Mathematics
Computer Science-2021.gz:143443
Computer Science
c7b7061be9145510346c2349da4dd596998f67a2
Sparse bounds for maximal rough singular integrals via the Fourier transform

We prove that the class of convolution-type kernels satisfying suitable decay conditions of the Fourier transform, appearing in the works of Christ, Christ-Rubio de Francia, and Duoandikoetxea-Rubio de Francia gives rise to maximally truncated singular integrals satisfying a sparse bound by $(1+\varepsilon,1+\varepsilon)$-averages for all $\varepsilon>0$, with linear growth in $\varepsilon^{-1}$. This is an extension of the sparse domination principle by Conde-Alonso, Culiuc, Ou and the first author to maximally truncated singular integrals. Our results cover the rough homogeneous singular integrals $T_\Omega$ on $\mathbb{R}^d$ with bounded angular part $\Omega $ having vanishing integral on the $(d-1)$-dimensional sphere. Consequences of our sparse bound include novel quantitative weighted norm estimates as well as Fefferman-Stein type inequalities. In particular, we obtain that the $L^2(w)$ norm of the maximal truncation of $T_\Omega$ depends quadratically on the Muckenhoupt constant $[w]_{A_2}$, extending a result originally by Roncal, Tapiola and the second author. A suitable convex-body valued version of the sparse bound is also deduced and employed towards novel matrix weighted norm inequalities for the maximal truncated rough homogeneous singular integrals. Our result is quantitative, but even the qualitative statement is new, and the present approach via sparse domination is the only one currently known for the matrix weighted bounds of this class of operators.


Francesco D P , T P. H K L
Sparse bounds for maximal rough singular integrals via the Fourier transform Tome , n o ( ), p. -.

Introduction and main results
Let η ∈ (0, 1). A countable collection S of cubes of R d is said to be η-sparse if there exist measurable sets {E I : I ∈ S} such that E I ⊂ I, |E I | η|I|, I, J ∈ S, I = J =⇒ E I ∩ E J = ∅.
Let T be a sublinear operator mapping the space L ∞ 0 (R d ) of complexvalued, bounded and compactly supported functions on R d into locally integrable functions. We say that T has the sparse (p 1 , p 2 ) bound [10] if there exists a constant C > 0 such that for all f 1 , f 2 ∈ L ∞ 0 (R d ) we may find a 1 2 -sparse collection S = S(f 1 , f 2 ) such that in which case we denote by T (p1,p2),sparse the least such constant C. As customary, Estimating the sparse norm(s) of a sublinear or multisublinear operator entails a sharp control over the behavior of such operator in weighted L pspaces; this theme has been recently pursued by several authors, see for instance [1,8,19,20,21,33]. This sharp control is exemplified in the following proposition, which is a collection of known facts from the indicated references.
Proposition 1.1. -Let T be a sublinear operator on R d mapping L ∞ 0 (R d ) to L 1 loc (R d ). Then the following hold.

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In this article, we are concerned with the sparse norms (1.1) of a class of convolution-type singular integrals whose systematic study dates back to the celebrated works by Christ [4], Christ-Rubio de Francia [5], and Duoandikoetxea-Rubio de Francia [12], admitting a decomposition with good decay properties of the Fourier transform. To wit, let {K s : R d → C, s ∈ Z} be a sequence of (smooth) functions with the properties that supp K s ⊂ A s := {x ∈ R d : 2 s−4 < |x| ∞ < 2 s−2 }, for some α > 0. We consider truncated singular integrals of the type  Theorem 1.2 entails immediately a variety of novel corollaries involving weighted norm inequalities for the maximally truncated operators T . In addition to, for instance, those obtained by suitably applying the points of Proposition 1.1, we also detail the quantitative estimates below, whose proof will be given in Section 7. (1) Let 1 < p < ∞, w ∈ A p be a Muckenhoupt weight and σ = w − 1 p−1 be its A p Muckenhoupt dual. We have the estimate (2) The Fefferman-Stein type inequality holds with implicit constant possibly depending on d only.
holds for 1 q < p < ∞ and w ∈ A q , with implicit constant possibly depending on p, q and d only.
(4) The following Coifman-Fefferman type inequality holds for all ε > 0 with implicit constant possibly depending on p and d only.
The above estimate, in particular, is stronger than the uniform weak type (1, 1) for the operators T Ω,δ , a result originally due to Seeger [31]. As the weak type (1, 1) of T Ω, under no additional smoothness assumption on Ω is a difficult open question, estimating the (1, 1 + ε) sparse norm of T Ω, as in (1.8) seems out of reach.
The study of sharp weighted norm inequalities for T Ω,δ (the uniformity in δ is of course relevant here) was initiated in the recent article [16] by

