positive function. Let~p= (p1, . . . , pm) with 1< pj <∞

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ISHWARI KUNWAR AND YUMENG OU

and 1/p= 1/p₁+· · ·+ 1/p_m. w~ is said to be amultilinear A_p_~ weight if [w]~ ^1/p_A

~

p := sup

Q⊂Rⁿ Q

ν_w_~

1/p m

Y

j=1 Q

w^1−p

0 j

j

1/p⁰_j

<∞, where

ν_w_~ :=

m

Y

j=1

w_j^p/p^j.

A particular example of such weights is w~ with w_j ∈ A_p_j, ∀j, which is referred to as aproduct multiple weight. The weight classA_~_p was first introduced in [17] where the authors show that it is the correct class of weights in multilinear Calder´on-Zygmund theory and is tied to the multilinear maximal function. In general, if w~ ∈ A_p_~, wj may not be a locally integrable function for anyj, but instead,

~

w∈A_~_p ⇐⇒

( w^1−p

0 j

j ∈A_mp⁰

j, j= 1, . . . , m, ν_w_~ ∈Amp.

In addition to the classical BMO space, we define the following weighted BMO space associated to weightν normed by

kbk_BMO(ν):= sup

Q⊂Rⁿ

1 ν(Q)

ˆ

Q

|b− hbi_Q|dx, where ν(Q) := ´

Qν(x)dx. Note that hbi_Q in the above still denotes the average value of b with respect to Lebesgue measure. This space was first introduced by Muckenhoupt and Wheeden [23] and Garc´ıa-Cuerva [7] inde- pendently, and is often referred to as the Bloom BMO space due to its role in connection with two-weight estimates for commutators studied by Bloom [2]. There are also dyadic versions of BMO and weighted BMO associated to certain dyadic grid D, which are denoted as BMO_D and BMOD(ν). These norms are defined similarly as above but with supremum taken over only dyadic cubes.

Our first theorem provides a quantitative two-weight upper bound for the first order multilinear commutator (1).

Theorem 1. Let ~p= (p₁, . . . , p_m),1< p_j <∞and1/p= 1/p₁+· · ·+1/p_m. Let ~µ = (µ1, . . . , µm) and ~λ = (λ1, . . . , λm) be vector weights such that µ_j, λ_j ∈ A_p_j, ∀j = 1, . . . , m. Define ν_j = (µ_j/λ_j)^1/p^j, ∀j = 1, . . . , m, and ν_~_λ =Qm

j=1λ^p/p_j ^j. If bj ∈BMO(νj) for j= 1, . . . , m, then there holds k[b_m,· · · ,[b1, T]1· · ·]m(f1,· · ·, fm)k_Lp(ν_~_λ)

.C(~µ, ~λ, ~p)

m

Y

j=1

kb_jk_BMO(ν

j) m

Y

j=1

kf_jk_Lpj(µj),

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whereT is am-linear Calder´on-Zygmund operator, Haar multiplier or paraproduct (with respect to any dyadic grid), and

C(~µ, ~λ, ~p) :=





m

Y

j=1

[µ_j]

max(1, ¹

pj−1) A_pj [λ_j]

1

pj max(pj,p⁰₁,...,p⁰_m) A_pj



.

We defer the definitions of m-linear Calder´on-Zygmund operators, Haar multipliers and paraproducts to Subsection 1.1.

Two-weight inequalities of this type have been extensively studied in the linear setting such as for Hilbert transform [2, 9], general CZ operators [10, 18] and multi-parameter CZ operators [11]. It is well known that weighted estimates in the multilinear setting has some intrinsic difference compared with the linear case, as there are usually quasi-Banach spaces involved and multilinear weights need not have each component being a classical linear Ap weight. Our result seems to be the first in the literature to extend it to the multilinear setting and it is very interesting to know whether it is possible to weaken the assumptions on the weights in Theorem 1. In general, two weight estimates are known to be quite challenging, especially in the multilinear setting. Our result further magnifies the fact that commutators usually have better properties than the operator that is being commuted.

Theorem 1 follows from a domination of the commutator by certain sparse operators (Proposition 9 and 15) which satisfy the desired two-weight estimate above, and the sparse domination can be obtained via a stopping time argument relying on the weak type endpoint estimate of certain maximal truncated operators. In the case of the multilinear CZ operators, such maximal truncated operator was first treated by Grafakos and Torres [8], while for the dyadic operators this seems to be new. A similar technique has been used in the linear setting [18] to obtain an analog of the two-weight estimate for the commutator. Sparse domination of the commutator is expected to be of independent interest, as it provides a fine quantification of its boundedness which should imply not only the weighted estimates that we explore in this article, but also weak type endpoint bounds, which we plan to study in a forthcoming article.

