Ann Univ Ferrara DOI 10.1007/s11565-016-0260-0

Multilinear dyadic operators and their commutators Ishwari Kunwar1

Received: 10 August 2016 / Accepted: 19 September 2016 © Università degli Studi di Ferrara 2016

Abstract We obtain a generalized paraproduct decomposition of the pointwise product of two or more functions that naturally gives rise to multilinear dyadic paraproducts and Haar multipliers. We then study the boundedness properties of these multilinear operators and their commutators with dyadic BMO functions. We also characterize the dyadic BMO functions via the boundedness of (a) certain paraproducts, and (b) the commutators of multilinear Haar multipliers and paraproduct operators. Keywords Multilinear paraproducts · Multilinear Haar multipliers · Dyadic BMO functions · Commutators Mathematics Subject Classification 42A45 · 42B20 · 42B25

Contents 1 Introduction and statement of main results . . . . . . . . 2 Notation and preliminaries . . . . . . . . . . . . . . . . . 3 Multilinear dyadic paraproducts . . . . . . . . . . . . . . 4 Multilinear Haar multipliers and multilinear commutators References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Ishwari Kunwar [email protected] School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, USA

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1 Introduction and statement of main results Dyadic operators have attracted a lot of attention in the recent years. The proof of so-called A2 theorem (see [7]) consisted in representing a general Calder´on-Zygmund operator as an average of dyadic shifts, and then verifying some testing conditions for those simpler dyadic operators. It seems reasonable to believe that, taking a similar approach, general multilinear Calder´on-Zygmund operators can be studied by studying multilinear dyadic operators. Regardless of this possibility, multilinear dyadic operators in their own right are an important class of objects in Harmonic Analysis. Statements regarding those operators can be translated into the non-dyadic world, and are sometimes simpler to prove. In this paper we introduce multilinear analogues of dyadic operators such as paraproducts and Haar multipliers, and study their boundedness properties. Corresponding theory of linear dyadic operators, which we will be using very often, can be found in [11]. In [1], the authors have studied boundedness properties of bilinear paraproducts defined in terms of so-called “smooth molecules”. The paraproduct operators we study are general multilinear operators defined in terms of indicators and Haar functions of dyadic intervals. In [3] Coifman, Rochberg and Weiss proved that the commutator of a B M O function with a singular integral operator is bounded in L p , 1 < p < ∞. The necessity of B M O condition for the boundedness of the commutator was also established for certain singular integral operators, such as the Hilbert transform. S. Janson [8] later studied its analogue for linear martingale transforms. In this paper we study commutators of multilinear dyadic operators, and characterize dyadic B M O functions via boundedness of these commutators. For the corresponding theory for general multilinear Calder´on-Zygmund operators we refer to [5] and [10]. We organize the paper as follows: In Sect. 2, we present an overview of some of the main tools we will be using in this paper. These include: the Haar system, linear Haar multipliers, dyadic maximal/square functions, linear/bilinear paraproduct operators and the space of dyadic B M O functions. For more details we refer to [11]. In Sect. 3, we obtain a decomposition of the pointwise product of m functions, m ≥ 2, which generalizes the paraproduct decomposition of two functions. This decomposition naturally gives rise to multilinear paraproducts. We investigate the boundedness properties of these multilinear paraproduct operators. We also define multilinear anologue of the linear paraproduct operator πb , and characterize dyadic B M O functions via the boundedness of certain multilinear paraproduct operators. In Sect. 4, we introduce multilinear Haar multipliers in a way consistent with the definition of linear Haar multipliers and multilinear paraproducts, and then investigate their boundedness properties. We also study boundedness properties of their commutators with dyadic B M O functions, and provide a characterization of dyadic B M O functions via the boundedness of those multilinear commutators. In particular, we show that the commutators of the multilinear paraproducts with a function b are bounded if and only if b is a dyadic B M O function. Our main results involve the following operators:

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• P α ( f1 , f2 , . . . , fm ) =



⎛ ⎝

I ∈D

• πbα ( f 1 , f 2 , . . . , f m ) =



σ (α)

f j (I, α j )⎠ h I

j=1

⎛

b, h I  ⎝

I ∈D

• Tα ( f 1 , f 2 , . . . , f m ) :=

⎞

m 



⎛ I ⎝

I ∈D

m  j=1

m 

, α ∈ {0, 1}m \{(1, 1, . . . , 1)}.

⎞ (α) f j (I, α j )⎠ h 1+σ , α ∈ {0, 1}m . I

⎞ σ (α)

f j (I, α j )⎠ h I

, α∈{0, 1}m \{(1, 1, . . . , 1)},

j=1

 = { I } I ∈D bounded. • [b, Tα ]i ( f 1 , f 2 , . . . , f m )(x) := b(x)Tα ( f 1 , f 2 , . . . , f m )(x) − Tα ( f 1 , . . . , b f i , . . . , f m )(x), 1 ≤ i ≤ m, α ∈ {0, 1}m \{(1, 1, . . . , 1)},  = { I } I ∈D bounded and b ∈ B M O d . In the above definitions, D := {[m2−k , (m + 1)2−k ) : m, k ∈ Z} is the standard 1  1 I+ − 1 I− , dyadic grid on R and h I ’s are the Haar functions defined by h I = |I |1/2 where I− and I+ are the left and right halves of I. With ,  denoting the standard inner product in L 2 (R), f i (I, 0) :=  f i , h I  and f i (I, 1) :=  f i , h 2I  = |I1| I f i , the f i (I ) and average of f i over I. The Haar coefficient  f i , h I  is sometimes denoted by  the average of f i over I by  f i  I . For α ∈ {0, 1}m , σ (α) denotes the number of 0 components in α. For convenience, we will denote the set {0, 1}m \{(1, 1, . . . , 1)} by Um . In the following main results L p stands for the Lebesgue space L p (R) :=



1/ p f :  f  p < ∞ with  f  p =  f  L p := R | f (x)| p d x . The Weak L p space, p,∞ , is the space of all functions f such that also denoted by L  f  L p,∞ (R) := sup t |{x ∈ R : f (x) > t}|1/ p < ∞. t>0

Moreover, b B M O d := sup I ∈D |I1| norm of b. We now state our main results:

I

|b(x) − b I | d x < ∞, is the dyadic B M O

m Theorem m 1 Let1 α = (α1 , α2 , . . . , αm ) ∈ {0, 1} and 1 < p1 , p2 , . . . , pm < ∞ with j=1 p j = r . Then  (a) For α = (1, 1, . . . , 1), P α ( f 1 , f 2 , . . . , f m )r  mj=1  f j  p j .    (b) For σ (α) ≤ 1, πbα ( f 1 , f 2 , . . . , f m )r  b B M O d mj=1  f j  p j , if and only if b ∈ B M Od .    (c) For σ (α) > 1, πbα ( f 1 , f 2 , . . . , f m )r ≤ Cb mj=1  f j  p j , if and only if |b, h I | sup √ < ∞. |I | I ∈D

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In each of the above cases, the paraproducts are weakly bounded if 1 ≤ p1 , p2 , . . . , pm < ∞. Theorem Let  = { I } I ∈D be a given sequence and let α = (α1 , α2 , . . . , αm ) ∈ Um . Let 1 < p1 , p2 , . . . , pm < ∞ with m  1 1 = . pj r j=1

Then Tα is bounded from L p1 × L p2 × · · · × L pm to L r if and only if ∞ := sup | I | < ∞. I ∈D

Moreover, Tα has the corresponding weak-type boundedness if 1 ≤ p1 , p2 , . . . , pm < ∞. Theorem Let α = (α1 , α2 , . . . , αm ) ∈ Um , 1 ≤ i ≤ m, and 1 < p1 , p2 , . . . , pm , r < ∞ with m  1 1 = . pj r j=1

Suppose b ∈ L p for some p ∈ (1, ∞). Then the following two statements are equivalent. (a) b ∈ B M O d . (b) [b, Tα ]i : L p1 × L p2 × · · · × L pm → L r is bounded for every bounded sequence  = { I } I ∈D . In particular, b ∈ B M O d if and only if [b, P α ]i : L p1 × L p2 × · · · × L pm → L r is bounded.

2 Notation and preliminaries 2.1 The Haar system and the Haar multipliers Let D denote the standard dyadic grid on R, D = {[m2−k , (m + 1)2−k ) : m, k ∈ Z}. Associated to each dyadic interval I there is a Haar function h I defined by h I (x) =

1  1 I+ − 1 I− , 1/2 |I |

where I− and I+ are the left and right halves of I.

