Paraproducts, Bloom BMO and Sparse BMO Functions

We address $L^p(\mu)\rightarrow L^p(\lambda)$ bounds for paraproducts in the Bloom setting. We introduce certain"sparse BMO"functions associated with sparse collections with no infinitely increasing chains, and use these to express sparse operators as sums of paraproducts and martingale transforms -- essentially, as Haar multipliers -- as well as to obtain an equivalence of norms between sparse operators $\mathcal{A}_\mathcal{S}$ and compositions of paraproducts $\Pi^*_a\Pi_b$.

In 1985, Steven Bloom proved [2] that the commutator where H is the Hilbert transform, is bounded L p (µ) → L p (λ), where µ, λ are two A p weights (1 < p < ∞), if and only if b is in a weighted BMO space determined by the two weights µ and λ, namely b ∈ BMO(ν), where ν := µ 1/p λ −1/p and b BMO(ν) := sup In [7] this result was extended to commutators [b, T ] in R n with Calderón-Zygmund operators T .
Soon after, [11] gave a different proof which yielded a quantitative result for the upper bound: (0.1) The proof in [7] took the route of Hytönen's representation theorem (the R n , Calderón-Zygmund operator generalization of Petermichl's result [14] on the Hilbert transform), and relied heavily on paraproduct decompositions.The proof in [11] used sparse operators and Lerner's median inequalities to obtain directly a sparse domination result for the commutator [b, T ] itself, avoiding paraproducts althogehter.
This paper addresses L p (µ) → L p (λ) bounds for the paraproducts.Based on the one-weight situation, we suspect that these bounds should be smaller than the ones for commutators: in the one-weight case , are both known to be sharp -see [3,12] and the references therein -(where throughout this paper A B is used to mean A ≤ C(n)B, with a constant depending on the dimension and maybe other quantities such as p or Carleson constants Λ of sparse collections, but in any case not depending on any A p characteristics of the weights involved).In the two-weight Bloom situation, we show in Theorem 3.3 that We do not know if this bound is sharp, and this is subject to future investigations -but the bound is smaller than the one in (0.1).In fact, it is strictly smaller with the exception of p = 2, when both bounds are [µ] A 2 [λ] A 2 .We can however show that our bound is sharp in one particular instance, namely when µ = w and λ = w −1 for some A 2 weight w.We show this in Section 3.1 via an appeal to the one-weight linear A 2 bound for the dyadic square function.
2010 Mathematics Subject Classification.42B20, 42B35, 47A30.I. Holmes Fay is supported by Simons Foundation: Mathematics and Physical Sciences-Collaboration Grants for Mathematicians, Award number 853930.
Obviously this bound does not recover the one-weight situation: letting µ = λ = w for some w ∈ A 2 , ν = 1 and our bound would give when we know that the optimal bound is linear in the A 2 characteristic.If the optimal Bloom paraproduct bound is to recover this one-weight situation, we suspect it would need a dependency on [ν] A 2 -as it would need to somehow account for the case µ = λ, or ν = 1.
The proof of the Bloom paraproduct bound above relies on dominating the paraproduct by a "Bloom sparse operator" A ν S f := Q∈S ν Q f Q 1 1 Q , where S is a sparse collection, and proving that A ν S satisfies the bound [µ] 1/(p−1) A p [λ] A p above.We do this in Theorem 2.6.The domination of the paraproduct is treated in Section 3.
Before all this however, we consider in Section 2 a special type of sparse collections, Υ D (R n ), which are sparse collections with no "infinitely increasing chains" (a terminology borrowed from [8]).We see that any such collection can be associated with a BMO function which satisfies b S BMO ≤ Λ, where Λ is the Carleson constant of S (we show this in Appendix A).Once we have a BMO function, we can immediately talk about paraproducts with symbol b S .
In fact, we see in Section 2.3 that these functions allow us to express any sparse operator A S , S ∈ Υ D (R n ), as a sum of paraproducts and a martingale transform: , where T τ S is a martingale transform: T τ L p (w)→L p (w) τ ∞ .
The process used to obtain the BMO function b S associated with S also works with weights, and obtaining a function in weighted BMO spaces associated with Repeating the process above, we try to express A S as a sum of the paraproducts associated with b w S and a martingale transform -but we discover instead the operator and its decomposition as While it would be interesting if the paraproducts and the martingale transform could somehow be "separated" above, giving an independent proof that these operators have the same dependency on [w] A p by showing each is equivalent to norms of A S , we are able to show that norms of sparse operators are equivalent to certain compositions of paraproducts.In Section 2.4, we see that where b S is another BMO function we can easily associate with S: This provides an upper bound: For the other direction, we show in Appendix B -using a bilinear form argument -that for all Bloom weights µ, λ, ν, BMO functions a ∈ BMO D , b ∈ BMO D (ν), and Λ > 1, Note that taking µ = λ = w above, for some w ∈ A p , we have the one-weight result Moreover, we obtain the equivalence of norms Section 3 gives a proof of a pointwise domination of paraproducts by sparse operators.It relies on first proving certain local pointwise domination results, which are then applied to BMO D (w) functions with finite Haar expansion, and extending to the general case.So this argument works whenever Π b acts between L p spaces where the Haar system is an unconditional basis -Lebesgue measure or A p weights.The argument also works with the weighted BMO norm, defined in terms of an L 1 (dx) quantity -the Haar system is not unconditional in L 1 (dx), but we can choose an ordering of the Haar system that ensures convergence in L 1 (dx).The choice to work with b rather than compactly supported f is motivated by the desire to obtain domination by sparse operators with no infinitely increasing chains.Specifically, we work with restricted paraproducts: and construct a sparse collection S(Q 0 ) ⊂ D(Q 0 ) which "ends" at Q 0 , and such that The second author deeply thanks Cristina Pereyra for several conversations about this work, and for her general support.