It is natural to consider higher order generalization of Theorem 1. We start with introducing the general definition of a higher order multilinear commutator. Given~k= (k₁, . . . , k_m). ∀1≤j ≤m, assume k_j ≥0, and let b~j = (b¹_j, . . . , b^k_j^j) where each bⁱ_j is a function on Rⁿ. A general multilinear higher order commutator can be defined as

(2) C^~^k,m

{b~j}(T) :=

h

b^k_m^m,· · · , h

b¹_m,· · ·, h

b^k₁¹,· · · , b¹₁, T

1· · ·i

m

· · ·i

m

. Note that C^~^k,m

{b~j}(T) is also a m-linear operator, and it is invariant under permutations of n

bⁱ_jo

i=1,...,kj

,∀1 ≤j ≤m. In the definition above,k_j = 0

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for somej∈ {1, . . . , m}simply means that there is no commutator structure in thej-th component. It is easy to see that [b, T]β defined at the beginning and the full first order commutator (1) are both special cases of C^~^k,m

{b~j}(T).

Remark 2. In fact, it is also natural to consider the analogue of Theorem 1 where in some components there is no commutator structure at all, i.e.

first order commutator C^~^k,m

{b~j}(T) with 0 ≤ k_j ≤ 1, ∀j = 1, . . . , m. Indeed, via almost the same method, we have the following slight generalization of Theorem 1.

Theorem 3. Let ~p= (p1, . . . , pm),1< pj <∞and1/p= 1/p1+· · ·+1/pm. Let I ={i₁, . . . , i_`} ⊂ {1, . . . , m} and b_i_s ∈BMO(ν_i_s), s= 1, . . . , `. For any vector weights ~µ = (µ1, . . . , µm) and ~λ = (λ1, . . . , λm) such that µis, λis ∈ A_p_is, ∀s= 1, . . . , `, while µ_j =λ_j, ∀j /∈I. Denote~q= (p_j)_{j /}_∈I, w~ = (µ_j)_{j /}_∈I and assume that the(m−`)-linear weightw~ ∈A_~_q. Then, let~ν = (νi1, . . . , νi`) satisfy ν_i_s = (µ_i_s/λ_i_s)^1/p^is, there holds

k[b_i₁,· · · ,[bi`, T]i`· · ·]i1(f1,· · · , fm)k_Lp(ν~λ)

.C(~µ, ~λ, ~p)

`

Y

s=1

kb_sk_BMO(ν

is) m

Y

j=1

kf_jk_Lpj(µj),

whereT is am-linear Calder´on-Zygmund operator, Haar multiplier or paraproduct (with respect to any dyadic grid), and

C(~µ, ~λ, ~p) :=

`

Y

s=1

[µis]

max(1, ¹

pis−1) A_pis [λis]

1

pismax(p_is,q,p⁰₁,...,p⁰_m) A_pis

! [w]~

1

qmax(q,p⁰₁,...,p⁰_m)

A_~_q ,

1/q :=X

j /∈I

1/q_j.

The statement of the theorem above seems quite complicated, while we introduced the extra notation for the subsetI ⊂ {1, . . . , m}simply to include the case of deficient commutators, i.e. when T is not commutated with any symbol in some components (for example [b1, T]1). Note that in the fully degenerate case I = ∅, i.e. when there is no commutator structure at all, the theorem degenerates to theA2 theorem for multilinear operators:

kT(f1, . . . , fm)k_Lp(ν_w_~) .[w]~

1

pmax(^p,p⁰1,...,p⁰_m)

A_~_p

m

Y

j=1

kf_jk_Lpj(wj), ∀w~ ∈A_~_p. which for T being a multilinear CZ operator was first proved by Li, Moen and Sun [21].

Next, we extend the two-weight estimate to higher order commutators through an abstract two-weight bootstrapping technique, which applies to arbitrary multilinear operators, not necessarily of Calder´on-Zygmund type.