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The collection of all Haar functions {h I : I ∈ D} is an orthonormal basis of L 2 (R), and an unconditional basis of L p for 1 < p < ∞. In fact, if a sequence  = { I } I ∈D is bounded, the operator T defined by T f (x) =



 I  f, h I h I

I ∈D

is bounded in L p for all 1 < p < ∞. The converse also holds. The operator T is called the Haar multiplier with symbol . 2.2 The dyadic maximal function Given a function f , the dyadic Hardy-Littlewood maximal function M d f is defined by 1 M f (x) := sup |I x∈I ∈D |

 | f (t)| dt.

d

I

For the convenience of notation, we will just write M to denote the dyadic maximal operator. Clearly, M is bounded on L ∞ . It is well-known that M is of weak type (1, 1) and strong type ( p, p) for all 1 < p < ∞. 2.3 The dyadic square function The dyadic Littlewood-Paley square function of a function f is defined by  S f (x) :=

 | f, h I |2 1 I (x) |I |

1/2 .

I ∈D

For f ∈ L p with 1 < p < ∞, we have S f  p ≈  f  p with equality when p = 2. 2.4 BMO space A locally integrable function b is said to be of bounded mean oscillation if b B M O

1 := sup |I | I

 |b(x) − b I | d x < ∞, I

where the supremum is taken over all intervals in R. The space of all functions of bounded mean oscillation is denoted by B M O. If we take the supremum over all dyadic intervals in R, we get a larger space of dyadic BMO functions which we denote by B M O d .

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For 0 < r < ∞, define

B M Or = b ∈ L rloc (R) : b B M Or < ∞ ,  1/r  1 where, b B M Or := sup |b(x) − b I |r d x . I |I | I For any 0 < r < ∞, the norms b B M Or and b B M O are equivalent. The equivalence of norms for r > 1 is well-known and follows from John-Nirenberg’s lemma (see [9]), while the equivalence for 0 < r < 1 has been proved by Hanks in [6]. (See also [12], page 179.) For r = 2, it follows from the orthogonality of Haar system that ⎛

b B M O d 2

⎞1/2  1 = ⎝ sup | b(J )|2 ⎠ . I ∈D |I | J ⊆I

2.5 The linear/bilinear paraproducts Given two functions f 1 and f 2 , the point-wise product f 1 f 2 can be decomposed into the sum of bilinear paraproducts: f 1 f 2 = P (0,0) ( f 1 , f 2 ) + P (0,1) ( f 1 , f 2 ) + P (1,0) ( f 1 , f 2 ), where for α = (α1 , α2 ) ∈ {0, 1}2 , P α ( f1 , f2 ) =



σ (α)

f 1 (I, α1 ) f 2 (I, α2 )h I

I ∈D

with f i (I, 0) =  f i , h I , f i (I, 1) =  f i  I , σ (α) = #{i : αi = 0}, and h σI (α) being the pointwise product h I h I . . . h I of σ (α) factors. The paraproduct P (0,1) ( f 1 , f 2 ) is also denoted by π f1 ( f 2 ), i.e., π f1 ( f 2 ) =



 f 1 , h I  f 2  I h I .

I ∈D

Observe that  π f1 ( f 2 ), g =

 I ∈D

123

  f 1 , h I  f 2  I h I , g =

 I ∈D

 f 1 , h I  f 2  I g, h I 

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which is equal to 

f2 , P

(0,0)





( f 1 , g) = =

f2 , 



  f 1 , h I g, h I h 2I

I ∈D

 f 1 , h I g, h I  f 2 , h 2I 

I ∈D

=



 f 1 , h I  f 2  I g, h I .

I ∈D

This shows that π ∗f1 = P (0,0) ( f 1 , ·) = P (0,0) (·, f 1 ). The ordinary multiplication operator Mb : f → b f can therefore be given by: Mb ( f ) = b f =P (0,0) (b, f )+P (0,1) (b, f ) + P (1,0) (b, f )=πb∗ ( f )+πb ( f ) + π f (b). The function b is required to be in L ∞ for the boundedness of Mb in L p . However, the paraproduct operator πb is bounded in L p for every 1 < p < ∞ if b ∈ B M O d . Note that B M O d properly contains L ∞ . Detailed information on the operator πb can be found in [11] or [2]. 2.6 Commutators of Haar multipliers The commutator of T with a locally integrable function b is defined by [b, T ]( f )(x) := T (b f )(x) − Mb (T ( f ))(x). It is well-known that for a bounded sequence  and 1 < p < ∞, the commutator [b, T ] is bounded in L p for all p ∈ (1, ∞) if b ∈ B M O d . These commutators have been studied in [13] in non-homogeneous martingale settings.

3 Multilinear dyadic paraproducts 3.1 Decomposition of pointwise product

m 

fj

j =1

 In this sub-section we obtain a decomposition of pointwise product mj=1 f j of m functions that is analogous to the following paraproduct decomposition: f 1 f 2 = P (0,0) ( f 1 , f 2 ) + P (0,1) ( f 1 , f 2 ) + P (1,0) ( f 1 , f 2 ).  The decomposition of mj=1 f j will be the basis for defining multi-linear paraproducts and m-linear Haar multipliers, and will also be very useful in proving boundedness properties of multilinear commutators.

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We first introduce the following notation:

• f (I, 0) :=  f (I ) =  f, h I  = R f (x)h I (x)d x.

• f (I, 1) :=  f  I = |I1| I f (x)d x. • Um := {(α1 , α2 , . . . , αm ) ∈ {0, 1}m : (α1 , α2 , . . . , αm ) = (1, 1, . . . , 1)} . • σ (α) = #{i : αi = 0} for α = (α1 , . . . , αm ) ∈ {0, 1}m . • (α, i) = (α1 , . . . , αm , i), (i, α) = (i, α1 , . . . , αm ) for α = (α1 , . . . , αm ) ∈ {0, 1}m . • PIα ( f 1 , . . . , f m ) =

m

σ (α)

for α ∈ Um and I ∈ D. m   f j (I, α j )h σI (α) for • P α ( f 1 , . . . , f m ) = I ∈D PIα ( f 1 , . . . , f m ) = j=1 I ∈D α ∈ Um . j=1

f j (I, α j )h I

With this notation, the paraproduct decomposition of f 1 f 2 takes the following form:  f 1 f 2 = P (0,0) ( f 1 , f 2 ) + P (0,1) ( f 1 , f 2 ) + P (1,0) ( f 1 , f 2 ) = P α ( f 1 , f 2 ). α∈U2

Note that Um = {(α, 1) : α ∈ Um−1 } ∪ {(α, 0) : α ∈ Um−1 } ∪ {(1, . . . , 1, 0)}. To obtain an analogous decomposition of lemma:

m j=1

(3.1)

f j , we need the following crucial

Lemma 3.1 Given m ≥ 2 and functions f 1 , f 2 , . . . , f m , with f i ∈ L pi , 1 < pi < ∞,we have m 

 f j J 1J =

 

PIα ( f 1 , f 2 , . . . , f m ) 1 J ,

α∈Um J I

j=1

for all J ∈ D. Proof We prove the lemma by induction on m. First assume that m = 2. We want to prove the following:  f1  J  f2  J 1 J =

 

PIα ( f 1 , f 2 )1 J

α∈U2 J  I

⎛ =⎝ ⎛ =⎝





(1,0) PI ( f 1 , f 2 )+



J I

J I

J I







J I

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(0,1) PI ( f 1 , f 2 )+

 f 1 (I ) f 2  I h I +

J I

 f1  I  f 2 (I )h I +

J I

⎞ (0,0) PI ( f 1 , f 2 )⎠ 1 J

⎞  f 2 (I )h 2I ⎠ f 1 (I ) 

1J .

