Bloom weighted bounds for sparse forms associated to commutators

In this paper we consider bilinear sparse forms intimately related to iterated commutators of a rather general class of operators. We establish Bloom weighted estimates for these forms in the full range of exponents, both in the diagonal and off-diagonal cases. As an application, we obtain new Bloom bounds for commutators of (maximal) rough homogeneous singular integrals and the Bochner-Riesz operator at the critical index. We also raise the question about the sharpness of our estimates. In particular we obtain the surprising fact that even in the case of Calder\'on--Zygmund operators, the previously known quantitative Bloom weighted estimates are not sharp for the second and higher order commutators.


Introduction
Let S be a sparse family of dyadic cubes, let b ∈ L 1 loc (R n ), m ∈ N and 1 ≤ r < s ≤ ∞.The key object of this paper is the bilinear sparse form defined by B m S,b,r,s (f, g) : This object appears naturally when one studies iterated commutators of various operators T and pointwise multiplication by a function b.Let r < p, q < s and let µ, λ be weights.Our goal is to obtain quantitative weighted L p (µ)×L q ′ (λ 1−q ′ )-bounds for B m S,b,r,s in the Bloom setting [2] both in the diagonal and off-diagonal cases.By the Bloom setting one means that an assumption on b is imposed in terms of the Bloom weight ν depending on µ and λ in a suitable way.
In the following cases the Bloom bounds for B m S,b,r,s have been considered before: • r = 1, s = ∞, m ≥ 1 and p = q [21,22].
The first author was supported by ISF grant no.1035/21.The third author was supported by FONCyT PICT 2018-02501 & by Spanish Government Ministry of Science grant PID2020-113048GB-I00.
Our results below are quantitative and cover all possible combinations of 1 ≤ r < p, q < s ≤ ∞ and m ≥ 1.In particular, the bounds we obtain are new in the following settings: • Limited range: r > 1 or s < ∞.
In the Bloom setting, prior works have been primarily focused on estimates for commutators of Calderón-Zygmund operators, which by [21,22] boil down to estimates for B m S,b,1,∞ .The study of the boundedness in the case m = 1 for these operators in the full range p, q ∈ (1, ∞) has recently been completed by Hänninen-Sinko and the second author [10].For a comprehensive overview of the development of both unweighted and (Bloom) weighted estimates for these commutators, as well a discussion on the necessity of the conditions on b, we direct the reader to the introductions of, e.g., [10,11,12].
Our key application is a quantitative Bloom weighted estimate for iterated commutators of a rather general class of operators.This class includes, for example, Calderón-Zygmund operators, (maximal) rough homogeneous singular integral operators and Bochner-Riesz operators at the critical index.We refer to [18,Remark 4.4] for a further list of operators that fall within the scope of our theory.Note that for various operators on this list, no Bloom weighted (or even unweighted) commutator estimates were known previously.
Our approach will build upon a sparse domination procedure for commutators developed by Rivera-Ríos, the first and third authors [21], with subsequent generalizations by various authors.We will proceed in 3 steps: (1) In Section 3, we will prove that iterated commutators of certain sublinear operators T can be dominated by two sparse forms: B m S,b,r,s (f, g) and the dual form B m S,b,s ′ ,r ′ (g, f ).Our key novel point here is in Lemma 3.4, which allows one to reduce m + 1 sparse forms to only 2 sparse forms.
(2) We prove Bloom weighted estimates for B m S,b,r,s (f, g).To do so, we first extend a result of Li [23] and Fackler-Hytönen [8] to certain fractional sparse forms in Section 4. Afterwards, in Section 5 we combine the proof strategy of Hänninen, Sinko and the second author [10] with the change of measure formula of Cascante-Ortega-Verbitsky [4] to estimate B m S,b,r,s (f, g) in the Bloom setting.Furthermore, in the case q ≤ p and m ≥ 2, we provide a second Bloom weighted estimate for B m S,b,r,s (f, g) using a proof strategy suggested by Li [24].In particular, Theorem 5.1 presents two incomparable quantitative bounds based on the approaches from [22] and [24].