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Theorem 4. Let ~ν = (ν₁, . . . , ν_m) be a fixed multiple weight on Rⁿ, ~p = (p1, . . . , pm), 1/p= 1/p1+· · ·+ 1/pm, 1< pj <∞, 1≤p <∞ and Te be a m-linear operator satisfying

kTek_L^p1(µ1)×···×L^pm(µm)→L^p(ν~λ) ≤C_n,m,~_p,

Te

[~µ]_A_p_~,[~λ]_A_~_p ,

where C_n,m,~_p,_T_e(·,·) is an increasing function of both components, with ~µ ∈ A_~_p, ~λ ∈ A_p_~ and µj/λj = ν_j^p^j, ∀j. For ~k = (k1, . . . , km) with kj ≥ 0,

∀1≤j≤m, let bⁱ_j ∈BMO(Rⁿ), ∀1≤i≤k_j, then there holds

C^~^k,m

{~b} (Te)

L^p¹(µ1)×···×L^pm(µm)→L^p(ν_~_λ)

≤C_n,m,~_p,~_k,

Te

[~µ]_A_~_p,[~λ]_A_~_pY^m

j=1 kj

Y

i=1

kbⁱ_jk_BMO(_Rn),

where if k_j = 0 for somej, then the corresponding BMO(Rⁿ) norms do not appear on the right hand side.

Combined with Theorem 3, this immediately implies the following two- weight upper bound estimate for higher order commutators.

Corollary 5. Let ~p = (p₁, . . . , p_m), 1 < p_j < ∞, 1 ≤ p < ∞ and 1/p = 1/p₁+· · ·+ 1/p_m. Let I ={i₁, . . . , i_`} ⊂ {1, . . . , m}, b¹_i_s ∈BMO(ν_i_s), and bⁱ_i_s ∈ BMO(Rⁿ), ∀2 ≤ i ≤ kis, s = 1, . . . , `. For any vector weights ~µ = (µ₁, . . . , µ_m) and ~λ= (λ₁, . . . , λ_m) such that µ_i_s, λ_i_s ∈A_p_is, ∀s = 1, . . . , `, while µj = λj, ∀j /∈ I. Denote ~q = (pj)_{j /}_∈I, w~ = (µj)_{j /}_∈I and assume that the (m−`)-linear weight w~ ∈ A_~_q. Then, let ~ν = (ν_i₁, . . . , ν_i_`) satisfy νis = (µis/λis)^1/p^is, and~k = (k1, . . . , km) with kis ≥1 for s= 1, . . . , ` and k_j = 0 for j /∈I. Then there holds

C^~^k,m

{~b} (T)

L^p¹(µ1)×···×L^pm(µm)→L^p(ν~λ)

≤C_n,m,~_p,~_k,T

[~µ]_A_~_p,[~λ]_A_p_~ Y^`

s=1

kb¹_i

sk_BMO(ν

is)

!



`

Y

s=1 k_is

Y

i=2

kbⁱ_i

sk_BMO(Rn)



, forT being am-linear Calder´on-Zygmund operator, Haar multiplier or paraproduct, where if kis = 1 for some s, then the corresponding BMO(Rⁿ) norms do not appear on the right hand side.

Remark 6. Note that the choice b¹_i

s above doesn’t play any particular role as the commutator is invariant under permutations of

n bⁱ_j

o

for each fixed j. Moreover, even though we don’t explicitly assume ~µ, ~λ∈A_~_p in the statement of the corollary, it follows immediately from H¨older’s inequality and the assumptions µis, λis ∈ Ap_is, ∀s = 1, . . . , `, w~ ∈ A_~_q that the vector weights ~µ, ~λ are indeed A_~_p.

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Theorem 4 is proven using a method involving the Cauchy integral formula, which goes back to the work of Coifman, Rochberg and Weiss [4]. In the linear, one-weight case when all functionsb_j are the same, this result was proved by Chung, Pereyra and Pérez in [3]. In the two-weight case, the only previously known result along this line of research is by Hytönen [13], where he applies the Cauchy integral method to obtain a linear version of Theorem 4 when bj =b∈BMO∩BMO(ν),∀j. Hytönen’s result was first proved by Holmes and Wick [12], where a different method involving decomposition into dyadic shift operators is applied. In the multilinear case, much less is known. The Cauchy integral formula (in a simple form) was first used by Pérez and Torres [26] to study certain first order multilinear commutators, and it was further extended to a very general setting by Bényi et al. [1]

where different types of multilinear operators (not necessarily CZ), weights and BMO spaces are considered. Our result continues the line of research and seems to be the first attempt to extend the Cauchy integral method to the multilinear two-weight setting.