(3.2)

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⎛ For 1 < pi < ∞,  f i  J 1 J = ⎝



⎞  f i (I )h I ⎠ 1 J . So,

J I

 f1  J  f2  J 1 J ⎞⎛ ⎞ ⎛     =⎝ f 1 (I )h I ⎠ ⎝ f 2 (K )h K ⎠ 1 J J I

=



 f 1 (I )h I ⎝

J I

=

=

=

=

⎛

⎧ ⎨ ⎩ J I ⎧ ⎨ ⎩ J I ⎧ ⎨ ⎩ J I ⎧ ⎨

⎩ J I ⎛  =⎝

J K



 f 2 (I )h I + f 2 (K )h K + 

I K

⎞



 f 2 (K )h K ⎠ 1 J

J K I

⎞⎫ ⎬     f 2 (I )h 2I + f 1 (I )  f 2  I h I + f 1 (I )  f 1 (I )h I ⎝ f 2 (K )h K ⎠ 1 J ⎭ J I J I J K I ⎛ ⎞⎫ ⎬        f 2 (I )h 2I + f 1 (I )  f 2  I h I + f 1 (I )  f 2 (K )h K ⎝ f 1 (I )h I ⎠ 1 J ⎭ J I J K K I ⎫ ⎬      f 2 (I )h 2I + f 1 (I )  f 2  I h I + f 1 (I )  f 2 (K )  f 1  K h K 1 J ⎭ J I J K ⎫ ⎬      f 2 (I )h 2I + f 1 (I )  f 2  I h I + f 1 (I )  f 2 (I )  f 1  I h I 1 J ⎭ J I J I ⎞      f1  I  f 2 (I )h 2I ⎠ 1 J . f 1 (I ) f 2  I h I + f 2 (I )h I + f 1 (I ) 

J I





J I

J I

⎛



Now assume m > 2 and that m−1 

 f j J 1J =

 

PIα ( f 1 , f 2 , . . . , f m−1 )1 J .

α∈Um−1 J I

j=1

Then, m 

⎛

m−1 

 f j J 1J = ⎝

j=1

⎞  f j  J 1 J ⎠  fm  J 1 J

j=1

=

 

⎛

PIα ( f 1 , f 2 , . . . , f m−1 ) ⎝

α∈Um−1 J I

=

 



×⎝

 I K

f m (K )h K ⎠ 1 J

J K

PIα ( f 1 , f 2 , . . . , f m−1 )

α∈Um−1 J I

⎛

⎞

f m (K )h K + f m (I )h I +



⎞ f m (K )h K ⎠ 1 J

J K I

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This gives m 

 

 f j J 1J =

PIα ( f 1 , f 2 , . . . , f m−1 ) f m  I 1 J

α∈Um−1 J I

j=1

 

+

PIα ( f 1 , f 2 , . . . , f m−1 ) f m (I )h I 1 J

α∈Um−1 J I

+

 

⎛ PIα ( f 1 , f 2 , . . . , f m−1 ) ⎝

α∈Um−1 J I

 

=

+

PI(α,1) ( f 1 , f 2 , . . . , f m )1 J + ⎛

f m (K )h K ⎝

J K

 

=







α∈Um−1 K I (α,1) PI ( f 1 , f 2 , . . . , f m )1 J

⎞

+

 

(α,0)

( f 1 , f 2 , . . . , f m )1 J

(α,0)

( f 1 , f 2 , . . . , f m )1 J

PI

α∈Um−1 J I

(α,1)

PI

( f 1 , f 2 , . . . , f m )1 J +

α∈Um−1 J I



PI(α,0) ( f 1 , f 2 , . . . , f m )1 J

f m (K )h K  f 1  K . . .  f m−1  K 1 J

 

+

 

PIα ( f 1 , f 2 , . . . , f m−1 )⎠ 1 J

J K

=

f m (K )h K ⎠ 1 J

α∈Um−1 J I

α∈Um−1 J I

+

⎞

J K I

α∈Um−1 J I





PI

α∈Um−1 J I

(1,...,1,0)

PI

 

( f 1 , f 2 , . . . , f m )1 J

J I

=

 

PIα ( f 1 , f 2 , . . . , f m )1 J .

α∈Um J I

The last equality follows from (3.1). Lemma 3.2 Given m ≥ 2 and functions f 1 , f 2 , . . . , f m , with f i ∈ L pi , 1 < pi < ∞,we have m 

fj =



P α ( f 1 , f 2 , . . . , f m ).

α∈Um

j=1

Proof We have already seen that it is true for m = 2. By induction, assume that m−1 

fj =



P α ( f 1 , f 2 , . . . , f m−1 )

α∈Um−1

j=1

=

 

α∈Um−1 I ∈D

123

PIα ( f 1 , f 2 , . . . , f m−1 )

Ann Univ Ferrara

Then, m 

⎛

fj = ⎝

j=1

⎞

m−1 

f j ⎠ fm

j=1

=



 

PIα ( f 1 , f 2 , . . . , f m−1 )

α∈Um−1 I ∈D

=

 



 f m (J )h J

J ∈D

PIα ( f 1 , f 2 , . . . , f m−1 )

α∈Um−1 I ∈D

⎛



×⎝

f m (J )h J + f m (I )h I +

I J

=

 



⎞ f m (J )h J ⎠

J I

PIα ( f 1 , f 2 , . . . , f m−1 ) f m  I

α∈Um−1 I ∈D

 

+

PIα ( f 1 , f 2 , . . . , f m−1 ) f m (I )h I

α∈Um−1 I ∈D

 

+

⎛ PIα ( f 1 , f 2 , . . . , f m−1 ) ⎝

=

PI(α,1) ( f 1 , f 2 , . . . , f m ) +

α∈Um−1 I ∈D

+



⎛

f m (J )h J ⎝

=

(α,1)

PI

 



( f1 , f2 , . . . , fm ) +

PI(α,1) ( f 1 , f 2 , . . . , f m ) +

α∈Um−1 I ∈D

=

⎞

 

(α,0)

PI

( f1 , f2 , . . . , fm )

f m (J )h J  f 1  J . . .  f m−1  J

 

+P 

PI(α,0) ( f 1 , f 2 , . . . , f m )

α∈Um−1 I ∈D

J

=

 

PIα ( f 1 , f 2 , . . . , f m−1 )⎠

α∈Um−1 I ∈D

+

f m (J )h J ⎠

α∈Um−1 I ∈D

α∈Um−1 J I

J

 

⎞

J I

α∈Um−1 I ∈D

 



(1,...,1,0)

 

PI(α,0) ( f 1 , f 2 , . . . , f m )

α∈Um−1 I ∈D

( f1 , f2 , . . . , fm )

α

P ( f 1 , f 2 , . . . , f m ).

α∈Um

Here the last equality follows from (3.1). 3.2 Multilinear dyadic paraproducts  On the basis of the decomposition of pointwise product mj=1 f j we now define multilinear dyadic paraproduct operators, and study their boundedness properties.

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Definition 3.1 For m ≥ 2 and α = (α1 , α2 , . . . , αm ) ∈ {0, 1}m , we define multilinear dyadic paraproduct operators by P α ( f1 , f2 , . . . , fm ) =

m 

f j (I, α j )h σI (α)

I ∈D j=1

where f i (I, 0) =  f i , h I , f i (I, 1) =  f i  I and σ (α) = #{i : αi = 0}. Observe that if β = (β1 , β2 , . . . , βm ) is some permutation of α = (α1 , α2 , . . . , αm ) and (g1 , g2 , . . . , gm ) is the corresponding permutation of ( f 1 , f 2 , . . . , f m ), then P α ( f 1 , f 2 , . . . , f m ) = P β (g1 , g2 , . . . , gm ). Also note that P (1,0) and P (0,1) are the standard bilinear paraproduct operators: P (0,1) ( f 1 , f 2 ) =



 f 1 , h I  f 2  I h I = P( f 1 , f 2 )

I ∈D

P (1,0) ( f 1 , f 2 ) =



 f 1  I  f 2 , h I h I = P( f 1 , f 2 ).

I ∈D

In terms of paraproducts, the decomposition of point-wise product obtained in the previous section takes the form m 



fj =

j=1

m j=1

f j we

P α ( f 1 , f 2 , . . . , f m ).

α∈{0,1}m α =(1,1,...,1)

Definition 3.2 For a given function b and α = (α1 , α2 , . . . , αm ) ∈ {0, 1}m , we define the paraproduct operators πbα by πbα ( f 1 , f 2 , . . . , f m ) = P (0,α) (b, f 1 , f 2 , . . . , f m ) =



b, h I 

I ∈D

m 

1+σ (α)

f j (I, α j ) h I

j=1

where (0, α) = (0, α1 , . . . , αm ) ∈ {0, 1}m+1 . Note that πb1 ( f ) = P (0,1) (b, f ) =

 I ∈D

b(I, 0) f (I, 1)h I =



b, h I  f  I h I = πb ( f ).

I ∈D

The rest of this section is devoted to the boundedness properties of these multilinear paraproduct operators P α and πbα .

