Sharp $A_1$ weighted estimates for vector valued operators

TL;DR

This paper establishes sharp weighted estimates for vector valued operators like maximal functions and Calderón-Zygmund operators using Aq weights, introducing new convex body domination techniques and a novel proof for scalar cases.

Contribution

It provides the first sharp weighted bounds for vector valued operators and develops convex body domination results, including a new proof for scalar operators.

Findings

Quantitative weighted L^p estimates for vector valued operators.

Convex body domination results for singular integrals.

New proof for scalar weighted estimates.

Abstract

Given $1\leq q<p<\infty$ quantitative weighted L^p estimates, in terms of Aq weights, for vector valued maximal functions, Calder\'on-Zygmund operators, commutators and maximal rough singular integrals are obtained. The results for singular operators will rely upon suitable convex body domination results, which in the case of commutators will be provided in this work, obtaining as a byproduct a new proof for the scalar case as well.

Equations

$$[w{]}_{A_p}=\underset{Q}{\sup }\frac{1}{|Q|}{\int }_{Q}w(x)dx{\left(\frac{1}{|Q|}{\int }_{Q}w(y{)}^{-\frac{p^{\prime }}{p}}dy\right)}^{\frac{p}{p^{\prime }}}<\infty .$$

$$[w{]}_{A_{\infty }}=\underset{Q}{\sup }\frac{1}{w(Q)}{\int }_{Q}M(w{\chi }_Q)(x)dx<\infty$$

$$\Vert f{\Vert }_{L^p(W)}={\left({\int }_{{\mathbb{R}}^d}{\left|W^{\frac{1}{p}}(x)f(x)\right|}^{p}dx\right)}^{\frac{1}{p}}.$$

$$[W{]}_{A_p}=\underset{Q}{\sup }\frac{1}{|Q|}{\int }_Q{\left(\frac{1}{|Q|}{\int }_Q{\Vert W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\Vert }^{p^{\prime }}dy\right)}^{\frac{p}{p^{\prime }}}dx<\infty .$$

$$[W{]}_{A_{q,\infty }^{sc}}=\underset{e\in {\mathbb{C}}^n}{\sup }{\left[{\left|W^{\frac{1}{q}}e\right|}^q\right]}_{A_{\infty }}<\infty .$$

$$\Vert T\vec{f}{\Vert }_{L^2(W)}\le c_{n,d,T}[W{]}_{A_2}^{\frac{1}{2}}[W{]}_{A_{2,\infty }^{sc}}^{\frac{1}{2}}[W^{-1}{]}_{A_{2,\infty }^{sc}}^{\frac{1}{2}}\Vert \vec{f}{\Vert }_{L^2(W)}\le c_{n,T}[W{]}_{A_2}^{\frac{3}{2}}\Vert f{\Vert }_{L^2(W)}.$$

$${\Vert \underset{\delta >0}{\sup }|W^{\frac{1}{2}}{T}_{\Omega ,\delta }\vec{f}|\Vert }_{L^2({\mathbb{R}}^d)}\le c_{n,d,T}[W{]}_{A_2}^{\frac{5}{2}}\Vert W^{\frac{1}{2}}\vec{f}{\Vert }_{L^2({\mathbb{R}}^d)}$$

$$T_{\Omega ,\delta }f(x)={\int }_{|x-y|>\delta }\frac{\Omega \left(\frac{x-y}{|x-y|}\right)}{|x-y{|}^d}f(y)dy.$$

$$\Vert T\vec{f}{\Vert }_{L^p(W)}\le c_{n,d,T}{\left[W\right]}_{A_p}{\left[W^{-\frac{p^{\prime }}{p}}\right]}_{A_{p^{\prime },\infty }^{sc}}^{\frac{1}{p}}{\left[W\right]}_{A_{p,\infty }^{sc}}^{\frac{1}{p^{\prime }}}\Vert f{\Vert }_{L^p(W)}\le c_{n,d,T}{\left[W\right]}_{A_p}^{1+\frac{1}{p-1}-\frac{1}{p}}\Vert \vec{f}{\Vert }_{L^p(W)}.$$