Furthermore, we obtain a lower bound estimate for the commutator, which provides a characterization of BMO via multilinear commutators.

Theorem 7. The following statements are equivalent.

(1) b∈BMO(Rⁿ).

(2) For all1/p= 1/p₁+· · ·+1/p_mwith1< p_j <∞,β ∈ {1, . . . , m}, and weight w~ = (w1, . . . , wm) with wβ ∈ Apβ and (m−1)-linear weight (w1, . . . ,wcβ, . . . , wm) ∈ A_~_q, ~q = (p1, . . . ,cpβ, . . . , pm), the following map is bounded:

[b, T]β :L^p¹(w1)× · · · ×L^p^m(wm)→L^p(ν_w_~),

where T is any m-linear Calder´on-Zygmund operator, Haar multiplier or paraproduct.

(3) For all dyadic grid D, there exist some choices of p,~ β as above, some w~ with w_j ∈A_p_j, ∀j, and some α, so that the following map~ is bounded:

[b, P_~^α^~]_β :L^p¹(w₁)× · · · ×L^p^m(w_m)→L^p(ν_w_~),

where P_~^~^α is any Haar multiplier defined in (3) with respect to D satisfying that {|_I|}_I are bounded from below uniformly and αj 6= 1 for somej ∈ {1, . . . ,β, . . . , m}.b

This result shows that multilinear dyadic commutator is a representative class for commutators of other multilinear operators: given a function b, if for each dyadic grid the commutator [b, P_~^~^α]β is bounded on weighted L^~^p for some choices of Haar multiplier P_~^α^~, parameter β, and multilinear weight, then commutator [b, T]_β⁰ is bounded on any weighted L^~^q for any choices of continuous or dyadic multilinear CZ operator T, parameter β⁰, and multilinear weight.

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In the linear setting, dyadic operators have been extensively studied as model cases and tools to handle various problems in the continuous setting, which is why we expect our results above concerning dyadic operators (Haar multipliers and paraproducts) to have some further applications in the advancement of the multilinear theory. For example, in a very recent work of the second author with Li, Martikainen and Vuorinen [20], a bilinear representation theorem is proved which enables one to represent bilinear CZ operators as averages of bilinear dyadic shifts (higher complexity version of Haar multipliers) and paraproducts. It would be interesting to know whether similar versions of Theorem 1, 3 and 7 hold for dyadic shifts as well. It is also worth noticing that multilinear Haar multipliers and paraproducts arise naturally in the decomposition of m-fold pointwise product of functionsf₁, . . . , f_m, which we refer to [15] for more details.

Remark 8. In Theorem 1,3,4,7above, if one is only interested in studying the dyadic operators with respect to dyadic gridD, one can replace the BMO spaces that appear by their dyadic versions and the same results remain valid.

The article is organized as follows. In Section 2, we prove a sparse bound for first order multilinear CZ commutators, which, combined with Subsec- tion 3.1, will complete the proof of Theorem 1 and 3 for CZ operators. In the rest of Section 3, Theorem 4 and 7 will be demonstrated. We then discuss the case with dyadic operators (Haar multipliers and paraproducts) of Theorem 1 and 3 in Appendix A, as it proceeds almost parallel to the continuous one.

Before finishing the Introduction, we state the definitions of the multilinear operators that appear in the main theorems.

1.1. Definitions of multilinear CZ and dyadic operators. LetT be a m-linear operator mapping (S(Rⁿ))^m intoS⁰(Rⁿ). T is called am-linear Calder´on-Zygmund operator if for appropriate functions {f₁, . . . , f_m},

T(f₁, . . . , f_m)(x) = ˆ

(Rⁿ)^m

K(x, y₁, . . . , y_m)f₁(y₁)· · ·f_m(y_m)dy₁· · ·dy_m, whenever x /∈ ∩^m_j=1suppf_j, where the kernel K is assumed to be locally integrable away from the diagonalx=y1 =· · ·=ym in (Rⁿ)^m+1, satisfying for someC, >0 the size estimate

|K(y₀, y1, . . . , ym)| ≤ C (Pm

k,`=0|y_k−y_`|)^nm

for all (y0, y1, . . . , ym)∈(Rⁿ)^m+1 away from the diagonal, and the smooth- ness estimate

|K(y₀, . . . , yj, . . . , ym)−K(y0, . . . , y⁰_j, . . . , ym)| ≤ C|y_j−y_j⁰| (Pm