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m 1 1 Lemma 3.3 Let 1 < p1 , p2 , . . . , pm , r < ∞ and j=1 p j = r . Then for α = (α1 , α2 , . . . , αm ) ∈ Um , the operators P α map L p1 × · · · × L pm → L r with estimates of the form: P α ( f 1 , f 2 , . . . , f m )r 

m 

 f j pj

j=1

Proof First we observe that, if x ∈ I ∈ D, then | f  I | ≤ | f | I ≤ M f (x) and that

! ! ! | f, h I | 1 !! f h I !! = √ ! √ |I | |I | R ! !  ! 1 !! = f 1 I+ − f 1 I− !! |I | ! R R    1 = |f|+ |f| |I | I I−  + 1 |f| ≤ |I | I ≤ M f (x).

Case I: σ (α) = 1. Let α j0 = 0. Then P α ( f1 , f2 , . . . , fm ) =

m  I ∈D j=1

⎛

=

⎜ ⎜ ⎝

I ∈D

σ (α)

f j (I, α j )h I

m  j=1 j = j0

⎞ ⎟  f j I ⎟ ⎠  f j0 , h I h I .

Using square function estimates, we obtain ⎛ ⎞1/2        m  ⎜   α ! !2 ⎟ 1 I  P ( f 1 , f 2 , . . . , f m )  ⎜ ! f j  I ! | f j , h I |2 ⎟  0 ⎝ r ⎠  |I |   I ∈D j=1   j = j0 r ⎛ ⎞   1/2      ⎜ m ⎟  1 2 I ⎜   ⎟ ≤ ⎝ M f j⎠ | f j0 , h I |  |I |   j=1 I ∈D   j = j 0

r

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Ann Univ Ferrara

⎛  ⎞      m ⎜  ⎟ ⎜ ⎟ (S f j ) = M f j 0  ⎝ ⎠  j=1   j = j  0

m 

≤



r

M f j  p j S f j0  p j0

j=1 j = j0 m 

 f j pj ,

j=1

where we have used H¨older inequality, and the boundedness of maximal and square function operators to obtain the last two inequalities. Case II: σ (α) > 1. Choose j  and j  such that α j  = α j  = 0. Then ! ! α ! P ( f 1 , f 2 , . . . , f m )(x)! ! ! ⎛ ⎞ ! ! ⎛ ⎞ ! ! ! ! ⎜   f , h ⎟  (x) 1 j I I ! ! ⎜ ⎟ ⎝ =!  f j I ⎠ ⎜ √ ! ⎟  f j  , h I  f j  , h I  ! ⎝ ⎠ |I | ! |I | j:α j =0 ! ! I ∈D j:α j =1 ! ! j = j  , j  ⎛ ⎞    1 I (x) ⎝ ⎠ ≤ M f j (x) | f j  , h I || f j  , h I | . |I |   I ∈D

j: j = j , j

By Cauchy-Schwarz inequality ! !! ! ! f j  , h I ! ! f j  , h I ! 1 I (x) |I |

I ∈D

≤



 I ∈D

| f j  , h I |

2 1 I (x)

1  2

|I |



| f j  , h I |

I ∈D

= S f j  (x) S f j  (x).

2 1 I (x)

1 2

|I | (3.3)

Therefore, ⎛ ! ! α ! P ( f 1 , f 2 , . . . , f m )(x)! ≤ ⎝

 j: j = j  , j 

123

⎞ M f j (x)⎠ S f j  (x) S f j  (x).

Ann Univ Ferrara

Now using generalized H¨older’s inequality and the boundedness properties of the maximal and square functions, we get ⎛  α   P ( f 1 , f 2 , . . . , f m ) ≤ ⎝ r 

 j: j = j  , j 

m 

⎞ M f j  p j ⎠ S f j   p j  S f j   p j 

 f j pj .

j=1

m Lemma m 1 3.4 1Let α = (α1 , . . . , αm ) ∈ {0, 1} and 1 < p1 , . . . , pm , r < ∞ with j=1 p j = r .

(a) For σ (α) ≤ 1, πbα is a bounded operator from L p1 × · · · × L pm to L r if and only if b ∈ B M Od . (b) For σ (α) > 1, πbα is a bounded operator from L p1 × · · · × L pm to L r if and only |b, h I | < ∞. if sup √ |I | I ∈D Proof (a) We prove this part first for σ (α) = 0, that is, for α1 = · · · = αm = 1. Assume that b ∈ B M O d . Then for ( f 1 , . . . , f m ) ∈ L p1 × · · · × L pm , we have πbα ( f 1 , . . . , f m ) = P (0,α) (b, f 1 , . . . , f m ) m   = b, h I   f j I h I I ∈D

=



j=1

πb ( f 1 ), h I 

I ∈D

m 

 f j I h I

j=2

= P (0,α2 ,...,αm ) (πb ( f 1 ), f 2 , . . . , f m ) . Since b ∈ B M O d and f 1 ∈ L p1 with p1 > 1, we have πb ( f 1 ) p1  b B M O d  f 1  p1 . So, πbα ( f 1 , . . . , f m )r = P (0,α2 ,...,αm ) (πb ( f 1 ), f 2 , . . . , f m ) r m   πb ( f 1 ) p1  f j pj  b B M O d

j=2 m 

 f j pj ,

j=1

where the first inequality follows from Lemma 3.3.

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Conversely, assume that πb f i = |J |

−

1 pi

: L p1 × · · · × L pm → L r is bounded. Then for

1 J (x) with J ∈ D,

      (1,1,...,1)   ( f 1 , f 2 , . . . , f m ) ≤ πb(1,1,...,1)  πb r

L p1 ×···×L pm →L r

,

since  f i  pi = 1 for all 1 ≤ i ≤ m. For such f i ,   % $    − 1 + 1 +···+ 1   (1,1,...,1)  (1,1,...,1) p1 p2 pm  ( f 1 , f 2 , . . . , f m ) = |J | πb (1 J , 1 J , . . . , 1 J ) πb  r r      1    = |J |− r  b(I )1 J mI h I  .   I ∈D

r

Taking  I = 1 if I ⊆ J and  I = 0 otherwise, we observe that               m       b(I )h I  =  b(I )1 J  I h I    J ⊇I ∈D    J ⊇I ∈D r r      = I  b(I )1 J mI h I    I ∈D  r        b(I )1 J mI h I  ,   I ∈D

r

where the last inequality follows from the boundedness of Haar multiplier T on L r . Thus, we have              −1/r   sup |J |−1/r  b(I )h I  b(I )1 J mI h I      sup |J |   J ∈D J ∈D  J ⊇I ∈D  I ∈D r r    (1,1,...,1)   πb ,  p p r L

1 ×···×L m →L

proving that b ∈ B M O d . Now the proof for σ (α) = 1 follows from the simple observation that πbα is a transpose (1,...,1) . For example, if σ (α) = 1 with α1 = 0 and α2 = · · · = αm = 1 and if r  of πb  is the conjugate exponent of r, then for g ∈ L r &

πbα ( f 1 , . . . , f m ), g

'

 =



b, h I  f 1 , h I 

I ∈D

=

 I ∈D

123

b, h I  f 1 , h I 

m 

j=2 m 

  f j  I h 2I , g

 f j  I g, h 2I 

j=2

Ann Univ Ferrara

=



b, h I  f 1 , h I 

I ∈D

 =

m 

 f j  I g I

j=1



b, h I g I

I ∈D

m 



 f j  I h I , f1

j=1

  = πb(1,...,1) (g, f 2 , . . . , f m ), f 1 . |b, h I | < ∞. For m = 2 we have √ |I | I ∈D

(b) Assume that b∗ ≡ sup

!r  !  !!  !r ! ! ! ! ! (0,0) 3 b, h I  f 1 , h I  f 2 , h I h I (x)! d x ! !πb ( f 1 , f 2 )! d x = ! ! R R I ∈D  r   1 I (x) |b, h I | | f 1 , h I | | f 2 , h I | 3/2 d x ≤ |I | R I ∈D  r  |b, h I |  1 I (x) | f 1 , h I | | f 2 , h I | dx ≤ sup √ |I | |I | R I ∈D I ∈D r   1 I (x) r | f 1 , h I | | f 2 , h I | d x. = b∗ |I | R I ∈D