(3) Combining the first two steps, in Section 6, we obtain quantitative Bloom weighted estimates for iterated commutators.We apply this result to Calderón-Zygmund operators, (maximal) rough homogeneous singular integral operators and Bochner-Riesz operators at the critical index.
Since our estimates are quantitative, it is natural to ask about their sharpness.Here we encounter with an interesting phenomenon, which is new even for Calderón-Zygmund operators.To be more precise, in [22], the first and the third authors jointly with Rivera-Ríos showed for a Calderón-Zygmund operator T and for m ≥ 1 (extending their previous work [21] for m = 1) that where BMO ν stands for the weighed BMO space with weight ν := (µ/λ) 1/pm , and T m b is the m-th order commutator of T with a locally integrable function b.
Observe that the notion of sharpness in the Bloom setting (or in the two-weight setting, in general) has not been defined before.It is easy to see that a bound by [λ] αp Ap [µ] βp Ap is stronger than a bound by [λ] p and β p ≤ β ′ p and at least one of these inequalities is strict.In the case if, for example, α p < α ′ p and β p > β ′ p , the bounds will be incomparable.This leads us to the following definition.Definition 1.1.Let p ∈ (1, ∞), µ, λ ∈ A p and let T be an operator.We say that the estimate βp Ap is sharp if neither of the exponents α p and β p can be decreased.
Having this definition at hand, we are ready to present our result about the sharpness of (1.1).This result comes as a surprise to us because it says that the estimate (1.1) is sharp for all 1 < p < ∞ only if m = 1.To be more precise, we have the following.].We shall see in Section 6 that a similar phenomenon with two incomparable bounds holds for a large class of operators.
Notation.We will make extensive use of the notation " " to indicate inequalities up to an implicit multiplicative constant.These implicit constants may depend on p, q, n, m, but not on any of the functions under consideration.If these implicit constants depend on the weights µ, λ, this will be denoted by " µ,λ ".

Preliminaries
2.1.Dyadic lattices.Denote by Q the set of all cubes Q ⊂ R n with sides parallel to the axes.For a cube Q ∈ Q with side length ℓ(Q) and α > 0 we denote the cube with the same center as Q and side length αℓ(Q) by αQ.
Given a cube Q ∈ Q, denote by D(Q) the set of all dyadic cubes with respect to Q, that is, the cubes obtained by repeated subdivision of Q and each of its descendants into 2 n congruent subcubes.Following [19, Definition 2.1], a dyadic lattice D in R n is any collection of cubes such that (i) Any child of Q ∈ D is in D as well, i.e.D(Q) ⊆ D.
(ii) Any Q ′ , Q ′′ ∈ D have a common ancestor, i.e. there exists a there exists a cube Q ∈ D containing K. Throughout the paper, D will always denote a dyadic lattice.Definition 2.1.Let η ∈ (0, 1) and let S ⊂ Q be a family of cubes.We say that S is η-sparse if, for every cube Q ∈ S, there exists a subset and the sets {E Q } Q∈S are pairwise disjoint.We will omit the sparseness number η when its value is nonessential.

For a cube
We define the maximal operator by 2.2.Weights.By a weight w we mean a non-negative w ∈ L 1 loc (R n ).For 1 < p < ∞ we say that w belongs to the Muckenhoupt A p -class and write For 1 ≤ r < ∞ we say that w belongs to reverse Hölder class and write Furthermore, we say that w belongs to the Muckenhoupt A ∞ -class and write We will frequently use that by the definition of the A p -constant, we have The following quantitative self-improvement lemma from [14] will play a key role in our applications.Proposition 2.2 ([14, Theorem 1.1 and 1.2]).There exists a constant c n > 0 such that for w ∈ A p with 1 < p < ∞ we have

A sparse domination principle for commutators
In this section we will prove a general sparse domination principle for iterated commutators, following the line of research started in [21] by Rivera-Ríos and the first and the third authors.In order to state our result, let us introduce some notation.