k,`=0|y_k−y_`|)^nm+

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whenever 0≤j≤mand|y_j−y⁰_j| ≤ ¹₂max0≤k≤m|y_j−y_k|. We also make the a priori assumption thatT mapsL^q¹× · · · ×L^q^m boundedly intoL^qfor some 1< q_j <∞satisfyingPm

j=11/q_j = 1/q. We refer the reader to [8, 16, 5, 19]

and the references therein for more details and properties of multilinear CZ operators. Note that our results remain valid for more general multilinear CZ operators with weaker kernel assumptions, for example kernels satisfying Dini type estimates orm-linearL^r-H¨ormander condition as studied in [19], as what matters here (more precisely, in the proof of Proposition 9 of Section 2) is the boundedness of the corresponding maximal truncated operators.

Next, we define the multilinear dyadic operators that appear in Theorem 1, 7. Let D be a fixed dyadic grid on Rⁿ. For all I ∈ D, {h^α_I^j} are the cancellative Haar functions for{0,1}ⁿ 3α_j 6=~1, andh¹_I :=h^~¹_I :=|I|^−1/2χ_I is the non-cancellative Haar function. First, define the multilinear Haar multiplier associated to the bounded sequence~={_I}_I∈D as

(3) P_~^α^~(f) :=~ X

I∈D

Ihf₁, h^α_I¹i · · · hf_m, h^α_I^mih^α_I^m+1|I|^−(m−1)/2,

where at least two of all {α₁, . . . , αm+1} are not equal to~1, i.e. there are at least two out of the m+ 1 Haar functions that are cancellative. Then, forg∈BMOD, define the multilinear dyadic paraproduct associated to the bounded sequence~={_I}_I∈D as

(4) π^α_g,~^~(f~) :=X

I∈D

_Ihg, h^α_I¹i





m

Y

j=1

hf_j, h^α_I^j+1i



h^α_I^m+2|I|^−m/2,

where α₁ 6= ~1 and at least one of the superscripts {α₂, . . . , α_m+2} is not equal to~1.

2. Sparse domination of multilinear commutators

In this section, we present the first step (Proposition 9 below) of the proof of Theorem 1 and 3 for multilinear Calder´on-Zygmund operators, which reduces it to the estimates of certain sparse operators. A similar version of this reduction for the dyadic operators is given in Appendix A (Proposition 15). A collection S of cubes in Rⁿ is calledη-sparse if there exists E_I ⊂I for all I ∈ S such that|E_I|> η|I|and {E_I}I∈S pairwise disjoint.

Proposition 9. Let T be a m-linear Calder´on-Zygmund operator and I = {i₁, . . . , i_`} ⊂ {1, . . . , m}. Given locally integrable functions~b= (b_i₁, . . . , b_i_`) onRⁿ, there exists a constantC=C(n, T)so that for any bounded functions f~ = (f1, . . . , fm) with compact support, there exists 3ⁿ sparse collections S_j =S_j(T, ~f ,~b) of dyadic cubes,j = 1, . . . ,3ⁿ such that

(5)

[bi1,· · ·,[bi_`, T]i_`· · ·]i1(f~) ≤C





3ⁿ

X

j=1

X

~ γ∈{1,2}^`

A^~^γ

S_j,~b(f)~



, a.e.

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where

A^~^γ

Sj,~b(f~) := X

Q∈Sj

`

Y

s=1

Γ(bis, fis, Q, γis)

!

 Y

j /∈I

h|f_j|i_Q



χQ,

Γ(b, f, Q, γ) :=

(|b− hbi_Q| h|f|i_Q ifγ = 1, h|(b− hbi_Q)f|i_Q ifγ = 2.

Proof. For the sake of brevity, we prove the proposition only in the bilinear setting and assume that the commutator is full (I = {1, . . . , m}). It will be easy to see that the argument extends to the multilinear setting in the obvious way, and the deficient commutator case is even easier to treat. Let m= 2, then (5) reduces to the domination

|[b₂,[b₁, T]₁]₂(f₁, f₂)| ≤C

3ⁿ

X

j=1

A^(1,1)

S_j,~b(f~) +A^(1,2)

S_j,~b(f~) +A^(2,1)

S_j,~b(f~) +A^(2,2)

S_j,~b(f)~

almost everywhere for some choices of sparse collectionsS_j. We claim that it suffices to show for any fixed cube Q₀ ⊂Rⁿ that there exists a ¹₂-sparse collection S ⊂ D(Q₀) such that for a.e. x∈Q0,

|[b₂,[b₁, T]₁]₂(f₁χ_3Q₀, f₂χ_3Q₀)|

≤CX

Q∈S



 X

~γ∈{1,2}²

Γ(be 1, f1, Q, γ1)eΓ(b2, f2, Q, γ2)