Using (3.3) and H¨older’s inequality we obtain  !  !r ! (0,0) ! r π ( f , f ) d x ≤ b (S f 1 )r (x) (S f 2 )r (x) d x ! b 1 2 ! ∗ R

R

 ≤

br∗  ×

R

R



p /r (S f 1 )r (x) 1 d x



p /r (S f 2 )r (x) 2 d x

r/ p1

r/ p2

≤ br∗ S f 1 rp1 S f 2 rp2  br∗  f 1 rp1  f 2 rp2 . Thus we have, πb(0,0) ( f 1 , f 2 )r  b∗  f 1  p1  f 2  p2 . Observe that πb(0,0) ( f 1 , f 2 )(I, 0) = πb(0,0) ( f 1 , f 2 ), h I  =

1 b, h I  f 1 , h I  f 2 , h I . |I |

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Now consider m > 2 and let σ (α) > 1. Without loss of generality we may assume that α1 = α2 = 0. Then     m    1+σ (α)  α   πb ( f 1 , f 2 , . . . , f m )r =  b, h I  f 1 , h I  f 2 , h I  f j (I, α j )h I   I ∈D  j=3 r     m   1 σ (α)−1   =  f , h  f , h  f (I, α )h b, h I 1 I 2 I j j I    I ∈D |I |  j=3 r     m   (0,0) σ (α)−1   = πb ( f 1 , f 2 ), h I  f j (I, α j )h I    I ∈D  j=3 r    β (0,0)  = P (πb ( f 1 , f 2 ), f 3 , . . . , f m ) r

(0,0)

 πb

m 

( f 1 , f 2 )q

 f j pj

j=3

 b∗

m 

 f j pj

j=1

where β = (0, α3 , . . . , αm ) ∈ {0, 1}m−1 and πb(0,0) ( f 1 , f 2 ) ∈ L q with 1 q,q

1 p1

+

1 p2

=

> r > 1. Conversely, assume that πbα : L p1 × · · · × L pm → L r is bounded and that σ (α) > 1. 1

Choose any J ∈ D, and take f j = |J | 2 α j = 1 so that  f j  p j = 1. Then

− p1

j

h J if α j = 0, and f j = |J |

    α π ( f 1 , . . . , f m ) ≤ π α  p b b L 1 ×···×L pm . r We also have  σ (α) m    α π ( f 1 , . . . , f m ) = |J | 2 − j=1 b  r =

= |J |

σ (α) 1 2 −r σ (α) 1 2 −r

|b, h J ||J |−

= |J | |b, h J ||J | |b, h J | . = √ |J |

123



1+σ (α)   b, h J h J  r   σ (α) 1   1+σ (α) − |J | 2 r |b, h J | h J  r 1 pj

1+σ (α) 2

− 1+σ2 (α)

1 J r 1

|J | r

− p1

j

1 J if

Ann Univ Ferrara

Thus

|b,h √ J | |J |

  ≤ πbα  L p1 ×···×L pm . Since it is true for any J ∈ D, we have  |b, h J |  ≤ πbα  L p1 ×···×L pm < ∞, √ |J | J ∈D sup

as desired. Now that we have obtained strong type L p1 ×· · ·× L pm → L r boundedness estimates for the paraproduct operators P α with α ∈ Um and πbα with α ∈ {0, 1}m in the case  when 1 < p1 , p2 , . . . , pm , r < ∞ and mj=1 p1j = r1 , we are interested to investigate estimates corresponding to m1 ≤ r < ∞. We will prove in Lemma 3.6 that we obtain weak type estimates if one or more pi ’s are equal to 1. In particular, we obtain 1 L 1 ×· · ·× L 1 → L m ,∞ estimates for those operators. Then it follows from multilinear interpolation that the paraproduct operators strongly bounded from L p1 ×· · ·×L pm are m r to L for 1 < p1 , p2 , . . . , pm < ∞ and j=1 p1j = r1 , even if m1 < r ≤ 1. We first prove the following general lemma, which when applied to the operators P α and πbα gives aforementioned weak type estimates. Lemma 3.5 Let T be a multilinear operator that is bounded from the product of Lebesgue spaces L p1 × · · · × L pm to L r,∞ for some 1 < p1 , p2 , . . . , pm < ∞ with m  1 1 = . pj r j=1

Suppose that for every I ∈ D, T ( f 1 , . . . , f m ) is supported in I if f i = h I for some i ∈ {1, 2, . . . , m}. Then T is bounded from L 1 × · · · × L 1 × L pk+1 × · · · × L pm → qk

L qk +1

,∞

for each k = 1, 2, . . . , m, where qk is given by 1 1 1 = (k − 1) + + ··· + . qk pk+1 pm 1

In particular, T is bounded from L 1 × · · · × L 1 to L m ,∞ . q1

Proof We first prove that T is bounded from L 1 × L p2 × · · · × L pm to L q1 +1 Let λ > 0 be given. We have to show that  |{x : |T ( f 1 , f 2 , . . . , f m )(x)| > λ}| 

 f 1 1

m

j=2  f j  p j



,∞

.

q1 1+q1

λ

for all ( f 1 , f 2 , . . . , f m ) ∈ L 1 × L p2 · · · × L pm . Without loss of generality, we assume  f 1 1 =  f 2  p2 = · · · =  f m  pm = 1, and

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Ann Univ Ferrara

prove that |{x : |T ( f 1 , f 2 , . . . , f m )(x)| > λ}|  λ

q

1 − 1+q

1

.

For this, we apply Calder´on-Zygmund decomposition to the function f 1 at height q1

λ q1 +1 to obtain ‘good’ and ‘bad’ functions g1 and b1 , and a sequence {I1, j } of disjoint dyadic intervals such that f 1 = g1 + b1 ;   b1 = b1, j with supp(b1, j ) ⊆ I1, j and

b1, j d x = 0; I1, j

j

and



|I1, j | ≤ λ

−q

q1 1 +1

 f 1 1 = λ

−q

q1 1 +1

.

j

(Recall that we have assumed  f 1 1 = 1.) Moreover, since 1 < p1 < ∞, the good function g1 ∈ L p1 with g1  p1

 1  p1 −1   q1 q1 p1 p1 1/ p1 q +1 q +1 ≤ 2λ 1  f 1 1 = 2λ 1 ,

where p1 is the H¨older conjugate of p1 . Multilinearity of T implies that |{x : |T ( f 1 , . . . , f m )(x)| > λ}| !( )! !( )! ! λ !! !! λ !! ! + x : |T (b1 , f 2 , . . . , f m )(x)| > . ≤ ! x : |T (g1 , f 2 , . . . , f m )(x)| > 2 ! ! 2 ! Since g1 ∈ L p1 and T is bounded from L p1 × · · · × L pm to L r,∞ , we have ⎛

m 

⎞r

f j (J, α j ) ⎟ ⎜ 2g1  p1 ⎜ ⎟ j=2 ⎜ ⎟ |{x : |T (g1 , f 2 , . . . , f m )(x)| > λ/2}|  ⎜ ⎟ ⎜ ⎟ λ ⎝ ⎠ ⎛   p1 −1 ⎞r q1 p1 ⎜ 2 2λ q1 +1 ⎟ ⎜ ⎟ ⎜ ⎟ ≤⎜ ⎟ λ ⎝ ⎠ $ r

λ

123

q1 ( p1 −1) p1 (q1 +1) −1

%

.

Ann Univ Ferrara

Now,

1 r

=

m

1 j=1 p j

 r

=

1 p1

+

1 q1

implies that r =

p1 q 1 p1 +q1 .

So,

   q1 ( p1 − 1) p1 q 1 − q 1 − p1 q 1 − p1 p1 q 1 −1 = p1 (q1 + 1) ( p1 + q 1 ) p1 (q1 + 1) p1 q1 (− p1 − q1 ) = ( p1 + q1 ) p1 (q1 + 1) q1 . =− q1 + 1

Thus we have: |{x : |T (g1 , f 2 , . . . , f m )(x)| > λ/2}|  λ

q

1 − 1+q

1

.

From the properties of ‘bad’ function b1 we deduce that b1 , h I  = 0 only if I ⊆ I1, j for some j. The hypothesis of the lemma on the support of T ( f 1 , . . . , f m ) then implies that supp (T (b1 , f 2 , . . . , f m )) ⊆ ∪ j I1, j . Thus, !( )! q1 ! ! ! ! ! x : |T (b1 , f 2 , . . . , f m )(x)| > λ ! ≤ !∪ j I1, j ! ≤ λ− 1+q1 . ! ! 2 Combining these estimates corresponding to g1 and b1 , we have the desired estimate |{x : |T ( f 1 , f 2 , . . . , f m )(x)| > λ}|  λ

q

1 − 1+q

q1

1

.