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Hytönen, Roncal and Tapiola. Improved quantifications have been obtained in [7] as a consequence of the domination result (1.8), and further weighted estimates (including a Coifman-Fefferman type inequality, that is a norm control of T Ω,δ by M on all L p (w), 0 < p < ∞ when w ∈ A ∞ ) have been later derived from (1.8) in the recent preprint by the third named author, Pérez, Roncal and Rivera-Rios [28].
Although (1.7) is a bit weaker than (1.8), we see from comparison of (1.4) from Theorem 1.3 with the results of [7,28] that the quantification of the L 2 (w)-norm dependence on [w] A2 entailed by the two estimates is the same (quadratic); on the contrary, for p = 2, (1.8) yields the better estimate T Ω,δ L p (w) [w] p Ap . We also observe that the proof of the mixed estimate (1.5) actually yields the following estimate for the non-maximally truncated operators, improving the previous estimate given in [28] T Finally, we emphasize that (1.7) also yields a precise dependence on p of the unweighted L p operator norms. Namely, from the sparse domination, we get with absolute dimensional implicit constant, which improves on the implicit constants in [12]. Moreover, we note that the main result of [29] implies that if (1.9) is sharp, then our quantitative weighted estimate (1.4) is also sharp.
Remark 1.5. -We note that Theorem 1.3 holds with T being the commutator of a Calderón-Zygmund operator with a BMO symbol. This follows by comparing Theorem 1.2 with the sparse domination formula for commutators from [24], with the help of the John-Nirenberg inequality.

Matrix weighted estimates for vector valued rough singular integrals
Let (e ) n =1 , · , · F n and | · | F n be the canonical basis, scalar product and norm on F n over F, where F ∈ {R, C}. A recent trend in Harmonic Analysis (see, among others, [2, 3,9,15,30]) is the study of quantitative matrix weighted norm inequalities for the canonical extension of the (integral) linear operator T x ∈ R d TOME 70 (2020), FASCICULE 5 to F n -valued functions f . In Section 6 of this paper, we introduce an L p , p > 1, version of the convex body averages first brought into the sparse domination context by Nazarov, Petermichl, Treil and Volberg [30], and use them to produce a vector valued version of Theorem 1.2. As a corollary, we obtain quantitative matrix weighted estimates for the maximal truncated vector valued extension of the rough singular integrals T Ω,δ from (1.6). In fact, the next corollary is a special case of the more precise Theorem 6.4 from Section 6.
Corollary 1.6. -Let W be a positive semidefinite and locally integrable L(F n )-valued function on R d and T Ω,δ be as in (1.6). Then with implicit constant depending on d, n only, where the matrix A 2 constant is given by .
As the left hand side of (1.10) dominates the matrix weighted norm of the vector valued maximal operator first studied by Christ and Goldberg in [6], the finiteness of [W ] A2 is actually necessary for the estimate to hold. To the best of the authors' knowledge, Theorem 6.4 has no predecessors, in the sense that no matrix weighted norm inequalities for vector rough singular integrals were known before, even in qualitative form. At this time we are unable to assess whether the power 5 2 appearing in (1.10) is optimal. For comparison, if the angular part Ω is Hölder continuous, the currently best known result [30] is that (1.10) holds with power 3 2 ; see also [9].