χQ, (6)

where

eΓ(b_i, f_i, Q, γ_i) :=

(

bi− hb_ii_R_Q

h|f_i|i_3Q ifγi= 1, (b_i− hb_ii_R_Q)f_i

3Q ifγ_i= 2,

and R_Q is a cube from one of the fixed 3ⁿ dyadic grids such that 3Q⊂R_Q and |R_Q| ≤ 9ⁿ|Q|. The reduction to estimate (6) is fairly standard, which we omit and refer to [18] for instance for a justification.

Estimate (6) will follow from iterating the following claim: there exists a disjoint collection of cubes P_j ∈ D(Q₀) such that P

j|P_j|< ¹₂|Q₀| and for a.e. x∈Q₀ there holds

|[b₂,[b1, T]1]2(f1χ3Q0, f2χ3Q0)(x)|

≤C



 X

~γ∈{1,2}²

Γ(be ₁, f₁, Q₀, γ₁)eΓ(b₂, f₂, Q₀, γ₂)





+X

j

[b₂,[b₁, T]₁]₂(f₁χ_3P_j, f₂χ_3P_j)(x)

χ_P_j(x).

(7)

(11)

By the disjointness of {P_j} (which will be constructed later), (7) can be deduced from the tail estimate

|[b₂,[b₁, T]₁]₂(f₁χ_3Q₀, f₂χ_3Q₀)|χ_Q₀_\^S

jPj

+X

j

[b2,[b1, T]1]2(f1χ3Q0, f2χ3Q0)−[b2,[b1, T]1]2(f1χ3Pj, f2χ3Pj) χPj

≤C



 X

~γ∈{1,2}²

Γ(be ₁, f₁, Q₀, γ₁)eΓ(b₂, f₂, Q₀, γ₂)



. (8)

To see (8), we consider the following multilinear maximal truncated operator

M_{T ,Q}₀(f1, f2)(x) := sup

Q:x∈Q⊂Q0

esssup

ξ∈Q

|T(f₁χ_3Q₀, f₂χ_3Q₀)(ξ)−T(f₁χ_3Q, f₂χ_3Q)(ξ)|. It is proven in [19] that M_{T ,Q}₀ :L¹×L¹ →L^1/2,∞ and for a.e. x∈Q₀, (9)

|T(f₁χ_3Q₀, f₂χ_3Q₀)(x)| ≤C_nkTk_L1×L¹→L^1/2,∞|f₁(x)f₂(x)|+M_{T ,Q}₀(f₁, f₂)(x).

Using the fact that [b_i, T]_i = [b_i−c, T]_ifor any constantc, one can unravel the commutator to bound the left hand side of (8) by

A₁+A₂+B₁+B₂+C₁+C₂+D₁+D₂ where

A₁:=

b₁− hb₁i_R_Q

0

b₂− hb₂i_R_Q

0

|T(f₁χ_3Q₀, f₂χ_3Q₀)|χ_Q₀_\^S

jPj, A₂ :=

b₁− hb₁i_R_Q

0

b₂− hb₂i_R_Q

0

· X

j

T(f1χ3Q0, f2χ3Q0)−T f1χ3Pj, f2χ3Pj

χPj, B1 :=

b2− hb₂i_R_Q

0

T

(b1− hb₁i_R_Q

0)f1χ3Q0, f2χ3Q0

χ_Q₀\S

jPj, B2 :=

b2− hb₂i_R_Q

0

X

j

T

(b1− hb₁i_R_Q

0)f1χ3Q0, f2χ3Q0

−T

(b₁− hb₁i_R_Q

0)f₁χ_3P_j, f₂χ_3P_j χ_P_j, C₁:=

b₁− hb₁i_R_Q

0

T

f₁χ_3Q₀,(b₂− hb₂i_R_Q

0)f₂χ_3Q₀ χ_Q₀_\^S

jPj, C2:=

b1− hb₁i_R_Q

0

X

j

T

f1χ3Q0,(b2− hb₂i_R_Q

0)f2χ3Q0

−T

f₁χ_3P_j,(b₂− hb₂i_R_Q

0)f₂χ_3P_j χ_P_j, D₁ :=

T

(b₁− hb₁i_R_Q

0)f₁χ_3Q₀,(b₂− hb₂i_R_Q

0)f₂χ_3Q₀ χ_Q₀_\^S

jPj,

(12)