S  N
1.1.Dyadic Grids.By a dyadic grid D on R n we mean a collection of cubes Q ⊂ R n that satisfies: In other words, two dyadic cubes intersect each other if and only if one contains the other.
For example, the standard dyadic grid on R n is: We assume such a collection D is fixed throughout the paper.For every Q ∈ D and positive integer k ≥ 1 we let Throughout this paper we let (•, •) denote inner product in L 2 ( dx), so we write for example where ( f, h I ) = f h I dx is the Haar coefficient of f corresponding to I.
In R n , we have 2 n − 1 cancellative Haar functions and one non-cancellative: for every dyadic cube Q = I 1 × I 2 × . . .I n , where every I k ∈ D is a dyadic interval with common length where k ∈ {0, 1} for all k, and = ( 1 , . . ., n ) is known as the signature of h Q .The function h Q is cancellative except in one case, when ≡ 1.As in R, the cancellative Haar functions form an orthonormal basis for L 2 (R n , dx), and an unconditional basis for L p (R n , dx), 1 < p < ∞.We often write omitting the signatures, and understanding that h Q always refers to a cancellative Haar function.
There is really only one instance for us where the signatures matter, and that is in the definition Note that whenever P Q for some dyadic cubes P, Q, the Haar function h Q will be constant on P. We denote this constant by h Q (P) := the constant value h Q takes on P Q.

It is easy to show that
where throughout the paper denotes average over Q, and sums such as P⊂Q or R⊃Q are understood to be over dyadic cubes.

A p weights.
A weight is a locally integrable, a.e.positive function w(x) on R n .Any such weight immediately gives a measure on R n via dw := w(x)dx and yields the obvious L p -spaces associated with the measure w.We denote these spaces by L p (w).
where the supremum is over cubes Q ⊂ R n , p denotes the Hölder conjugate of p: and In fact, w ∈ A p if and only if the conjugate weight w is in A p , with We restrict our attention to dyadic A p weights, denoted A D p , and defined in the same way except the supremum is only over dyadic cubes Q ∈ D. Sometimes we use the standard L p -duality (L p (w)) * = L p (w) with inner product (•, •) dw , and other times we think of (L p (w)) * L p (w ) with regular Lebesgue inner product (•, •).We refer the reader to Chapter 9 of [6] for a thorough treatment of A p weights.
where the supremum is over cubes We similarly restrict our attention to dyadic BMO spaces, BMO D and BMO D (w) for the weighted version, both defined in the same way except the supremum is over dyadic cubes Q ∈ D.
In R, we have two paraproducts: They have the property that b f = Π b f + Π * b f + Π f b, and their boundedness is usually characterized by some BMO-type norm of the symbol b.
In R n we have three paraproducts: and Γ b is self-adjoint.Generally, in the L p -situation, we still have where we are thinking of Banach space duality in terms of (L p (µ)) * L p (µ ) and (L p (λ)) * L p (λ ), both with regular Lebesgue inner product (•, •).