,∞

Now beginning with the L 1 × L p2 × · · · × L pm → L q1 +1 estimate, we use the same argument to lower the second exponent to 1 proving that T is bounded from q2

L 1 × L 1 × L p3 × · · · × L pm to L q2 +1

,∞

, where q2 is given by

1 q2

= 1+

1 p3

+· · ·+

1 pm . qm qm +1 ,∞

We continue the same process until we obtain L 1 × L 1 × · · · × L 1 → L boundedness of T with q1m = 1 + 1 + · · · + 1 (m − 1 terms) = m − 1. This completes the proof since

qm qm +1

=

1 m.

m Lemma m 1 3.6 1Let α = (α1 , . . . , αm ) ∈ {0, 1} , 1 ≤ p1 , . . . , pm < ∞ and = . Then j=1 p j r

(a) For α = (1, 1, . . . , 1), P α is bounded from L p1 × · · · × L pm to L r,∞ . (b) If b ∈ B M O d and σ (α) ≤ 1, πbα is bounded from L p1 × · · · × L pm to L r,∞ . |b, h I | < ∞ and σ (α) > 1, πbα is bounded from L p1 × · · · × L pm to (c) If sup √ |I | I ∈D L r,∞ .

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Proof By orthogonality of Haar functions, h I (J, 0) = h I , h J  = 0 for any two distinct dyadic intervals I and J. The Haar functions have mean value 0, so it is easy to see that h I  J = 0 only if J  I since any two dyadic intervals are either disjoint or one is contained in the other. Consequently, if some f i = h I , then P α ( f1 , f2 , . . . , fm ) =

m 

σ (α)

f j (J, α j )h J

J ⊆I j=1

and, πbα ( f 1 , f 2 , . . . , f m ) =

 J ⊆I

b, h J 

m 

1+σ (α)

f j (J, α j )h J

,

j=1

which are both supported in I. Since the paraproducts are strongly (and hence weakly) bounded from L p1 × · · · × L pm → L r , the proof follows immediately from Lemma 3.5. Combining the results of Lemmas 3.3, 3.4 and 3.6, and using multilinear interpolation (see [4]), we have the following theorem: Theorem 3.1 Let α = (α1 , . . . , αm ) ∈ {0, 1}m and 1 < p1 , . . . , pm < ∞ with  m 1 1 j=1 p j = r . Then  (a) For α = (1, 1, . . . , 1), P α ( f 1 , f 2 , . . . , f m )r  mj=1  f j  p j .    (b) For σ (α) ≤ 1, πbα ( f 1 , f 2 , . . . , f m )r  b B M O d mj=1  f j  p j , if and only if b ∈ B M Od .    (c) For σ (α) > 1, πbα ( f 1 , f 2 , . . . , f m )r ≤ Cb mj=1  f j  p j , if and only if |b, h I | sup √ < ∞. |I | I ∈D In each of the above cases, the paraproducts are weakly bounded if 1 ≤ p1 , p2 , . . . , pm < ∞.

4 Multilinear Haar multipliers and multilinear commutators 4.1 Multilinear Haar multipliers In this subsection we introduce multilinear Haar multipliers, and study their boundedness properties.

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Definition 4.1 Given α = (α1 , α2 , . . . , αm ) ∈ {0, 1}m , and a symbol sequence  = { I } I ∈D , we define m-linear Haar multipliers by Tα ( f 1 , f 2 , . . . , f m )

≡



I

I ∈D

m 

f j (I, α j )h σI (α) .

j=1

Theorem 4.1 Let  = { I } I ∈D be a given sequence and let α = (α1 , α2 , . . . , αm ) ∈ Um . Let 1 < p1 , p2 , . . . , pm < ∞ with m  1 1 = . pj r j=1

Then Tα is bounded from L p1 × L p2 × · · · × L pm to L r if and only if ∞ := sup | I | < ∞. I ∈D

Moreover, Tα has the corresponding weak-type boundedness if 1 ≤ p1 , p2 , . . . , pm < ∞. Proof To prove this lemma we use the fact that the linear Haar multiplier  T ( f ) =  I  f, h I h I I ∈D

is bounded on L p for all 1 < p < ∞ if ∞ := sup I ∈D | I | < ∞, and that T ( f ), h I  =  I  f, h I . By assumption σ (α) ≥ 1. Without loss of generality we may assume that αi = 0 if 1 ≤ i ≤ σ (α) and αi = 1 if σ (α) < i ≤ m. In particular, we have α1 = 0. Then  I f 1 (I, α1 ) =  I  f 1 , h I  = T ( f 1 ), h I  = T ( f 1 )(I, α1 ). First assume that ∞ := sup I ∈D | I | < ∞. Then,      m  σ (α)    f (I, α )h Tα ( f 1 , f 2 , . . . , f m )r =  I j j I    I ∈D j=1  r     m   σ (α)   = T ( f )(I, α ) f (I, α )h  1 1 j j I    I ∈D  j=2 = P α (T ( f 1 ), f 2 , . . . , f m )r m   T ( f 1 ) p1  f j pj

r

j=2



m 

 f j pj .

j=1

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Conversely, assume that Tα : L p1 × L p2 × · · · × L pm → L r is bounded, and let σ (α) = k. Recall that αi = 0 if 1 ≤ i ≤ σ (α) = k and αi = 1 if k = σ (α) < i ≤ m. Taking f i = h I if 1 ≤ i ≤ k and f i = 1 I if k < i ≤ m, we observe that Tα ( f 1 , f 2 , . . . , f m )r =

 R

| I | = |I |1/r , |I |k/2 and m 

 f j pj =

j=1

k   i=1

R

1/r | I h kI (x)|r d x

|h I (x)| pi d x

k  

1/r  | I |r 1 (x)d x I |I |kr/2 R

1/ pi  m  j=k+1



 =

R

|1 I (x)| p j d x

1/ pi  m 

1 = 1 I (x)d x |I | pi /2 R i=1 j=k+1   k m   1 = |I |1/ pi |I |1/ p j 1/2 |I | i=1

=

R

1/ p j

1 I (x)d x

1/ p j

j=k+1

|I |1/r |I |k/2

Since ( f 1 , f 2 , . . . , f m ) ∈ L p1 × L p2 × · · · × L pm , the boundedness of T implies that Tα ( f 1 , f 2 , . . . , f m )r ≤ Tα  L p1 ×···×L pm →L r

m 

 f j pj .

j=1

That is, | I | |I |1/r 1/r α |I | ≤ T  , p p m 1  L ×···×L |I |k/2 |I |k/2 for all I ∈ D. Consequently, ∞ = sup I ∈D | I | ≤ Tα  L p1 ×···×L pm < ∞, as desired. If 1 ≤ p1 , p2 , . . . , pm < ∞, the weak-type boundedness of Tα follows from Lemma 3.5. 4.2 Multilinear commutators In this subsection we study boundedness properties of the commutators of Tα with the multiplication operator Mb when b ∈ B M O d . For convenience we denote the operator Mb by b itself. We are interested in the following commutators: [b, Tα ]i ( f 1 , f 2 , . . . , f m )(x) ≡ (Tα ( f 1 , . . . , b f i , . . . , f m ) − bTα ( f 1 , f 2 , . . . , f m ))(x) where 1 ≤ i ≤ m.