D₂ :=X

j

T

(b₁− hb₁i_R_Q

0)f₁χ_3Q₀,(b₂− hb₂i_R_Q

0)f₂χ_3Q₀

−T

(b1− hb₁i_R_Q

0)f1χ_3P_j,(b2− hb₂i_R_Q

0)f2χ_3P_j χ_P_j. We now define the exceptional setE =S4

j=1Ej where

E₁ :={x∈Q₀ : max (|f₁f₂|(x),M_{T ,Q}₀(f₁, f₂)(x))> Ch|f₁|i_3Q₀h|f₂|i_3Q₀}, E₂ :=n

x∈Q₀: max

b₁− hb₁i_R_Q

0

f₁f₂

(x), M_{T ,Q}₀

b₁− hb₁i_R_Q

0

f₁, f₂ (x)

> C D

b1− hb₁i_R_Q

0

f1

E

3Q0

h|f₂|i_3Q₀o , E₃ :=n

x∈Q₀: max f₁

b₂− hb₂i_R_Q

0

f₂

(x), M_{T ,Q}₀

f1, b2− hb₂i_R_Q

0

f2

(x)

> Ch|f₁|i_3Q₀D

b2− hb₂i_R_Q

0

f2

E

3Q0

o , E₄ :=n

x∈Q₀: max

b₁− hb₁i_R_Q

0 b₂− hb₂i_R_Q

0

f₁f₂

(x), M_T,Q₀

b1− hb₁i_R_Q

0

f1, b2− hb₂i_R_Q

0

f2

(x)

> C D

b1− hb₁i_R_Q

0

f1

E

3Q0

D

b2− hb₂i_R_Q

0

f2

E

3Q0

o . For C chosen sufficiently large, there holds |E| ≤ ¹

2ⁿ⁺²|Q₀|. The collection{P_j} are constructed by the stopping cubes obtained via the Calder´on- Zygmund decomposition of function χE at level λ = ₂n+1¹ . In particular, there holds for eachj that ₂n+1¹ |P_j| ≤ |P_j∩E| ≤ ¹₂|P_j|and |E\S

jP_j|= 0.

Hence,P

j|P_j|< ¹₂|Q₀|and P_j∩E^c6=∅.

It’s thus left to show that the terms A₁, . . . , D₂ are bounded by the right hand side of (8). TakingA1, A2 as examples, ifx∈Q0\S

jPj, thenx /∈E1, hence

A₁(x)≤CeΓ(b₁, f₁, Q₀,1)eΓ(b₂, f₂, Q₀,1) implied by property (9). If x∈Pj, by definition ofM_{T ,Q}₀,

A2(x)≤

b1− hb₁i_R_Q

0

b2− hb₂i_R_Q

0

· esssup

ξ∈P_j

T(f1χ3Q0, f2χ3Q0)−T f1χ3Pj, f2χ3Pj

(ξ)

≤CeΓ(b₁, f₁, Q₀,1)eΓ(b₂, f₂, Q₀,1).

The rest of the terms can be estimated similarly, thus the proof is complete.

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3. Bloom inequalities for multilinear commutators

3.1. First order multilinear commutators. According to Proposition 9 and 15, Theorem 1 and 3 would follow from the two-weight inequality below for the multilinear sparse operator adapted to the symbol functions~b of the commutator.

Lemma 10. Let ~p = (p₁, . . . , p_m) with 1 < p₁, . . . , p_m < ∞ and 1/p = Pm

j=11/pj. For any I ={i₁, . . . , i`} ⊂ {1, . . . , m}, let~b= (bi1, . . . , bi`) with bis ∈ BMO(νis), s = 1, . . . , `. Given vector weights ~µ = (µ1, . . . , µm) and

~λ = (λ₁, . . . , λ_m) such that µ_i_s, λ_i_s ∈ A_p_j, ∀s = 1, . . . , ` while µ_j = λ_j,

∀j /∈I. Denote~q = (p_j)_{j /}_∈I, w~ = (µ_j)_{j /}_∈I and assume that the(m−`)-linear weight w~ ∈A_~_q. Let ~ν = (ν_i₁, . . . , ν_i_`) satisfy ν_i_s = (µ_i_s/λ_i_s)^1/p^is. Then for any sparse collection S, and~γ ∈ {1,2}^`, there holds

kA^~^γ

S,~b(f~)k_Lp(ν_~_λ).C(~µ, ~λ, ~p)

`

Y

s=1

kb_i_sk_BMO(ν_is₎

m

Y

j=1

kf_jk_Lpj(µj), where

C(~µ, ~λ, ~p) =

`

Y

s=1

[µis]

max(1, ¹

pis−1) A_pis [λis]

1

pismax(p_is,q,p⁰₁,...,p⁰_m) A_pis

! [w]~

1

qmax(q,p⁰₁,...,p⁰_m)

A~q ,

1/q :=X

j /∈I

1/q_j.