S BMO F
2.1.Sparse Families.Let 0 < η < 1.A collection S ⊂ D is said to be η-sparse if for every Q ∈ S there is a measurable subset E Q ⊂ Q such that the sets {E Q } Q∈S are pairwise disjoint, and satisfy It is easy to see that it suffices to impose this condition only on Q ∈ S. It is also easy to see that any η-sparse collection is 1/η-Carleson.Far less obvious is the remarkable property that any Λ-Carleson collection is 1/Λ-sparse, which is proved in the now classic work [10].
A special type of sparse collection which appears most frequently in practice is defined in terms of so-called "S-children."Suppose a family S ⊂ D has the property that where α ∈ (0, 1) and ch S (Q), the S-children of Q, is the collection of maximal P ∈ S such that P Q.Then S is (1 − α)-sparse: let which are clearly pairwise disjoint, and satisfy A collection that is sparse with respect to Lebesgue measure is also sparse with respect to any A p measure w.Recall that (see [6], Proposition 9.1.5)an equivalent definition for [w] A p is , where

and
(2.1) 2. Sparse BMO Functions.We borrow the following terminology from [8]: we say a collection S ⊂ D has an infinitely increasing chain if there exist The following Lemma is also found in [8]:

If a collection S ⊂ D has no infinitely increasing chains, then every
These types of collections will be important for us, so we let denote the set of all sparse collections in D which have no infinitely increasing chains.
Lemma 2.2.Let S ∈ Υ D (R n ) be a sparse collection with no infinitely increasing chains.Then the set of points contained in infinitely many elements of S has measure 0.
Proof.Let S * denote the collection of maximal elements of S. Since S ∈ Υ D (R n ), every Q ∈ S is contained in a unique Q * ∈ S * .Any x which belongs to infinitely may elements of S must then belong to an infinitely decreasing chain terminating at some maximal Q * ∈ S * .Fix any such chain and let A be the set of points contained in all By Lemma 2.2 we know that b S is almost everywhere finite: if x is contained in infinitely many elements of S, then b S (x) = ∞, but this can only happen on a set of measure zero.
Note also that b S is locally integrable: for some Then, for some Q 0 ∈ D: In fact, we can reduce this further to (2.2) However, a more careful estimate is possible.We prove the following in Appendix A.

sparse collection with no infinitely increasing chains and Carleson constant Λ. Then the function b
This process works to yield a weighted BMO function as well: with any S ∈ Υ D (R n ) and w ∈ A D p we associate the function S is a.e.finite, locally integrable, and which then easily gives 1

Sparse Operators as Sums of Paraproducts and Martingale Transform.
For ease of notation we work in R below, but the obvious analog for R n follows easily in the same way.Consider where S ∈ Υ D (R) and w is an , where ν := µ 1/p λ −1/p for two weights µ, λ ∈ A D p .We treat this operator in more detail in Section 2.5.

Using the b w
S function associated with S and w, we write (2.3) where So: The second term can be further explored as Returning to (2.3): where the first two terms are the paraproducts with symbol b w S , the sparse BMO D (w) function associated with S and w, and the third term is Remark 2.1.In case w ≡ 1, we obtain the unweighted situation (2.5) , where T τ S is a martingale transform: Remark 2.2.In fact, (2.4) expresses sparse operators as Haar multipliers: recall that a Haar multiplier is an operator of the form where {φ J (x)} J∈D is a sequence of functions indexed by D. It is known that (see [1]): So, from (2.4): Look more closely now at (2.5): This gives an upper bound for A S : L p (w) → L p (w) in terms of the norms of paraproducts and martingale transform -when usually it is the norms of sparse operators that are used as upper bounds: Divide above by Λ (S) := Λ, the Carleson constant of S, and recall that b S BMO D ≤ Λ, as well as τ S ∞ ≤ Λ: from which we can deduce that, for all Λ > 1: Given the well-known domination results [9] for the martingale transform and paraproducts: Remark 2.3.It would be interesting if the martingale and paraproducts can be "separated" somehow, and to obtain independently that paraproducts and martingale transforms have the same dependency on [w] A p by showing they are both equivalent to A S .However, we can show that the norms of A S are equivalent to norms of certain compositions of paraproducts.We do this next.