Strategy of proof of the main results
We will obtain Theorem 1.2 by an application of an abstract sparse domination principle, Theorem 3.3 from Section 3, which is a modification of [7,Theorem C]. At the core of our approach lies a special configuration of stopping cubes, the so-called stopping collections Q, and their related atomic spaces. The necessary definitions, together with a useful interpolation principle for the atomic spaces, appear in Section 2. In essence, Theorem 3.3 can be summarized by the inequality where the supremum is taken over all stopping collections Q and all measurable linearizations of the truncation parameters t 1 , t 2 , and Q t2 t1 are suitably adapted localizations of (the adjoint form to the linearized versions of) T . In Section 4, we prove the required uniform estimates for the localizations Q t2 t1 coming from Dini-smooth kernels. The proof of Theorem 1.2 is given in Section 5, relying upon the estimates of Section 4 and the Littlewood-Paley decomposition of the convolution kernels (1.2) whose first appearance dates back to [12].
Remark 1.7. -We remark that while this article was being finalized, an alternative proof of (1.8) was given by Lerner [23]. It is of interest whether the strategy of [23], relying on bumped bilinear grand local maximal functions, can be applied towards estimate (1.7) as well.

Notation
With q = q q−1 we indicate the Lebesgue dual exponent to q ∈ (1, ∞), with the usual extension 1 = ∞, ∞ = 1. The center and the (dyadic) scale of a cube Q ∈ R d will be denoted by c Q and s Q respectively, so that |Q| = 2 s Q d . We use the notation for the p-Hardy Littlewood maximal function and write M in place of M 1 . Unless otherwise specified, the almost inequality signs imply absolute dimensional constants which may be different at each occurrence.

Stopping collections and interpolation in localized spaces
The notion of stopping collection Q with the top (dyadic) cube Q has been introduced in [7, Section 2]: we proceed by recalling the relevant definitions. A stopping collection Q with top Q is a collection of pairwise disjoint dyadic cubes contained in 3Q and satisfying suitable Whitney type properties. More precisely, We proceed with the definition of the localized spaces Y p (Q), X p (Q), where L stands for the (non-dyadic) 2 5 -fold dilate of L, and that X p (Q) is the subspace of Y p (Q) of functions satisfying Finally, we write b ∈Ẋ p (Q) if b ∈ X p (Q) and each b L has mean zero. We will omit (Q) from the subscript of the norms whenever the stopping collection Q is clear from context. There is a natural interpolation procedure involving the Y p -spaces. We do not strive for the most general result but restrict ourselves to proving a significant example, which is also of use to us in the proof of Theorem 1.2.
Proposition 2.1. -Let B be a bisublinear form and A 1 , A 2 be positive constants such that the estimates

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Proof. -We may assume A 2 < A 1 , otherwise there is nothing to prove. We are allowed to normalize A 1 = 1. Fixing now 0 < ε < 1, so that 1 < p < 2, it will suffice to prove the estimate implied constant depending on dimension only. Let λ 1 to be chosen later. Using the notation f >λ := f 1 |f |>λ , we introduce the decompositions which verify the properties We have used that b 1 is supported on the union of the cubes Q ∈ Q and has mean zero on each Q, and therefore g 1 has the same property, given that f 1 ∈Ẋ p (Q). Therefore which yields (2.6) with the choice λ = A −1 2 .

A sparse domination principle for maximal truncations
We consider families of functions and associate to them the linear operators FASCICULE 5 and their sublinear maximal versions We assume that there exists 1 < r < ∞ such that For pairs of bounded measurable functions t 1 , t 2 : R d → Z, we also consider the linear operators In consequence, for the linearized versions defined in (3.4) we have . A related word on notation: we will be using linearizations of the type With this we mean we are using the constant function equal to s Q as our upper truncation function. Finally, we will be using the notations t 2 ∧s Q for the linearizing function x → min{t 2 (x), s Q } and t 1 ∨ s L for the linearizing function x → max{t 1 (x), s L }.
Given two bounded measurable functions t 1 , t 2 and a stopping collection Q with top Q, we define the localized truncated bilinear forms Remark 3.2. -Note that we have normalized by the measure of Q, unlike the definitions in [7, Section 2]. Observe that as a consequence of the support assumptions in (3.1) and of the largest allowed scale being s Q , we have Within the above framework, we have the following abstract theorem.