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Note that if b is a constant function, [b, Tα ]i ( f 1 , f 2 , . . . , f m )(x) = 0 for all x. Our α p1 p2 pm approach to study the boundedness properties m of1 [b, T1 ]i : L × L × · · · × L → r L with 1 < p1 , p2 , . . . , pm < ∞ and j=1 p j = r for non-constant b requires us to assume that b ∈ L p for some p ∈ (1, ∞), and that r > 1. However, this restricted unweighted theory turns out to be sufficient to obtain a weighted theory, which in turn implies the unrestricted unweighted theory of these multilinear commutators. We will present the weighted theory of these commutators in a subsequent paper. Theorem 4.2 Let α = (α1 , α2 , . . . , αm ) ∈ Um . If b ∈ B M O d ∩ L p for some 1 < p < ∞ and ∞ := sup I ∈D | I | < ∞, then each commutator [b, Tα ]i is bounded from L p1 × L p2 × · · · × L pm → L r for all 1 < p1 , p2 , . . . , pm , r < ∞ with m  1 1 = , pj r j=1

with estimates of the form: [b, Tα ]i ( f 1 , f 2 , . . . , f m )r  b B M O d

m 

 f j pj .

j=1

Proof It suffices to prove boundedness of [b, Tα ]1 , as the others are identical. Moreover, we may assume that each f i is bounded and has compact support, since such functions are dense in the L p spaces. Writing b f 1 = πb ( f 1 ) + πb∗ ( f 1 ) + π f1 (b) and using multilinearity of Tα , we have Tα (b f 1 , f 2 , . . . , f m ) = Tα (πb ( f 1 ), f 2 , . . . , f m ) + Tα (πb∗ ( f 1 ), f 2 , . . . , f m ) +Tα (π f1 (b), f 2 , . . . , f m ). On the other hand, bTα ( f 1 , f 2 , . . . , f m )

=



I

I ∈D

=



m 

I  b(I )

+ =





m 

1+σ (α)

f j (I, α j )h I

j=1

I

m 

I ∈D

j=1



m 

I

  b(J )h J

J ∈D

j=1

I ∈D

+

 f j (I, α j )h σI (α)

⎛

f j (I, α j )h σI (α) ⎝ ⎛ f j (I, α j )h σI (α) ⎝

j=1 I ∈D α πb ( f 1 , . . . , T ( f i ), . . . , f m )

 I J



⎞  b(J )h J ⎠ ⎞  b(J )h J ⎠

J I

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+

 I ∈D

+

 J ∈D

 I b I

m 

σ (α)

f j (I, α j )h I

j=1

⎛

 b(J )h J ⎝



I

J I

m 

⎞ f j (I, α j )h σI (α) ⎠

j=1

for some i with αi = 0. Indeed, some αi equals 0 by assumption, and for such i, we have T ( f i )(I, αi ) = T f i (I ) =  I f i (I, αi ).  ( f i )(I ) =  I  For ( f 1 , f 2 , . . . , f m ) ∈ L p1 × L p2 × · · · × L pm , we have Tα (πb ( f 1 ), f 2 , . . . , f m )r  πb ( f 1 ) p1  b B M O d

m 

 f j pj

j=1 m 

Tα (πb∗ ( f 1 ), f 2 , . . . , f m )r  πb∗ ( f 1 ) p1  b B M O d

 f j pj

j=2 m 

 f j pj

j=2 m 

 f j pj .

j=1

and, πbα ( f 1 , . . . , T ( f i ), . . . , f m )r  b B M O d  f 1  p1 · · · T ( f i ) pi · · ·  f m  pm m   b B M O d  f j pj . j=1

So, to prove boundedness of [b, Tα ]1 , is suffices to show similar control over the terms:  ⎛ ⎞   m     σ (α)    ⎝ ⎠ I f j (I, α j )h I (4.1) b(J )h J     J ∈D j=1 J I r

and,

    m    α σ (α)  T (π f (b), f 2 , . . . , f m ) −  .  b f (I, α )h I I j j I 1      j=1 I ∈D r

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(4.2)

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Estimation of (4.1) Case I: σ (α) odd. In this case, m m    1−σ (α)  I f j (I, α j )h σI (α) =  I |I | 2 f j (I, α j )h I . Tα ( f 1 , f 2 , . . . , f m ) = I ∈D

I ∈D

j=1

j=1

So, Tα ( f 1 , f 2 , . . . , f m ), h I h I =  I |I |

1−σ (α) 2

m 

f j (I, α j )h I =  I

j=1

m 

σ (α)

f j (I, α j )h I

.

j=1

This implies that  ⎛ ⎞      α   ⎝ ⎠ (4.1) =  T ( f 1 , f 2 , . . . , f m ), h I h I  b(J )h J   J ∈D  J I r        = b(J )Tα ( f 1 , f 2 , . . . , f m ) J h J    J ∈D r   α    = πb T ( f 1 , f 2 , . . . , f m ) r   α  b d  T ( f 1 , f 2 , . . . , f m ) BMO

 b B M O d



m 

r

 f j pj .

j=1

Case II: σ (α) even. In this case at least two αi s are equal to 0. Without loss of generality we may assume that α1 = 0. Then denoting T ( f 1 ) by g1 , P (α2 ,...,αm ) ( f 2 , . . . , f m ) by g2 , and using the fact that ⎛ ⎞    g1 (I )g2  I h I + g1  I g2 (I )h I + g1 (I )g2 (I )h 2I ⎠ 1 J , g1  J g2  J 1 J = ⎝ J I

J I

J I

we have  ⎛ ⎞   m     σ (α)    ⎝ ⎠ I f j (I, α j )h I b(J )h J     J ∈D j=1 J I  ⎛ ⎞ r      2 ⎠   ⎝ = g1 (I )g2 (I )h I  b(J )h J  J ∈D  J I r  ⎛ ⎞         ⎝ ⎠ = g1 (I )g2  I h I − g1  I g2 (I )h I  b(J )h J g1  J g2  J 1 J −    J ∈D J I J I r

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            (0,1)   ≤ (g1 , g2 ) J h J  b(J )g1  J g2  J h J  +  b(J )P     J ∈D J ∈D r r        + b(J )P (1,0) (g1 , g2 ) J h J    J ∈D

r

 b B M O d g1  p1 g2 q +b B M O d P (0,1) (g1 , g2 )r + b B M O d P (1,0) (g1 , g2 )r  b B M O d g1  p1 g2 q m   b B M O d  f j pj . j=1

m 1 where, q is given by q1 = j=2 p j . Here the last three inequalities follow from Lemmas 3.3 and 3.4, and the fact that g1  p1 = T ( f 1 ) p1   f 1  p1 . Estimation of (4.2): Case I: α1 = 0. This case is easy as we observe that Tα (π f1 (b), f 2 , . . . , f m ) − =



 I π f 1 (b)(I )

I ∈D

=



 I b I  f 1 (I )

I ∈D

m  j=2 m 



 I b I

m 

I ∈D

σ (α)

f j (I, α j )h I

j=1 σ (α)

f j (I, α j )h I

−



 I b I  f 1 (I )

I ∈D σ (α)

f j (I, α j )h I

−



 I b I  f 1 (I )

I ∈D

j=2

m  j=2 m 

σ (α)

f j (I, α j )h I

σ (α)

f j (I, α j )h I

j=2

= 0. So there is nothing to estimate. Case II: α1 = 1. In this case, Tα (π f1 (b), f 2 , . . . , f m ) − =

 I ∈D

=

 I ∈D

123

 I π f1 (b) I

m 



 I b I

I ∈D

m 

σ (α)

f j (I, α j )h I

j=1

f j (I, α j )h σI (α) −

 I b I  f 1  I

I ∈D

j=2

  I π f1 (b) I − b I  f 1  I



m  j=2

m  j=2

σ (α)

f j (I, α j )h I

f j (I, α j )h σI (α)

Ann Univ Ferrara

We have assumed that b ∈ L p for some p ∈ (1, ∞). So, using Lemma 3.1, we have b I  f 1  I 1 I =



 b(J ) f 1  J h J 1 I +

I J



b J  f 1 (J )h J 1 I +

I J

= πb ( f 1 ) I 1 I + π f1 (b) I 1 I +





 b(J )  f 1 (J )h 2J 1 I

I J

 b(J )  f 1 (J )h 2J 1 I .

I J

Hence, b I  f 1  I 1 I − π f1 (b) I 1 I = πb ( f 1 ) I 1 I +



 b(J )  f 1 (J )h 2J 1 I .