Sparse Operators and Compositions of Paraproducts. Consider the composition
We show in Appendix B, using a bilinear form argument, that: Theorem 2.5.There is a dimensional constant C(n) such that for all Bloom weights µ, λ Some immediate observations about this result: • From Theorem 2.6: • Take µ = λ = w, for some w ∈ A p .Then ν = 1 and we obtain in the one-weight situation: (2.6) We associated with S the BMO function b S = Q∈S 1 1 Q .There is another, even more obvious BMO function we can associate with S: For any Q 0 ∈ D: Moreover, so we may express the sparse operator A S as Then which means that for all Λ > 1: Combined with (2.6), we have

The Bloom Sparse Operator
for a sparse collection S ⊂ D(R n ), where µ, λ ∈ A p (1 < p < ∞) and ν := µ 1/p λ −1/p are Bloom weights.In looking to bound this operator L p (µ) → L p (λ), the first obvious route is to appeal to the known one-weight bounds for the usual, unweighted sparse operator , and we use duality to express So we look for a bound of the type This yields the same dependency on the A p characteristics of µ, λ as obtained in [11] for commutators: We give another proof, inspired by the beautiful proof in [4] of the A 2 conjecture for usual unweighted sparse operators, which yields a smaller bound.
Theorem 2.6.Let S ⊂ D be a sparse collection of dyadic cubes, µ, λ ∈ A D p , 1 < p < ∞ be two A p weights on R n , and ν := µ 1/p λ −1/p .Then the Bloom sparse operator (2.7) Proof.In looking for a bound of the type , so we look instead for a bound of the type . Using the standard L p (λ) − L p (λ) duality with (•, •) dλ inner product, we write meaning we finally look for a bound of the type As in [4], we make use of the weighted dyadic maximal function: and its property of being L q (u)-bounded with a constant independent of u: Theorem 2.7.For any locally finite Borel measure u on R n and any q ∈ (1, ∞): (2.8) See, for example, [8] for a proof of this fact.Now: We express the averages involving f and g as weighted averages: Apply the fact that (an easy consequence of Hölder's inequality), and the fact that for any A p weight w, we have to go further: Now apply (2.1): so we may later use disjointness of the sets {E Q } Q∈S .
Putting these estimates together: which proves the theorem.

P  B BMO
We show the following pointwise domination result, inspired by ideas in [9] on pointwise domination of the martingale transform.

Theorem 3.1. There is a dimensional constant C(n) such that: for every
The same holds for the other paraproducts Π * b and Γ b .
Assuming this, return to the Bloom situation for a moment and say b ∈ BMO D (ν) has finite Haar expansion.Then there are at most 2 n disjoint dyadic cubes where S is a sparse collection with Carleson constant Λ and no infinitely increasing chains: holds for all b ∈ BMO D (ν) with finite Haar expansion -and thus for all b.
Corollary 3.2.Given Bloom weights µ, λ ∈ A D p , ν = µ 1/p λ −1/p , for all b ∈ BMO D (ν): The same holds for the other paraproducts Π * b and Γ b .In light of the bound for A ν S in Theorem 2.6, pick some value for Λ, say Λ = 2, and we have: Remark 3.2.As discussed in the introduction, we do not know if this bound is sharp -but we can show that one particular instance of this inequality is sharp -namely when µ = w and λ = w −1 for some A D 2 weight w, in which case the "intermediary" Bloom weight is also ν = w: (3.1) To see this, if Q is a cube: So we may look at the paraproducts with symbol w: in R these are 2 , these are bounded Recall the decomposition (we will show this in a moment) and we have further that w BMO(w) , which yields Finally, the fact that shows that any smaller bound in (3.1) would imply a bound for S D : L 2 (w) → L 2 (w) smaller than [w] A 2 , which is well-known to be false.
Going back to (3.2), it is easy to show that Then 1 We will need the following result, which may be found in Lemma 2.10 of [13].
then T is of weak (1, 1) type, with where C n is a dimensional constant and B := Now we prove some properties of Π b .
Proposition 3.5.The maximal truncation defined above satisfies the following: ii. Π b is dominated by M D Π b : iii.Π b is strong (2, 2): where for every k ∈ Z, Q k is the unique cube in D with side length 2 k that contains x.Fix m ∈ Z: Taking m → −∞ finishes the proof.
ii.Let P ∈ D and define F P (x) : If x P, then there is a unique k ≥ 0 such that So, there is a unique , P 0 P (k) , such that x ∈ P 0 .Then: " (%'() . This therefore holds for all x ∈ R n and all P ∈ D, which proves ii.
iii.This follows immediately from ii and the well-known bound for Π b in the unweighted case: iv.Once we verify supp( Π b ) ⊂ Q for all Q ∈ D, we use iii and Proposition 3.4 to conclude iv.
which is clearly 0 if x Q.