Proof. Let 0≤`1≤`2 ≤m and consider the sparse operator A(f~) =X

Q∈S

`1

Y

s=1

|b_s− hb_si_Q|h|f_s|i_Q

!



`2

Y

s=`1+1

h|(b_s− hb_si_Q)fs|i_Q



·





m

Y

s=`2+1

h|f_s|i_Q



χ_Q,

where we have omitted the dependence onSand~bfor the sake of brevity. By symmetry, it suffices to prove the lemma for A (with`=`₂) and we assume 0< `₁< `₂< mas this is the most difficult case.

According to Lemma 5.1 in [18], there exists a sparse collection Se of dyadic cubes such thatS ⊂S,e and for a.e. x∈Q∈ S,

|b(x)− hbi_Q| ≤C1

X

J∈S:eJ⊆Q

h|b− hbi_J|i_JχJ.

Applying this result iteratively for (`₂ −`₁) times, one can find a sparse collection (still denoted as S) such thate S ⊂Seand for a.e. x∈Q∈ S,

|b_s(x)− hb_si_Q| ≤C1

X

J∈S:eJ⊆Q

h|b_s− hb_si_J|i_JχJ, ∀`₁+ 1≤s≤`2.

(14)

Therefore, for `₁+ 1≤s≤`₂, h|(b_s− hb_si_Q)fs|i_Q . 1

|Q|

ˆ

Q

X

J∈S,J⊆Qe

h|b_s− hb_si_J|i_J|f_s|χ_J

≤ 1

|Q|kb_sk_BMO(ν

s)

X

J∈S,Je ⊆Q

h|f_s|i_Jν_s(J)

= kb_sk_BMO(ν

s)

D A¹

Se(fs)νs

E

Q, whereA¹

Se(f) :=P

Q∈Seh|f|i_Qχ_Q is the classical linear sparse operator. This implies that

A(f~).X

Q∈S

`1

Y

s=1

|b_s− hb_si_Q|h|f_s|i_Q

!



`2

Y

s=`1+1

kb_sk_BMO(ν_s₎D A¹

Se(f_s)ν_sE

Q



·





m

Y

s=`2+1

h|f_s|i_Q



χQ. Next, for any 1≤s≤`1 and Q∈ S, the trivial estimate

|b_s− hb_si_Q|h|f_s|i_Qχ_Q ≤X

J∈S

|b_s− hb_si_J|h|f_s|i_Jχ_J implies thatA(f~) admits the further domination

A(f~).

`1

Y

s=1

X

J∈S

|b_s− hb_si_J|h|f_s|i_JχJ

!!

·

X

Q∈S





`2

Y

s=`1+1

kb_sk_BMO(ν_s₎D A¹

Se(fs)νs

E

Q









m

Y

s=`2+1

h|f_s|i_Q



χQ

=:

`1

Y

s=1

T_b_s(f_s)

!

A^m−`_S ¹ A¹

Se(f_`₁₊₁)ν_`₁₊₁, . . . ,A¹

Se(f_`₂)ν_`₂, f_`₂₊₁, . . . , f_m

·





`2

Y

s=`1+1

kb_sk_BMO(ν_s₎





=:

`1

Y

s=1

T_b_s(f_s)

!



`2

Y

s=`1+1

kb_sk_BMO(ν_s₎



B,

whereA^m−`_S ¹ denotes the classical (m−`₁)-linear sparse operator.

Fix 1 < p₁, . . . , p_m < ∞, 1/p = 1/p₁ +· · ·+ 1/p_m. Let weights ~λ = (λ1, . . . , λ`2, µ`2+1, . . . , µm), ~µ = (µ1, . . . , µm) be such that λs, µs ∈ Aps, s = 1, . . . , `2 and w~ = (µ_`₂₊₁, . . . , µm) ∈ A_~_q is a (m −`2)-linear weight