$$[b,G]f(x)=b(x)Gf(x)-G(bf)(x).$$

$$\Vert Tf{\Vert }_{L^p(w)}\le c_{n,T}p{p}^{\prime }[w{]}_{A_1}^{\frac{1}{p}}[w{]}_{A_{\infty }}^{\frac{1}{p^{\prime }}}\Vert f{\Vert }_{L^p(w)}.$$

$$\Vert [b,T]f{\Vert }_{L^p(w)}\le c_{n,T}\Vert b{\Vert }_{BMO}{\left(p{p}^{\prime }\right)}^2[w{]}_{A_1}^{\frac{1}{p}}[w{]}_{A_{\infty }}^{1+\frac{1}{p^{\prime }}}\Vert f{\Vert }_{L^p(w)}.$$

$$\Vert Tf{\Vert }_{L^{1,\infty }(w)}\le c\phi ([w{]}_{A_1})\Vert f{\Vert }_{L^1(w)}.$$

$$\Vert Tf{\Vert }_{L^{p,\infty }(w)}\le c\phi ([w{]}_{A_p})\Vert f{\Vert }_{L^p(w)}.$$

$$\Vert T{\Vert }_{L^2(w)}\le c_{n,T}[w{]}_{A_2}+c_{n,T}\left(\Vert T{\Vert }_{L^2(w)\to L^{2,\infty }(w)}+\Vert T{\Vert }_{L^2(w^{-1})\to L^{2,\infty }(w^{-1})}\right).$$

$$[W{]}_{A_1}=\underset{Q}{\sup }{ess\; sup}_{y\in Q}\frac{1}{|Q|}{\int }_Q\Vert W(x)W^{-1}(y)\Vert dx<\infty$$

$$M_{W,p}(\vec{f})(x)=\underset{x\in Q}{\sup }\frac{1}{|Q|}{\int }_Q\left|W^{\frac{1}{p}}(x)W^{-\frac{1}{p}}(y)\vec{f}(y)\right|dy$$

$$M_{W,p}^{\prime }(\vec{f})(x)=\underset{x\in Q}{\sup }\frac{1}{|Q|}{\int }_Q\left|{\mathcal{W}}_{Q,p}{W}^{-\frac{1}{p}}(y)\vec{f}(y)\right|dy$$

$$\Vert M_{W,p}{\Vert }_{L^p({\mathbb{R}}^d;{\mathbb{C}}^n)\to L^p({\mathbb{R}}^d)}$$

$$\Vert M_{W,p}^{\prime }{\Vert }_{L^p({\mathbb{R}}^d;{\mathbb{C}}^n)\to L^p({\mathbb{R}}^d)}$$

$$\Vert T{\Vert }_{L^p(W)\to L^p(W)}\le c_{n,p,T}[W{]}_{A_1}^{\frac{1}{p}}[W{]}_{A_{1,\infty }^{sc}}^{\frac{1}{p^{\prime }}}\le c_{n,p,T}[W{]}_{A_1}$$

$${\Vert T_{\Omega ,W,p}^{\ast }\Vert }_{L^p({\mathbb{R}}^d;{\mathbb{C}}^n)\to L^p({\mathbb{R}}^d)}\le c_{n,d,\Omega ,p}[W{]}_{A_1}^{\frac{1}{p}}[W{]}_{A_{1,\infty }^{sc}}^{1+\frac{1}{p^{\prime }}}\le c_{n,d,\Omega ,p}[W{]}_{A_1}^2$$

$$\Vert [b,T]{\Vert }_{L^p(W)\to L^p(W)}\le c_{n,p,T}\Vert b{\Vert }_{BMO}[W{]}_{A_1}^{\frac{1}{p}}[W{]}_{A_{1,\infty }^{sc}}^{1+\frac{1}{p^{\prime }}}\le c_{n,p,T}[W{]}_{A_1}^2$$

$$\Vert Gf{\Vert }_{L^p(w)}\le c_{T,p,q}\phi ([w{]}_{A_1})\Vert f{\Vert }_{L^p(w)},$$

$$\Vert Gf{\Vert }_{L^p(w)}\le c_{T,p,q}\phi ([w{]}_{A_q})\Vert f{\Vert }_{L^p(w)},$$