Given a linear operator T and b ∈ L 1 loc (R n ), define the first order commutator T 1 b by Next, for m ∈ N, m ≥ 2, define higher order commutators T m b inductively by It is easy to see that Assume now that T is a general, not necessarily linear, operator.Then we use formula (3.1) as the definition of T m b .For 1 ≤ s ≤ ∞ we define the sharp grand maximal truncation operator where and the supremum is taken over all Q ∈ Q containing x.
We will use the following boundedness property of T and M # T,s .Definition 3.1.Given an operator T and r ∈ [1, ∞), we say that T is locally weak L r -bounded if there exists a non-increasing function ϕ T,r : (0, 1) → [0, ∞) such that for any cube This definition was given in [20] and was called the W r property of T .Note that the usual weak L r -boundedness of T implies the local weak L r -boundedness of T with Moreover, if T is locally weak L r 0 -bounded for some r 0 ∈ [1, ∞), it is locally weak L r -bounded for all r > r 0 by Hölder's inequality with ϕ T,r (λ) = ϕ T,r 0 (λ).
The main result of this section is the following abstract sparse domination principle for iterated commutators.Theorem 3.2.Let 1 ≤ r < s ≤ ∞, m ∈ N and let T be a sublinear operator.Assume that T and M # T,s are locally weak L r -bounded.Then there exist C m,n > 1 and λ m,n < 1 so that, for any f, g ) .We refer to [18,Remark 4.4] for a list of operators satisfying the assumptions of Theorem 3.2.Theorem 3.2 is an immediate corollary of the following two statements.Theorem 3.3.Under the assumptions of Theorem 3.2 we have where C is given by (3.2) Indeed, note that Lemma 3.4 allows us to reduce the summation over k = 0, . . ., m in Theorem 3.3 to the two extreme terms k = 0 and k = m, yielding the formulation of Theorem 3.2.
Before turning to the proofs, let us mention a brief history of the above results.
• In the case where T is a Dini-continuous Calderón-Zygmund operator, m = 1, r = 1 and s = ∞, Theorem 3.2 goes back to Rivera-Ríos and the first and the third authors [21].• In the case where m ≥ 1 and T is a generalized Hörmander singular integral operator, the corresponding version of Theorem 3.3 was obtained by Ibañez-Firnkorn and Rivera-Ríos [15].• The closest precursors of Theorem 3.2 were obtained by -Rivera-Ríos [28, Theorem 3.1] in the case m = 1.We note that in this work the bilinear maximal operator M T (f, g), introduced in [17], was used instead of M # T,s .-Ibañez-Firnkorn and Rivera-Ríos [16,Theorem 4.4] in the case m ≥ 1 and with M # T,∞ .• For a general account of similar sparse domination results we refer to our recent work [18].Comparing to [28,Theorem 3.1] and [16,Theorem 4.4], our novel points are the following.
T,∞ , we deal with a more flexible operator M # T,s .Here we continue the line of research originated in [18,20,26], where various variants of M # T,s were considered.
• We use a local weak L r -boundedness assumption, originating from [20], rather than the usual weak L r -boundedness assumption on T and M # T,s .• Our most important novel point in this section is in Lemma 3.4, which seems to be new.This lemma allows us to significantly simplify the main applications of Theorem 3.2 to quantitative weighted norm inequalities.The proof of Lemma 3.4 is quite elementary and we therefore present it first.
Then c k for 0 ≤ k ≤ m can be written in the form .
From this, we obtain the conclusion by using the estimate along with Minkowski's inequality.
Next we turn to the proof of Theorem 3.3.Its proof is based on the well-known ideas developed in the previous works (e.g., [17,18,20,26]).
Proof of Theorem 3.3.Let Q ∈ Q be a cube that contains the supports of f and g.We will show that there exists a 1  2 -sparse family where C is given by (3.2).Taking S = {3P : P ∈ F } afterwards yields the result.We construct the family F ⊂ D(Q) inductively.Set F 0 = {Q}.Next, given a collection of pairwise disjoint cubes F j , let us describe how to construct F j+1 .