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hold uniformly over all bounded measurable functions t 1 , t 2 , and all stopping collections Q. Then Proof. -The proof follows essentially the same scheme of [7, Theorem C]; for this reason, we limit ourselves to providing an outline of the main steps.
Step 1. Auxiliary estimate. -First of all, an immediate consequence of the assumptions of the Theorem is that the estimate Step 2. Initialization. -The argument begins as follows. Fixing f j ∈ L pj (R d ), j = 1, 2 with compact support, we may find measurable functions t 1 , t 2 which are bounded above and below and a large enough dyadic cube Q 0 from one of the canonical 3 d dyadic systems such that supp and we clearly can replace f 2 by |f 2 | in what follows.
Step 3. Iterative process. -Then, the argument proceeds via iteration over k of the following construction, which follows from (3.9) and the Calderón-Zygmund decomposition and is initialized by taking S k = {Q 0 } for k = 0. Given a disjoint collection of dyadic cubes Q ∈ S k with the further Whitney property that (2.3) holds for S k in place of Q, there exists a further collection of disjoint dyadic cubes L ∈ S k+1 such that TOME 70 (2020), FASCICULE 5 • (2.3) for S k in place of Q continues to hold, • each subcollection S k+1 (Q) = {L ∈ S k+1 : L ⊂ 3Q} is a stopping collection with top Q, and for which for all Q ∈ S k there holds More precisely, S k+1 is composed by the maximal dyadic cubes L such that for a suitably chosen absolute large dimensional constant C. This construction, as well as the Whitney property (2.3) results into guaranteeing that T k := ∪ k κ=0 S κ is a sparse collection for all k. When k =k is such that inf{s Q : Q ∈ Sk} < inf t 1 , the iteration stops and the estimate is reached. This completes the proof of Theorem 3.3.

Preliminary localized estimates for the truncated forms (3.5)
We begin by introducing our notation for the Dini constant of a family of kernels [K] as in (3.1). We write The estimates contained within the lemmata that follow are meant to be uniform over all measurable functions t 1 , t 2 and all stopping collections Q. The first one is an immediate consequence of the definitions: for a full proof, see [7,Lemma 2.3].
The second one is a variant of [7, Lemma 3.2]; we provide a full proof.
Proof. -We consider the family K fixed and use the simplified notation Q t2 t1 in place of Q[K] t2 t1 , and similarly for the truncated operators T [K]. By horizontal rescaling we can assume |Q| = 1. Let b ∈Ẋ 1 . Recalling the definition (2.5) and using bilinearity of Q t2 t1 it suffices for each stopping cube R ∈ Q to prove that as b R 1 |R| b Ẋ 1 , and conclude by summing up over the disjoint R ∈ Q, whose union is contained in 3Q. We may further assume R ⊂ Q; otherwise Q t2 t1 (b R , f ) = 0. In addition we can assume f is positive, by repeating the same argument below with the real and imaginary, and positive and negative parts of f . Using the definition of the truncated forms (3.5) and the disjointness of L ∈ Q, Thus, if R s denotes the cube concentric to R and whose sidelength is 2 10+s , using the support and Dini conditions which is bounded by the right hand side of (4.2).
The third localized estimate is new. However, its roots lie in the wellknown principle that the maximal truncations of a Dini-continuous kernel to scales larger than s do not oscillate too much on a ball of radius 2 s , see (4.7). This was recently employed, for instance, in [13,16,18].
Proof. -We use similar notation as in the previous proof and again we rescale to |Q| = 1, and work with positive b ∈ X 1 . We can of course assume that supp f ⊂ Q. We begin by removing an error term; namely, referring to notation (2.1), if The first inequality holds because dist(supp f, supp b o ) > 2 s Q −1 , so at most the s Q scale may contribute, and in particular no contribution comes from cubes L Q. The second inequality is a trivial estimate, see [7, Appendix A] for more details. Thus we may assume b R = 0 whenever R ∈ c Q. We begin the main argument by fixing R ∈ c Q. Then by support considerations

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Similarly, In fact, using (2.1) we learn that dist(sh Q, R) > 2 s R , whence the first equality. Therefore, subtracting and adding the last display to obtain the second equality,