$$\Vert M_{W,p}{\Vert }_{L^p({\mathbb{R}}^d;{\mathbb{C}}^n)\to L^p({\mathbb{R}}^d)}$$

$$\Vert M_{W,p}^{\prime }{\Vert }_{L^p({\mathbb{R}}^d;{\mathbb{C}}^n)\to L^p({\mathbb{R}}^d)}$$

$$\Vert T{\Vert }_{L^p(W)\to L^p(W)}\le c_{n,p,q,T}[W{]}_{A_q}^{\frac{1}{p}}[W{]}_{A_{q,\infty }^{sc}}^{\frac{1}{p^{\prime }}}\le c_{n,p,T}[W{]}_{A_q}$$

$${\Vert T_{\Omega ,W,p}^{\ast }\Vert }_{L^p({\mathbb{R}}^d;{\mathbb{C}}^n)\to L^p({\mathbb{R}}^d)}\le c_{n,d,\Omega ,p}[W{]}_{A_q}^{\frac{1}{p}}[W{]}_{A_{q,\infty }^{sc}}^{1+\frac{1}{p^{\prime }}}\le c_{n,d,\Omega ,p}[W{]}_{A_q}^2$$

$$\Vert [b,T]{\Vert }_{L^p(W)\to L^p(W)}\le c_{n,p,q,T}\Vert b{\Vert }_{BMO}[W{]}_{A_q}^{\frac{1}{p}}[W{]}_{A_{q,\infty }^{sc}}^{1+\frac{1}{p^{\prime }}}\le c_{n,p,q,T}\Vert b{\Vert }_{BMO}[W{]}_{A_q}^2$$

$$\sum _{P\in \mathcal{S},P\subseteq Q}|P|\le \Lambda |Q|.$$

$$\sum _{P\in \mathcal{S},P\subseteq Q}|P|\le {\eta }^{-1}\sum _{P\in \mathcal{S},P\subseteq Q}|E_P|\le {\Lambda }^{-1}|Q|.$$

$$⟨⟨\vec{f}⟩{⟩}_{r,Q}=\left\{\frac{1}{|Q|}{\int }_{Q}f\phi dx:\phi :Q\to \mathbb{R},\phi \in B_{L^{r^{\prime }}}(Q)\right\}$$

Full text

1. Introduction

We recall that a weight, namely, a non negative locally integrable function $w$ belongs to $A_{p}$ for $1<p<\infty$ if

[TABLE]

The $A_{p}$ class of weights characterizes the $L^{p}(w)$ boundedness of the maximal function as B. Muckenhoupt established in the 70s. Subsequent works of B. Muckenhoupt himself R. Wheeden, R. Hunt, R. R. Coifman and C. Fefferman were devoted to explore the connection of the $A_{p}$ class with weighted estimates for singular integrals. However, it was not until the 2000s that the quantitative dependence on the so called $A_{p}$ constant, namely $[w]_{A_{p}}$, became a trending topic. Probably the paradigmatic question in that line of research was the $A_{2}$ theorem finally established by T. Hytönen [10].

Now we recall that the $A_{p}$ classes are increasing, so it is natural to define $A_{\infty}=\bigcup_{p\geq 1}A_{p}$. T. Hytönen and C. Pérez [12] proved that

[TABLE]

is the smallest constant characterizing $A_{\infty}$, at least up until now, and provided a number of quantitative estimates in terms of $[w]_{A_{\infty}}$. Nevertheless, it is worth noting that essentially the same constant had already appeared in works by Fujii [7] and Wilson [38]. After [12] several papers have been devoted to the study of quantitative weigthed estimates in terms of the $A_{p}$ and the $A_{\infty}$ constants.

Among the possible extensions of the classical scalar theory of Calderón-Zygmund operators, vector valued extensions have received an increasing degree of attention in the last years. Let $W:\mathbb{R}^{d}\rightarrow\mathbb{C}^{n\times n}$ a matrix weight, namely, a matrix function such that $W(x)$ is positive definite a.e. Given $f:\mathbb{R}^{d}\rightarrow\mathbb{C}^{n}$ and $1<p<\infty$, we define

[TABLE]

Let $1<p<\infty$. We say that a matrix weight $W$ is an $A_{p}$ weight if

[TABLE]

Matrix $A_{p}$ weights were introduced by S. Treil and A. Volberg in [36]. In the late 90s it was shown in a series of works by M. Goldberg [8], F. Nazarov and S. Treil [27] and A. Volberg [37] that if $W$ is a matrix $A_{p}$ weight and $T$ is a Calderón-Zygumund operator, then $T$ is bounded on $L^{p}(W)$. The definition of $A_{p}$ that we have presented here is due to S. Roudenko [35] and is equivalent to the definitions in the aforementioned works.