Fix a cube P ∈ F j .For k = 0, . . ., m denote and consider the sets |P |, and the same bound holds for |M k (P )|.Since the maximal operator M r is weak L r -bounded with constant independent of r, there exists a c n,m > 0 such that We apply the local Calderón-Zygmund decomposition to χ Ω(P ) at height 1  2 n+1 .We obtain a family of pairwise disjoint cubes S P ⊆ D(P ) such that |Ω(P ) \ P ′ ∈S P P ′ | = 0 and for every P ′ ∈ S P , (3.4) In particular, it follows that (3.5) We define F j+1 = ∪ P ∈F j S P .Setting F = ∪ ∞ j=0 F j , we note by (3.5) that F is 1 2 -sparse.Now, by iteration, to prove (3.3) it suffices to show for j ∈ N and where C is given by (3.2).Set F j := ∪ P ∈F j P .Noting that it thus suffices to show that We first consider the first term on the left-hand side of (3.6).Since where ).Now consider the second term in (3.6).Fix P ′ ∈ F j+1 such that P ′ ⊆ P and denote Then, for y ∈ P ′ to be specified later we have (3.8) and consider the sets Set Ω(P ′ ) := ∪ m k=0 Ω k (P ′ ), for which we have | Ω(P ′ )| ≤ 1 4 |P ′ |.Now, define the good part of the cube P ′ as Then, by (3.4), we have and for all y ∈ G P ′ we have Hence, for all y ∈ G P ′ , we have From this, integrating (3.8) over y ∈ G P ′ , using Hölder's inequality and the definition of the set M(P ), we obtain where ) .By Hölder's inequality, for any q ∈ [1, ∞), Therefore, which, along with (3.7), proves (3.6).This completes the proof.
Remark 3.5.Under the assumptions of Theorem 3.2 and by the "three lattice theorem" (see, e.g., [19]), there exist 3 n dyadic lattices D j so that for any f, g where C is given by (3.2).

Weighted estimates for fractional sparse forms
In the next section, we will prove quantitative Bloom weighted estimates for the sparse forms in the conclusion of Theorem 3.2.As a preparation, we establish a weighted estimate for fractional sparse forms in this section.
Remark 4.2.Following the notation of Nieraeth [27], for 0 < r < p < s ≤ ∞ and a weight w, define Upon inspection of the proof, it is clear that the conclusion of Theorem 4.1 can be more symmetrically phrased as for all weights w such that [w] q,(r,s) < ∞.
As a direct corollary of Theorem 4.1, in the case r = 1 and s = ∞, we recover [10, Lemma 3.2], which is a special case of [8, Theorem 1.1]: p − 1 q and let w ∈ A q .For any sparse family of cubes S ⊂ D and f ∈ L p (w p/q ) we have In particular, we recover the well-known bound The proof of Theorem 4.1 is based on three main ingredients, the first of which is a very slight generalization of a result of Li [23].
Theorem 4.4 ([23, Theorem 1.2]).Let 1 < p ≤ q < s ≤ ∞ and r ∈ (0, p).Let w and σ be weights and λ Q ≥ 0 for any Q ∈ D. Let S ⊂ D be a sparse family of cubes and suppose that N is the best constant such that Proof.In the case r ≥ 1, the theorem is exactly [23, Theorem 1.2].The case r < 1 is proven analogously.Indeed, only the proof of the equivalence [23, (2.1)] ⇔ [23, (2.2)] needs to be adapted.To handle the average of f in the implication ⇐, one replaces the maximal operator argument by Hölder's inequality.Conversely, for the implication ⇒, one replaces Hölder's inequality by a maximal operator argument, using the boundedness of M S 1,u on L p (u).In order to estimate the two terms in (4.2), we will use the following norm equivalence from Cascante-Ortega-Verbitsky [4].
The final ingredient in the proof of Theorem 4.1 is the following result from Fackler-Hytönen [8].