Proof of Theorem 3.1.
Proof.I.The BMO decomposition.We make use of the following modification to the Calderón-Zygmund decomposition used in [5] to essentially reduce a weighted BMO function to a regular BMO function.Given a weight w on R n , a function b ∈ BMO D (w), a fixed dyadic cube Q 0 ∈ D, and ∈ (0, 1), let the collection: and put This is the collection from the usual CZ-decomposition of w, restricted to Q 0 , so we have But instead of defining the usual "good function" for w, we let As shown in [5], this function is in unweighted BMO, with: Moreover, so whenever dealing with a cube Q E, we can replace any average or Haar coefficient of b -the function in weighted BMO -with the average or Haar coefficient of a -the function in unweighted BMO.This has many advantages, since any usage of inequalities involving a will not add any extra A p characteristics.For instance, we can use the well-known bound for Haar coefficients of BMO functions (resulting from applying the John-Nirenberg theorem to replace the L 1 norm in the BMO definition with the L 2 norm): It also allows us to use the results on Π a f from the previous section.
II. Use the properties of the maximal truncation of unweighted BMO paraproducts.We claim that there exists a constant C 0 , depending on the dimension n and on , such that the set: Let then the collection First use the well-known weak (1, 1) inequality for the dyadic maximal function: Since a ∈ BMO D we can apply the weak (1, 1) inequality for Π a according to Proposition 3.5: and let again ϕ = f 1 1 Q 0 .By the definition of a, in this case, Π a f sums only over Then, as we wished, if we choose C 0 large enough: Join the collections E and F into: which then satisfies We show that: (3.4) Once we have this, we recurse on the terms of the second sum, and repeat the argument: for each So we construct the collection S(Q 0 ) recursively, starting with Q 0 as its first element, its S-children are G and so on.We have .
The collection S(Q 0 ) satisfies the S-children definition of sparse collections: Λ and we have the desired sparse collection with Carleson constant Λ such that III. Proof of (3.4).We start by noting that , so we may decompose Π b,Q 0 f as Now, we have to account for the relationship to the collection F and its union F.
In this case, Π a f (x) ≤ C 0 a BMO D | f | Q 0 , and since Π a dominates Π a : • Case 1a: and which gives (3.4) in this case.
• Case 1b: If x E, then the second part of the sum is 0 and we are done, having simply

Case 1b
Case 2: x ∈ F. Then there is a unique P ∈ F such that x ∈ P. Look first at the term . Since x ∈ P, this can be expressed as where P denotes the dyadic parent of P. The first term we split into two: for all y ∈ P.This would force Π a f (y) > C 0 a BMO D | f | Q 0 for all y ∈ P, so P ⊂ F -but this contradicts maximality of P in F .Therefore • Let us now look at the term B.
all y ∈ P, which would force P ⊂ F, again contradicting maximality of P in F .

So
where the term C is defined as We claim that where R 0 is the unique element of G such that x ∈ R 0 : • Case 2a: the second term is 0).
• Case 2b: -Case 2b.i:If P contains some elements of E, then again R 0 = P and we can "fill in the blanks" in the first term with the Π b,R 's from the second term: -Case 2b.ii:If P ⊂ S 0 for some S 0 ∈ E, then R 0 = S 0 and the first term in C is 0 (because P ⊂ E), and the second term is This concludes the proof.
Remark 3.3.One can also use Theorem 3.1 to obtain a full R n domination, losing the requirement for no infinitely increasing chains.Say f is such that supp( f ) ⊂ Q 0 for some Q 0 ∈ D (or, for general compactly supported functions, supp( f Note that, as an application of the modified CZ-decomposition used in Part I of the proof above, one can obtain To see this, let Q ∈ D and apply the decomposition to b over Q: Returning to Π b f , suppose first that x Q 0 .Then there is a unique k ≥ 1 such that x ∈ Q (k) 0 \Q (k−1) 0 , and If, on the other hand, x ∈ Q 0 , So a sparse collection S as follows: Moreover the associated sparse operator appears exactly in the previous inequalities, which can be expressed as: So, applying Theorem 3.1 to the function f ≡ 1 essentially gives us that local mean oscillations of functions in BMO D (w) can be dominated by one of the sparse BMO functions in Section 2.2: Proof.Let Q 0 ∈ D be fixed.We wish to estimate 1 In fact, With Q 0 ∈ D fixed, here we are only looking at S(Q 0 ) := {Q ∈ S : Q ⊂ Q 0 }.We define the collections as sets: so S 2 are the "S-grandchildren" of Q 0 , the second generation of S-cubes in Q 0 .Generally, Note that: k=1 is exactly the set of all x contained in infinitely many elements of S(Q 0 ).We can also see this directly, as the series For ease of notation, denote for now We have: 1 We can apply the same reasoning to each Q 2 ∈ S 2 : and we can conclude inductively Suppose for a moment that θ ≤ 1.Then (A.1) becomes Remark A.1.Thoroughly, we have above a sequence of partial sums Generally, if n < θ ≤ (n + 1) for some n ∈ N: the right hand side of (A.