Contrary to what happpens in the scalar setting, the $A_{2}$ conjecture remains an open problem in the vector valued setting. In [1], K. Bickel, S. Petermichl and B. Wick proved that the dependence of the norm of the martingale and Hilbert transform on the $A_{2}$ constant of the weight $W$ is at most $[W]_{A_{2}}^{\frac{3}{2}}\log([W]_{A_{2}})$. The second author and A. Stoica [32] established that the dependence of the norm of all Calderón-Zygmund operators with cancellation on $[W]_{A_{2}}$ coincides with the one for the matrix martingale transform, hence reducing the $A_{2}$ conjecture for those operators to the proof of the linear bound for the latter.

Given $1\leq q<\infty$, we say that $W\in A_{q,\infty}^{sc}$ if

[TABLE]

Quite recently F. Nazarov, S. Petermichl, S. Treil and A. Volberg [25] established the following quantitative estimate for $W\in A_{2}$,

[TABLE]

The preceding estimate is obtained using the so called convex body domination. In that work the linear dependence on the $A_{2}$ constant is conjectured. In the case of maximal rough singular integrals with $\Omega\in L^{\infty}(\mathbb{S}^{n-1})$, the following estimate, in the case $p=2$, was quite recently provided by F. Di Plinio, K. Li and T. Hytönen [4],

[TABLE]

where the scalar operator $T_{\Omega,\delta}$ is defined as follows

[TABLE]

Very recently D. Cruz-Uribe, J. Isralowitz and K. Moen [3] extended (1.1) to every $1<p<\infty$, providing the following estimate

[TABLE]

Some sharp estimates have been obtained as well in the vector valued setting. T. Hytönen, S. Petermichl and A. Volberg [14], and Isralowitz, Kwon, and the first author [15] established the linear upper bound on $[W]_{A_{2}}$ for the matrix-weighted square function and the matrix-weighted maximal function, respectively (namely, $M_{W,p}$ defined as in Section 2).

We recall that given a linear operator $G$ and a locally integrable function $b$, the commutator $[b,G]$ is defined by

[TABLE]

At this point we turn our attention back to the scalar setting. A. Lerner, S. Ombrosi and C. Pérez [19, 21] established the following result for Calderón-Zygmund operators. Given a Calderón-Zygmund operator $T$ and $w\in A_{1}$ we have that

[TABLE]

In the case of commutators, for $b\in BMO$ and $T$ a Calderón-Zygmund operator, C. Ortiz-Caraballo [29] proved that

[TABLE]

One of the motivations to obtain such a precise estimate for Calderón-Zygmund operators was to provide a proof of the $A_{2}$ constant. Assume that for every $w\in A_{1}$

[TABLE]

Then we also have that for every $1<p<\infty$ and every $w\in A_{p}$ ([19])

[TABLE]

We observe that in [19] it was proved that $\varphi(t)\leq t\log(e+t)$ using (1.2) as a main ingredient and it was also conjectured that $\varphi(t)\simeq t$. Being true the latter would have led to a proof of the $A_{2}$ conjecture, since in [31] it was established that

[TABLE]

However, the fact that $\varphi(t)\simeq t$ was disproved in [26], furthermore, in [18] it was established that $\varphi(t)\simeq t\log(e+t)$, and consequently the estimate in [19] is sharp.

2. Main Results

One of the main purposes of this paper is to provide vector valued counterparts of (1.2) and (1.3). To provide that kind of estimates we rely upon the definition of the matrix $A_{1}$ class that M. Frazier and S. Roudenko introduced in [6].