Combining these three ingredients with the proof strategy from [8], we can now prove Theorem 4.1.
Proof of Theorem 4.1.A direct computation shows that, in the notation of Theorem 4.4, we have u = w − r q−r , v = w s s−q and Let us first consider first the testing condition in (4.2).We will show that By Lemma 4.5, we have where For δ > 0 we have Our goal now is to use Lemma 4.6.Its assumptions imply the following restrictions on δ: Moreover, the assumption α + β + γ ≥ 1 of Lemma 4.6 holds trivially because α + β + γ = 1 + α r .We conclude that δ ∈ 1 q − 1 s , 1 − 1 s and the set of δ satisfying all restrictions will be non-empty if It is easily seen that this estimate is true for all r < q < s and α ≥ 0.
Taking δ > 0 satisfying the above restrictions and applying Lemma 4.6, we obtain Therefore, Further, which, along with the previous estimate, proves (4.3).Now consider the second testing condition in (4.2).Let us show that Again by Lemma 4.5, we have where Ψ(Q) is as in (4.4).By the above estimate for Ψ(Q), from which (4.5) follows.
Combining the two estimates for the testing conditions, inequalities (4.3) and (4.5), we obtain , where N is as in Theorem 4.4.From this, since RHt , we obtain the conclusion with , 1 q−r .

Bloom weighted bounds for sparse forms associated to commutators
In this section we consider one of the sparse forms in the conclusion of Theorem 3.2, namely, B m S,b,r,s (f, g) as defined in the introduction.Let us start with some definitions.Given b ∈ L 1 loc (R n ), a weight ν and α ≥ 0, define the weighted, fractional BMO-seminorm as We omit α from our notation if α = 0. Furthermore, given a cube and the weighted sharp maximal function Assume that µ ∈ A p/r and λ ∈ A q/r ∩ RH (s/q) ′ .Set α := − 1 t := 1 pm − 1 qm , α + := max{α, 0} and define the Bloom weight where for any p, q, m (5.1) [µ] Observe that the sense of (5.2) is that in the case q ≤ p and m ≥ 2 it provides an additional bound for C(µ, λ), which is incomparable with (5.1), in general.See Section 6 for a further discussion of this phenomenon.
Before turning to the proof, let us discuss some particular cases of Theorem 5.1.
Remark 5.3.Suppose that s = ∞ and thus [λ] RH (s/q) ′ = 1.In the diagonal case p = q, we have α = 0. So, in this case, If we additionally assume that rm ∈ N, we get In particular, for r = 1, we have the first of which was obtained in [22].Note that the second estimate is, in general, incomparable to the first.
Remark 5.4.In the spirit of Remark 4.2, we note that the conclusion of Theorem 5.1 can be replaced by for all weights µ such that [µ] p,(r,∞) < ∞ and weights λ such that [λ] q,(r,s) < ∞.
Several statements below will be needed to prove Theorem 5.1, starting with the following lemma from Rivera-Ríos and the first and third authors [21].We also need the following additional result from Cascante-Ortega-Verbitsky [4].
We are now ready to prove Theorem 5.1 in the case p ≤ q.
For δ ≥ 0 we denote in which we omit δ if δ = 0. Using Lemma 4.5 with the weight w = |f | r and Minkowski's inequality, we have =:
From this and from the above expression for C(µ, λ), the values of β µ 1 , and β λ 1 follow by direct computation.
We now turn to the case q ≤ p.We start with the estimate in (5.1), which works for any m ≥ 1.