Now, the summation in the above display is then bounded in absolute value by
using that |s L − s R | 8 whenever L ∈ N (R). Therefore when R ∈ c Q Since b is positive, using (4.3), summing (4.4) over R ∈ c Q and using that this is a pairwise disjoint collection, we obtain that TOME 70 (2020), FASCICULE 5 Therefore, we are left with bounding F, b . This is actually done using both the L r estimate and the Dini cancellation condition. In fact, decompose and we are left to control | F, z |. We recall from [16, Lemma 2.3] the inequality where R s is the cube concentric with R and sidelength 2 s , whence for suitable absolute constant C and taking averages there holds Finally, using the above display and the fact that each z R has zero average and is supported on R, Remark 4.5 (Calderón-Zygmund theory). -Let T be an L 2 (R d )-bounded singular integral operator with Dini-continuous kernel K. Then its maximal truncations obey the estimate where the smooth radial function ψ satisfies We know from classical theory [32, Ch. I.7] that Therefore, in consequence of (4.9) and of the bound M (1,1),sparse 1, an application of Theorem 3.3 in conjunction with Lemmata 4.2 and 4.3 yields that This is a well-known result. The dual pointwise version was first obtained in this form in [16] quantifying the initial result of Lacey [18]; see also [22]. An extension to multilinear operators with less regular kernels was recently obtained in [27].

Proof of Theorem 1.2
In this section, we will prove Theorem 1.2 by appealing to Theorem 3.3 for the family [K] = {(x, y) → K s (x − y) : s ∈ Z} of (1.2). First of all, we notice that the assumption (3.1) is a direct consequence of (1.2). It is known from e.g. [12] (and our work below actually reproves this) that, with reference to (1.3) which is assumption (3.3) with r = 2. Therefore, for an application of Theorem 3.3 with we are left with verifying the corresponding stopping estimates (3.6) hold with C L [K](p, p) ε −1 . We do so by means of a Littlewood-Paley decomposition, as follows. Let ϕ be a smooth radial function on R d with support in a sufficiently small ball containing the origin, having mean zero and such that Also define for some large integer ∆ which will be specified during the proof. Unless otherwise specified, the implied constants appearing below are independent of ∆ but may depend on α > 0 from (1.2) and on the dimension. Note that K s,j are supported in {|x| < 2 s }. Define now for all j 0 and note that, with unconditional convergence The following computation is carried out in [16, Section 3].
however, we need a stronger estimate on the pointwise maximal truncations, which is implicit in [12].

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Lemma 5.2. -There holds Proof. -Let β be a smooth compactly supported function on R d normalized to have β, 1 = 1, and write β s ( · ) = 2 −sd β(2 −s · ). By usual arguments it suffices to estimate the L 2 (R d ) operator norm of uniformly over t 1 , t 2 ∈ Z. We then have Next we estimate the second and third contribution in (5.5). We have, using the third assertion in (1.2), that t1<k s Thus by Plancherel, for m = 2, 3 we have We are now ready to verify the assumptions (3.6) for the truncated forms Q[K] t2 t1 associated to a family [K] satisfying the assumptions (1.2). By virtue of Lemma 5.1 and 5.2, Lemma 4.4 applied to the families [K j ] for the value ∆ = 2ε −1 α −1 yields that Therefore using linearity in the kernel family [K] of the truncated forms Q t2 t1 [K] and the decomposition (5.1)-(5.2) which, together with the previous observations, completes the proof of Theorem 1.2.

Extension to vector-valued functions
In this section, we suitably extend the abstract domination principle Theorem 3.3 to (a suitably defined) F n -valued extension, with F ∈ {R, C}, of the singular integrals of Section 3. In fact, the C n -valued case can be recovered by suitable interpretation of the R 2n -valued one; thus, it suffices to consider F = R.