Definition 1**.**

We say that a weight $W\in A_{1}$ if

[TABLE]

Before presenting our first result we would like to discuss briefly the definition of the maximal function. Due to the non-linearity of the maximal function, when it comes to study weighted estimates for it, the approach that has been mainly considered in the literature is to study weighted variants of it (see [8, 15]). In what follows we will deal with the following weighted maximal functions.

[TABLE]

[TABLE]

We remit the reader to Section 4 for the definition of $\mathcal{W}_{Q}$.

Theorem 1**.**

Let $W\in A_{1}$ and $1<p<\infty$. Then

[TABLE]

Let $T$ a Calderón-Zygmund operator, $b\in BMO$ and $\Omega\in L^{\infty}(\mathbb{S}^{d-1})$ with $\int_{\mathbb{S}^{d-1}}\Omega=0$, if $W\in A_{1}$ and $1<p<\infty$,

[TABLE]

where $T_{\Omega,W,p}^{*}f=\sup_{\delta>0}\left|W^{\frac{1}{p}}T_{\Omega,\delta}\left(W^{-\frac{1}{p}}\vec{f}\right)\right|$.

Coming back once again to the scalar setting, it is a known fact that an extrapolation argument [5, Corollary 4.3] allows to prove that if we have that

[TABLE]

for every $A_{1}$ weight, then the same dependence holds as well for every $w\in A_{q}$ with $1\leq q<p$, namely,

[TABLE]

Extrapolation arguments, in case of being feasible, have not been developed yet in this setting so we provide a direct proof of the preceding result in the cases considered in Theorem 1. We observe that we recover again the linear dependence already available in the scalar case. We wonder whether it is possible to provide some estimate analogous to the one supremmum estimates obtained in [23] and [34].

Theorem 2**.**

Let $1<q<p<\infty$ and $W\in A_{q}$. Then

[TABLE]

Let $T$ a Calderón-Zygmund operator, $b\in BMO$ and $\Omega\in L^{\infty}(\mathbb{S}^{d-1})$ with $\int_{\mathbb{S}^{d-1}}\Omega=0$, if $W\in A_{1}$ and $1<p<\infty$,

[TABLE]

where $T_{\Omega,W,p}^{*}f=\sup_{\delta>0}\left|W^{\frac{1}{p}}T_{\Omega,\delta}\left(W^{-\frac{1}{p}}f\right)\right|$.

We would like to note that both in Theorems 1 and 2 the dependences obtained are the same as the best known ones in the scalar case, and therefore, besides the case of the maximal rough singular integral, sharp.

The rest of the paper is organized as follows. In Section 3 we present a convex body domination result for commutators. We provide some extra facts about matrix $A_{p}$ weights in Section 4. Finally, in Section 5 we settle Theorems 1 and 2.

3. Convex body domination for Commutators

We begin the section borrowing some definitions from [17]. We say that a family of cubes $\mathcal{D}$ is a dyadic lattice if it satisfies the following properties

1. (1)

If $Q\in\mathcal{D}$ every dyadic child of $Q$ belongs to $\mathcal{D}$. In other words, if $\mathcal{D}(Q)$ is the standard grid of dyadic cubes of $Q$ and $Q\in\mathcal{D}$ then $\mathcal{D}(Q)\subseteq\mathcal{D}$. 2. (2)

If $Q_{1},Q_{2}\in\mathcal{D}$ there exists a common ancestor in $\mathcal{D}$ that is there exists $Q\in\mathcal{D}$ such that $Q_{1},Q_{2}\in\mathcal{D}(Q)$. 3. (3)

For every compact set $K\subseteq\mathbb{R}^{d}$ there exists $Q\in\mathcal{D}$ such that $K\subseteq Q$.

Given $\eta\in(0,1)$ we say that $\mathcal{S}\subset\mathscr{D}$ is a $\eta$-sparse family if for every $Q\in\mathcal{S}$ there exists a measurable subset $E_{Q}\subset Q$ such that

1. (1)

$\eta|Q|\leq|E_{Q}|.$ 2. (2)

The sets $E_{Q}$ are pairwise disjoint.