Proof of (5.1) in Theorem 5.1 in the case q ≤ p.By Lemma 5.5, there exists a sparse collection of cubes S ⊆ S ′ ⊂ D such that for any Q ∈ S, Let k := ⌊rm⌋ and γ := rm−(k −1) ∈ [1, 2).Applying subsequently Lemma 5.6 (k − 1) times yields where we omitted the assumption P 1 , . . ., P k ∈ S ′ from our notation for brevity.Define . By Lemma 4.5 and Hölder's inequality, we can estimate

Weighted bounds for commutators
In this final section we will apply the results from the previous sections to concrete operators.Let us first formulate a general result, in a qualitative form, which is an immediate corollary of Theorems 3.2 and 5.1 and duality.Theorem 6.1.Let 1 ≤ r < p, q < s ≤ ∞ and m ∈ N. Let T be a sublinear operator and b ∈ L 1 loc (R n ).Assume the following conditions: . By Remark 3.5, there exist 3 n dyadic lattices D j and sparse families S j ⊂ D j such that Therefore, the claims follow from applying Theorem 5.1 twice, directly and dually.We observe that in order to apply Theorem 5.1 to the dual terms B m S j ,b,s ′ ,r ′ (g, f ) in the dual spaces, we need the conditions which follow directly from our assumptions on µ and λ.
We refer to [18,Remark 4.4] for a list of operators satisfying the assumptions, and thus the conclusion, of Theorem 6.1.Note that even unweighted bounds for commutators with some of the operators on that list were previously unknown.
Next, we examine a quantitative form of Theorem 6.1 in an important particular case of interest.Theorem 6.2.Let 1 < p < ∞ and m ∈ N. Let T be a sublinear operator and b ∈ L 1 loc (R n ).Assume the following conditions: • Suppose that for all 1 < r < 2 < s < ∞, both T and M # T,s are locally weak L r -bounded, and where λ m,n > 0 is the constant provided by Theorem 3.2 and . By Remark 3.5, for any 1 < r < 2 < s < ∞ there exist 3 n dyadic lattices D j and sparse families S j ⊂ D j such that (6.1) Suppose that 1 < r < min( m+1 m , p) and max(p, m + 1) < s < ∞.Then ⌊rm⌋ = m and ⌊s ′ m⌋ = m, and hence, by Theorem 5.1, or, if m ≥ 2, alternatively Moreover, Theorem 5.1 also yields . Now, let c n > 0 be the constant in Proposition 2.2 and define Then we have for all 1 < r ≤ r and s ≤ s < ∞ that and, using (2.1), also Therefore, if r = 1 2 (min( m+1 m , p, r) + 1) and s = 2 max(p, m + 1, s), combining the above estimates with (6.1) completes the proof.
If T and M # T,s are both locally weak L 1 -bounded, we can take ψ in Theorem 6.2 constant in the first coordinate, which we record as the following corollary.Corollary 6.3.Let 1 < p < ∞ and m ∈ N. Let T be a sublinear operator and b ∈ L 1 loc (R n ).Assume the following conditions: • Suppose that for all 2 < s < ∞, both T and M # T,s are locally weak L 1 -bounded, and , where λ m,n is a constant provided by Theorem 3. 2 and ψ • Let µ, λ ∈ A p and define the Bloom weight ν := ( µ λ ) BMOν , where K p (µ, λ) is as in Theorem 6.2 and Proof.Since T and M # T,s are locally weak L 1 -bounded, they are locally weak L r bounded for all r > 1, and Applying Theorem 6.2 with ψ(r ′ , s) := ϕ T,1 (λ m,n ) + ψ(s) finishes the proof.
6.1.Calderón-Zygmund operators.As discussed in the introduction, Bloom weighted estimates for commutators have been widely studied for Calderón-Zygmund operators.As a first application of our results, we will compare the weighted estimates that we obtained in the diagonal p = q case for Calderón-Zygmmund operators and discuss their sharpness in the sense of Definition 1.1.In particular, let us prove Theorem 1.2.