Convex body domination
Our sparse domination principle for vector valued rough maximal truncations involves a generalization of the convex body domination first introduced by Nazarov, Petermichl, Treil and Volberg in [30]: we send to Remark 6.3 below for a more detailed account.
Let 1 p < ∞. To each f ∈ L p loc (R d ; R n ) and each cube Q in R d , we associate the closed convex symmetric subset of R n where we used the notation Φ q (Q) := {ϕ : Q → R, ϕ q,Q 1}. It is easy to see that sup where · p,Q on the right hand side is being interpreted in the usual fashion. A slightly less obvious fact that we will use below is recorded in the following simple lemma, which involves the notion of John ellipsoid of a closed convex symmetric set K. This set, which we denote by E K , stands for the solid ellipsoid of largest volume contained in K; in particular, the John ellipsoid of K has the property that where, if A ⊂ R n and c 0, by cA we mean the set {ca : a ∈ A}. We also apply this notion in the degenerate case as follows: if the linear span of K is a k-dimensional subspace V of R n , we denote by E K the solid ellipsoid of largest k-dimensional volume contained in K. In this case, (6.2) holds with √ k in place of √ n, but then it also holds as stated, since k n and E K is also convex and symmetric.
Proof. -By definition of · p,Q for scalar functions, for a suitable ϕ ∈ Φ p (Q). Thus f j p,Q is the j-th component of the vector TOME 70 (2020), FASCICULE 5 By (6.2), and in consequence of the assumption, f ϕ ∈ B √ n , which proves the assertion.
We now define the sparse (p 1 , p 2 ) norm of a linear operator T mapping the space L ∞ 0 (R d ; R n ) into locally integrable, R n -valued functions, as the least constant C > 0 such that for each pair f 1 , f 2 ∈ L ∞ 0 (R d ; R n ) we may find a 1 2 -sparse collection S such that We interpret the rightmost product in the above display as the right endpoint of the Minkowski product AB = { a, b R n : a ∈ A, b ∈ B} of the closed convex symmetric sets A, B ⊂ R n , which is a closed symmetric interval. We use the same familiar notation T (p1,p2),sparse for such norm. Within such framework, we have the following extension of Theorem 3.3.
with implicit constant possibly depending on r, p 1 , p 2 and the dimensions d, n only, and in particular uniform over bounded measurable truncation functions t 1 , t 2 . Remark 6.3. -The sets (6.1) for p = 1 have been introduced in this context by Nazarov, Petermichl, Treil and Volberg [30], where sparse domination of vector valued singular integrals by the Minkowski sum of convex bodies (6.1) is employed towards matrix-weighted norm inequalities. In [9], a similar result, but in the dual form (6.3) with p 1 = p 2 = 1 is proved for dyadic shifts via a different iterative technique which is a basic version of the proof of Theorem 3.3. Subsequent developments in vector valued sparse domination include the sharp estimate for the dyadic square function [15]. The usage of exponents p > 1 in (6.1), necessary to effectively tackle rough singular integral operators, is a novelty of this paper.

Matrix-weighted norm inequalities
We now detail an application of Theorem 6.2 to matrix-weighted norm inequalities for maximally truncated, rough singular integrals. In particular,

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Corollary 1.6 from the introduction is a particular case of Theorem 6.4 below.
The classes of weights we are concerned with are the following. A pair of matrix-valued weights W, V ∈ L 1 loc (R d ; L(R n )) is said to satisfy the (joint) supremum being taken over all cubes Q ⊂ R d and We where the second inequality is the content of [30,Lemma 4.3].
) be a pair of matrix weights, and T Ω,δ be defined by (1.6), with in particular Ω ∞ 1. Then We now explain how an application of Theorem 6.2 reduces Theorem 6.4 to a weighted square function-type estimate for convex-body valued sparse operators. First of all, fix f of unit norm in L 2 (R d ; R n ). We may then find g of unit norm in L 2 (R d ; R n ) and bounded measurable functions t 1 , t 2 such that the left hand side of (6.5) is bounded by twice the sum of (6.6) where [K] is the decomposition of the kernel of T Ω performed in [16,Section 3]. As we already remarked in the scalar valued case, such decomposition satisfies (1.2). The latter expression is (the norm of) a two-weight version of the matrix weighted maximal function of Christ and Goldberg [6]. In the one weight case when V = W −1 ∈ A 2 , its boundedness has been proved in [6] and quantified in [17], which contains the explicit bound and the implicit improvement where c d,n is a dimensional constant. A straightforward modification of the same argument, using the splitting on the right of the previous display, gives the bound 1/2 A∞ f L 2 (R d ;R n ) in the two weight case. Roughly speaking, the first factor is controlled by the two weight A 2 condition and the second one by the A ∞ property of V .
By virtue of the localized estimate (5.8) for [K], an application of Theorem 6.2 tells us that (6.6) is bounded by C/ε times a sparse sublinear form as in (6.3) with p 1 = p 2 = 1 + ε, f 1 = V 1 2 f and f 2 = W 1 2 g for all ε > 0. Finally, we gather that where the supremum is being taken over 1 2 -sparse collections S, and the proof of Theorem 6.4 is completed by the following proposition.
holds uniformly over all f, g of unit norm in L 2 (R d ; R n ) and all 1 2 -sparse collections S.
Proof. -There is no loss in generality with assuming that the sparse collection S is a subset of a standard dyadic grid in R d , and we do so. Fix ANNALES DE L'INSTITUT FOURIER ε > 0. By standard reductions, we have that (6.8) sup having defined the square function as a result of the sharp reverse Hölder inequality and of the Carleson embedding theorem there holds cf. [30, Proof of Lemma 5.2], and a similar argument applies to S V,ε . Therefore, for ε < 2 −10 min{t W , t V }, (6.8) turns into which in turn proves Proposition 6.5.
Remark 6.6. -We may derive a slightly stronger weighted estimate than (6.7) for the non-maximally truncated rough integrals T Ω,δ , by applying Theorem 6.2 in conjunction with the (1, 1 + ε) localized estimates proved in [7]. Namely, the estimate holds uniformly in δ > 0 for all f of unit norm in L 2 (R d ; R n ). Repeating the proof of Proposition 6.5 then yields the slightly improved weighted TOME 70 (2020), FASCICULE 5