|Q|
and the inner product Within Q 0 we form the local CZ-decompositions of f and g, and the BMO decomposition of b: Based on E 3 we define Moreover, (b, Finally, let For every Q ⊂ Q 0 , Q E, we have: and (b, h Q ) = ( b, h Q ), so: , where C(n) is the dimensional constant arising from using the John-Nirenberg Theorem.Finally, we have Now we recurse on the R∈E terms in (B.1) and form S(Q 0 ) by adding Q 0 first, E are the Schildren of Q 0 , and so on.The collection S(Q 0 ) satisfies the S-children definition of sparseness, with R∈ch S (Q) |R| ≤ |Q| for all Q ∈ S(Q 0 ), so it is 1  1− -Carleson.So, if we choose = Λ Λ−1 , we have We summarize this below: Proposition B.1.There is a dimensional constant C(n) such that for all a ∈ BMO D , b ∈ BMO D (w), where w is a weight on R n , fixed Q 0 ∈ D and Λ > 1, there is a Λ-Carleson sparse collection S(Q 0 ) ⊂ D(Q 0 ) such that holds for all a with finite Haar expansion, and therefore for all a.This proves Theorem 2.5.

1 |J|
where (τ S ) J := I∈S,I J |I| ≤ Λ, ∀J ∈ D. As discussed in Section 2.3, this gives us an upper bound for norms of sparse operators in terms of norms of paraproducts and martingale transforms, and in fact the equivalence sup S∈Υ D A S L p (w)→L p (w) n,p,Λ sup b∈BMO D Π b L p (w)→L p (w) Since the Haar expansion of b effectively dictates the Haar expansion of Π b (as well as Π * b and Γ b ), this will lead from finite Haar expansion b's to collections in Υ D (R n ).

1. 2 . 1 √|I| (1 1 I 1 I := 1 √ |I| 1 1 I
Haar Functions.Given a dyadic grid D on R, we associate to each I ∈ D the cancellative Haar function h I := h 0 I = + − 1 1 I − ), where I + and I − are the right and left halves of I, respectively.The non-cancellative Haar function is h .The cancellative Haar functions {h I } I∈D form an orthonormal basis for L 2 (R, dx), and an unconditional basis for L p (R), 1 < p < ∞.

1. 4 .
Paraproducts and BMO.We say b ∈ BMO where Λ is the Carleson constant of S. So |A| ≤ 1 k Λ|Q * | for all k ∈ N, and then |A| = 0.Alternatively, since {Q k } is a decreasing nest of sets, |A| = lim k→∞ |Q k |, and lim k→∞ |Q k | = 0 because the series ∞ k=1 |Q k | ≤ Q∈S,Q⊂Q * |Q| ≤ Λ|Q * | converges.* The lemma above ensures that the following definition is sound: with every sparse collection S ∈ Υ D (R n ) with no infinitely increasing chains we associate the function b S := Q∈S 1 1 Q .

1 p− 1 A
p [λ] A p .The same holds for the other paraproducts Π * b and Γ b .Remark 3.1.The result actually follows immediately for Π * b , since

1 )
is sharp (via the one-weight linear A 2 bound for the dyadic square function).The starting point is a simple observation: Given a weight w on R n , the weight itself belongs to BMO(w), with w BMO(w) ≤ 2.

3 . 2 .
which gives us b BMO D (w) ≤ 2 Π b : L 2 (w) → L 2 (w −1 ) , ∀b ∈ BMO D (w).* Now we proceed with the proof of Theorem 3.1, focusing on Π b , with the other paraproducts following similarly.Maximal Truncation of Paraproducts.Let b ∈ BMO D (R n ).Define the maximal truncation of the paraproduct Π b : A A. P  T . Recall that we are given S ∈ Υ D (R n ) and the associated function b S := Q∈S 1 1 Q , and we wish to show that b S BMO D ≤ Λ, where Λ is the Carleson constant of S.