I J

So we have Tα (π f1 (b), f 2 , . . . , f m ) − =−



⎛



 I b I

I ∈D

 I ⎝πb ( f 1 ) I 1 I +



σ (α)

f j (I, α j )h I

j=1



⎞

 b(J )  f 1 (J )h 2J ⎠

I J

I ∈D

=−

m 

 I πb ( f 1 ) I

I ∈D

m  j=2

−

 I ∈D

m 

σ (α)

f j (I, α j )h I

j=2 σ (α)

f j (I, α j )h I ⎛

I ⎝



⎞  b(J )  f 1 (J )h 2J ⎠

I J

= −T (πb ( f 1 ), f 2 , . . . , f m ) −



m 

f j (I, α j )h σI (α)

j=2

⎛  b(J )  f 1 (J )h 2J ⎝

 I J

J ∈D

I

m 

⎞ f j (I, α j )h σI (α) ⎠ .

j=2

Since T (πb ( f 1 ), f 2 , . . . , f m )r  πb ( f 1 ) p1

m 

f j (J, α j )  b B M O d

j=2

m 

 f j pj ,

j=1

we are left with controlling  ⎛ ⎞   m     σ (α) 2 ⎝  .    ⎠ I f j (I, α j )h I b(J ) f 1 (J )h J     J ∈D j=2 I J r

For this we observe that m     (α2 ,...,αm )  ( f 2 , . . . , f m )   f j pj , T q

j=2

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and that πb∗ ( f 1 ) T(α2 ,...,αm ) ( f 2 , . . . , f m ) =

⎛



 b(J )  f 1 (J )h 2J ⎝



J b(J )  f 1 (J )

J ∈D

+



m 

I

I J

J ∈D

+

 m 

⎞ f j (I, α j )h σI (α) ⎠

j=2 2+σ (α)

f j (J, α j )h J

j=2

⎛  b(J )  f 1 (J )h 2J ⎝



I

J I

J ∈D

m 

⎞ σ (α) ⎠

f j (I, α j )h I

j=2

Now, following the same technique we used to control (4.1), we obtain  ⎛ ⎞   m m      σ (α) 2    b   ⎝ ⎠ (J )h  f (I, α )h  f j pj . b(J ) f d 1 I j j BMO J I    J ∈D  j=2 j=1 J I r

We also have        ∗    πb ( f 1 ) T(α2 ,...,αm ) ( f 2 , . . . , f m ) ≤ πb∗ ( f 1 ) p1 T(α2 ,...,αm ) ( f 2 , . . . , f m ) r

q

 b B M O d

m 

 f j pj

j=1

and,     m m     2+σ (α)      J b(J ) f 1 (J ) f j (J, α j )h J  f j pj .    b B M O d  J ∈D  j=2 j=1 r

So we conclude that  ⎛ ⎞   m m     σ (α) ⎠ 2 ⎝   b    (J )h  f (I, α )h  f j pj . b(J ) f d 1 I j j B M O J I     J ∈D j=2 j=1 I J r

Thus we have strong type boundedness of [b, Tα ]1 → L p1 × L p2 × · · · × L pm → L r for all 1 < p1 , p2 , . . . , pm , r < ∞ with m  1 1 = . pj r j=1

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Note that Tα = P α if  I = 1 for all I ∈ D. The following theorem shows that the BMO condition on b is necessary for the boundedness of the commutator [b, P α ]i . Theorem 4.3 Let α = (α1 , α2 , . . . , αm ) ∈ Um , and 1 < p1 , p2 , . . . , pm , r < ∞ with m  1 1 = . pj r j=1

Assume that for given b and i, α

[b, P ]i ( f 1 , f 2 , . . . , f m )r ≤ C

m 

 f j pj ,

(4.3)

j=1

for all f j ∈ L p j . Then b ∈ B M O d . Proof Without loss of generality we may assume that i = 1. Fix I0 ∈ D. Case I: α1 = 0, σ (α) = 1. (1) Take f 1 = 1 I0 and f i = h I (1) for i > 1, where I0 is the parent of I0 . Then, 0

P α ( f 1 , f 2 , . . . , f m )) =

 I ∈D

1 I0 , h I h I (1) m−1 h I = 0, I 0

and, P α (b f 1 , f 2 , . . . , . . . , f m ) =

 I ∈D

=

b1 I0 , h I h I (1) m−1 hI I

 I ⊆I0

⎛

0

⎞m−1 (1) ⎜ K (I0 , I0 ) ⎟ ⎟ ⎜ ⎛

b1 I0 , h I  ⎝ *! ! ⎠ ! (1) ! ! I0 !

hI

⎞m−1

⎜ K (I0 , I0(1) ) ⎟ ⎟ ⎜

= ⎝ *! ! ⎠ ! (1) ! !I0 !



b, h I h I ,

I ⊆I0

(1)

where K (I0 , I0 ) is either 1 or −1 depending on whether I0 is the right or left half of (1) I0 . (1) For the second to last equality we observe that, if I is not a proper subset of I0 , (1) h I (1)  I = 0, and that if I is a proper subset of I0 but is not a subset of I0 , then 0



b1 I0 , h I  = 0. Moreover, for I ⊆ I0 , b1 I0 , h I  = R b1 I0 h I = R bh I = b, h I .

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Now from inequality (4.3), we get  ⎛ ⎞m−1   1   m (1) pi  1  |I  ⎜ K (I0 , I0(1) ) ⎟ 0 | p  ⎜ * ⎟ 1 + b, h I h I  ≤ C|I0 | ! ! ⎠ ⎝ ! (1) !   I ⊆I0 i=2 |I0(1) | !I0 !   r      1 1   1 p +···+ pm C|I | r . i.e.  b, h I h I  0   ≤2 2  I ⊆I0  r

Thus for every I0 ∈ D,     1 1   p +···+ pm C,  b, h I h I  1   ≤2 2  |I0 | r  I ⊆I0 1

r

and hence b ∈ B M O d . Case II: α1 = , 0 or σ (α) > 1. h I0 , if αi = 0 Taking f i = we observe that 1 I0 , if αi = 1, σ (α)

P α ( f 1 , f 2 , . . . , f m )) = h I0

and

σ (α)

P α (b f 1 , f 2 , . . . , . . . , f m ) = (b f 1 )(I0 , α1 )h I0 .

If α1 = 0, (b f 1 )(I0 , α1 ) = bh I0 (I0 , 0) =  bh I0 (I0 ) =

 R

bh I0 h I0 =

 1 b1 I0 = b I0 . |I0 | R

If α1 = 1, (b f 1 )(I0 , α1 ) = b1 I0 (I0 , 1) = b1 I0  I0 = b I0 . So in each case,   [b, P α ]1 ( f 1 , f 2 , . . . , f m )r = b P α ( f 1 , f 2 , . . . , f m ) − P α (b f 1 , f 2 , . . . , . . . , f m )r    σ (α) σ (α)  = bh I0 − b I0 h I0    r  σ (α)  = (b − b I0 )h I0  r

1 = √ (b − b I0 )1 I0 r . ( |I0 |)σ (α)

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On the other hand, m 

1 1 1 1 +···+ p1m  f j pj = √ |I0 | p1 = √ |I0 | r . σ (α) σ (α) ( |I0 |) ( |I0 |) j=1

Inequality (4.3) then gives 1 1 1 (b − b I0 )1 I0 r ≤ C √ |I0 | r √ ( |I0 |)σ (α) ( |I0 |)σ (α) 1 i.e. (b − b I0 )1 I0 r ≤ C. 1 |I0 | r

Since this is true for any I0 ∈ D, we have b ∈ B M O d . Combining the results from Theorems 4.2 and 4.3, we have the following characterization of the dyadic BMO functions. Theorem 4.4 Let α = (α1 , α2 , . . . , αm ) ∈ Um , 1 ≤ i ≤ m, and 1 < p1 , p2 , . . . , pm , r < ∞ with m  1 1 = . pj r j=1

Suppose b ∈ L p for some p ∈ (1, ∞). Then the following two statements are equivalent. (a) b ∈ B M O d . (b) [b, Tα ]i : L p1 × L p2 × · · · × L pm → L r is bounded for every bounded sequence  = { I } I ∈D . In particular, b ∈ B M O d if and only if [b, P α ]i : L p1 × L p2 × · · · × L pm → L r is bounded. Acknowledgments The author would like to thank Brett Wick for suggesting him this research project, and for providing valuable suggestions.

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Ann Univ Ferrara 7. Tuomas P. Hytönen: Representation of singular integrals by dyadic operators, and the A2 theorem. arXiv:1108.5119 (2011) 8. Janson, S.: BMO and commutators of martingale transforms. Ann. Inst. Fourier 31(1), 265–270 (1981) 9. John, F., Nirenberg, L.: On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14, (1961) 10. Lerner, A.K., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo-González, R.: New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Adv. Math. 220(4), 1222–1264 (2009) 11. Pereyra, M.C.: Lecture notes on dyadic harmonic analysis. Contemp. Math. 289, 1–60 (2001) 12. Stein, E.M.: Harmonic analysis: real variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton (1993) 13. Treil, S.: Commutators, paraproducts and BMO in non-homogeneous martingale settings. http://arxiv. org/pdf/1007.1210v1.pdf

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