Proof of Theorem 6.2
The proof of Theorem 6.2 is formally identical to the argument for the scalar valued case, provided that estimate (3.10) and the definition of E Q given in (3.11) are replaced by suitable vector valued versions. We begin with the second tool. The proof, which is a minor variation on [9, Lemma 3.3], is given below Lemma 6.7. -Let 0 < η 1, Q be a dyadic cube and f j ∈ L pj (R d ; R n ), j = 1, 2. Then the set for some cube L ⊂ R d with x ∈ L satisfies |E Q | Cη min{p1,p2} |Q| for some absolute dimensional constant C.
Proof. -We may assume that supp f j ⊂ 3Q. It is certainly enough to estimate the measure of each j ∈ {1, 2} component of E Q by Cη pj |Q|, and we do so: we fix j and are thus free to write f j = f, p j = p. Let Fix such a disjoint collection L 1 , . . . L m . Notice that if A ∈ GL(R n ) then L Af = L f . By action of GL(R n ) we may thus reduce to the case where E f p,3Q = B 1 , and in particular (6.10) B 1 ⊂ f p,3Q ⊂ B √ n .
By membership of each L µ ∈ L f , we know that η f p,Lµ ⊂ B 1 . A fortiori, there exists ϕ µ ∈ Φ p (L µ ) and a coordinate index µ ∈ {1, . . . , n} such that Using the membership ϕ µ ∈ Φ p (L µ ) for the first inequality and the disjointness of the supports for the second equality so that ϕ ∈ Φ p (3Q). In particular, beginning with the right inclusion in (6.10) and using (6.11) in the last inequality which rearranging yields (6.12) with C = n p , thus completing the proof.
At this point, let S k be a collection of pairwise disjoint cubes as in Step 3 of the proof of Theorem 3.3. The elements of the collection S k+1 are defined to be the maximal dyadic cubes L such that the same condition as in (3.11) holds, provided the definition of E Q therein is replaced with the one in (6.9). By virtue of Lemma 6.7, (3.12) still holds provided η is chosen small enough. And, we still obtain that S k+1 (Q) = {L ∈ S k+1 : L ⊂ 3Q} is a stopping collection. By definition of S k+1 , it must be that (6.13) f j 1 3Q pj ,K ⊂ C f j pj ,3Q whenever the (not necessarily dyadic) cube K is such that a moderate dilate CK of K contains 2 5 L for some L ∈ S k+1 (Q). Fix Q for a moment and let A j = (A mµ j : 1 m, µ n) ∈ GL(R n ), j = 1, 2 be chosen such that the John ellipsoid of f j 3Q,pj is B 1 , or its intersection with a lower dimensional subspace in a degenerate case, and A j f j := f j . It follows from (6.13) that if 2 5 L ⊂ CK then f j 1 3Q pj ,K ⊂ B C . This fact, together with Lemma 6.1 readily yields the estimates (6.14) f j Yp j (Q) 1, j = 1, 2.
We are ready to obtain a substitute for (3.10). In fact where we also employed (3.8) coupled with (6.14) in the last line. Assembling together the last two displays yields the claimed vector-valued version of (3.10), and finishes the proof of Theorem 6.2.