Recall that a linear operator T is called Dini-continuous Calderón-Zygmund operator if it is L 2 -bounded and for f ∈ L ∞ c (R n ) has a representation where Proof of Theorem 1.2.It is well-known that T and M # T,∞ are of weak L 1 -bounded (see, e.g., [9,20]).Therefore, after normalizing such that b BMOν = 1, by Corollary 6.3 we have .
and, if m ≥ 2, also Ap .First let us take m ≥ 2. Then (6.3) implies for p ≥ 2 that , and therefore we obtain that (6.2) is not sharp in the sense of Definition 1.1.Moreover, for 1 < p ≤ 2 (6.3) implies p−1 .Therefore, we again obtain that (6.2) is not sharp.Suppose now that m = 1.Let us show that in this case (6.2) is sharp for all p ∈ (1, ∞).By duality, the sharpness of the exponent max ( δ 1+1/p .Therefore, the sharpness of the exponent 1 p−1 would follow if we show that (6.4) By the unweighted L p boundedness of H, On the other hand, 1 δ |x| δ−1 |h 2 (x)| for all x ∈ (0, 1), and therefore which proves (6.4).
Let us show now that the exponent 1 p−1 of [µ] Ap cannot be decreased.The example is very similar.Define λ := 1 and µ(x Therefore, the sharpness of the exponent 1 p−1 would follow if we show that (6.5) Exactly as above, h 2 L p 1 δ 1/p .On the other hand, 1 δ |x| δ−1 p h 1 (x) for all x ∈ (0, 1), and therefore, which proves (6.5).This completes the proof.
As we mentioned in the introduction, Theorem 1.2 leaves open a question about the sharpness of (1.1) when m ≥ 2 and p ∈ [ 1+3m 2m , 1+3m m+1 ].Indeed, our example is based on the obvious fact that ν ∈ BMO ν .In the case m = 1 the choice b = ν shows the sharpness of (1.1) for all 1 < p < ∞.However, it is easy to check that in the case m ≥ 2 the same choice is not enough in order to show the sharpness of (1.1).6.2.Further applications.We conclude this paper by applying our results to several other concrete examples of operators, for which (quantitative) Bloom weighted bounds of their commutators have not been known before.
Given an operator T and s ≥ 1, define the (non-sharp) grand maximal truncation operator M T,s by Observe that (6.6) M # T,s f ≤ 2 M T,s f.Example 6.4.Consider a class of rough homogeneous singular integrals defined by for Ω ∈ L ∞ (S n−1 ) with zero average over the sphere.
It is a well-known result of Seeger [29] that T Ω is of weak type (1, 1).Moreover, it was shown by the first author [17] that for s > 2, the grand maximal truncation operator M T Ω ,s is weak L 1 -bounded with Therefore, by (6.6), M # T Ω ,s satisfies the same bound.From this, by Corollary 6.3, we obtain for µ, λ ∈ A p and ν := ( µ λ ) Let us deduce from the recent work of Tao-Hu [32] that (6.8) ϕ M #
Then we obtain that where Using that |T i f | Mf for i = 1, 2, we obtain M B (n−1)/2 ,s f (x) Mf (x) + M T,s f (x), x ∈ R n .
Therefore, it suffices to prove (6.9) for M T,s .In order to do that, we fix an integer N and further decompose T = S 1 + S 2 , where f * (ψ j K)(x).
Let ε := 2 −N .It was shown in [31] that S 1 is a Dini-continuous Calderón-Zygmund operator with Dini-constant bounded by log 1 ε , and that S 2 L 2 →L 2 ε α for some α ∈ (0, 1].Moreover, the kernels ψ j K have radial smoothness (as was observed in [25]), which makes it possible to apply Seeger's machinery from [29].Using all these ingredients, the proof goes through as in [17] by the first author.Remark 6.7.Let p, q ∈ (1, ∞), m ∈ N and b ∈ L 1 loc (R n ).By a similar argument as employed in Subsection 6.1 and Examples 6.4, 6.5 and 6.6, now using Theorem 6.1, we also get for all µ ∈ A p and λ ∈ A q and ν 1+ 1

Lemma 5 . 5 (
[21, Lemma 5.1]).Let b ∈ L 1 loc (R n ) and let S ⊂ D be a sparse family.There exists a sparse family S ′ ⊂ D such that S ⊆ S ′ and for every cubeQ ∈ S ′ , |b − b Q |χ Q P ∈S ′ :P ⊆Q |b − b P | P χ P .