Further, given $\Lambda>1$ we say that $\mathcal{S}\subset\mathscr{D}$ is a $\Lambda$ Carleson family if for every $Q\in\mathcal{S}$,

[TABLE]

Clearly every $\eta$-sparse family is $\eta^{-1}$ Carleson, since

[TABLE]

Though less obvious, the converse is true. Every $\Lambda$ Carleson family is $\Lambda^{-1}$ sparse [17, Lemma 6.3]. We will also use without further comment the fact that every $\Lambda$ Carleson family can be written as a union of $m$ Carleson families, each of which is $1+\frac{\Lambda-1}{m}$ Carleson [17, Lemma 6.6]. Hereafter we will sometimes refer to a family as sparse or Carleson without reference to $\eta$ or $\Lambda$ if the specific values of these constants are unimportant.

Convex body domination was introduced by F. Nazarov, S. Petermichl and A. Volberg in [25]. That notion provides a suitable counterpart to sparse domination in the vector-valued setting. Let $f:\mathbb{R}^{d}\longrightarrow\mathbb{C}^{n}$. Given a cube $Q$ if additionally $f\in L^{r}(Q)$ where $1\leq r<\infty$ and $r^{\prime}=\infty$ if $r=1$, we define

[TABLE]

where $B_{L^{r^{\prime}}(Q)}=\left\{\phi\in L^{r^{\prime}}(Q)\,:\,\|\phi\|_{L^{r^{\prime}}}\leq 1\right\}$. We will drop the subscript $r$ in the case $r=1$. In [25] it was established that $\langle\langle f\rangle\rangle_{Q}$ is a symetric, convex and compact set in $\mathbb{C}^{n}$ and in [4] that property was extended to the case $r>1$.

We recall that given $T$ a linear operator, the grand-maximal operator $M_{T}$ was defined for first as follows in [16]

[TABLE]

In [25] the authors proved the following result (see also [11]).

Theorem 3**.**

Let $T:L^{1}(\mathbb{R}^{d})\rightarrow L^{1,\infty}(\mathbb{R}^{d})$ be a linear operator such that also $M_{T}:L^{1}(\mathbb{R}^{d})\rightarrow L^{1,\infty}(\mathbb{R}^{d})$. For $\vec{f}\in L_{c}^{\infty}(\mathbb{R}^{d};{\mathbb{C}^{n}})$ and $\varepsilon\in(0,1)$ there exists a $(1-\varepsilon)$-sparse collection of dyadic cubes such that

[TABLE]

where $c_{T}=\|T\|_{L^{1}\rightarrow L^{1,\infty}}+\|M_{T}\|_{L^{1}\rightarrow L^{1,\infty}}$. More precisely, there exist functions $k_{Q}\in B_{L^{\infty}(Q\times Q)}$ such that

[TABLE]

Our purpose in this section is to establish the following vector-valued counterpart for commutators extending [22, Theorem 1.1].

Theorem 4**.**

Let $T:L^{1}(\mathbb{R}^{d})\rightarrow L^{1,\infty}(\mathbb{R}^{d})$ be a linear operator such that also $M_{T}:L^{1}(\mathbb{R}^{d})\rightarrow L^{1,\infty}(\mathbb{R}^{d})$. For $f\in L_{c}^{\infty}(\mathbb{R}^{d};{\mathbb{C}^{n}})$, every $b\in L_{loc}^{1}$ and $\varepsilon\in(0,1)$ there exists a $(1-\varepsilon)$-sparse collection of dyadic cubes such that

[TABLE]

where each $c_{d,n}$ is a constant depending on $n$ and $d$ and $c_{T}=\|T\|_{L^{1}\rightarrow L^{1,\infty}}+\|M_{T}\|_{L^{1}\rightarrow L^{1,\infty}}$. More precisely, there exist functions $k_{Q},\,k_{Q}^{*}\in B_{L^{\infty}(Q\times Q)}$ such that

[TABLE]

We observe that A. Lerner [16] proved for Calderón-Zygmund operators that

[TABLE]

and it is also a known fact that

[TABLE]

Consequently Theorems 3 and 4 hold in the case that $T$ is a Calderón-Zygmund operator with

[TABLE]

3.1. Proof of Theorem 4

$f\in L_{c}^{\infty}(\mathbb{R}^{d};{\mathbb{C}^{n}})$ and $b\in L_{loc}^{1}$, we further assume that $b\in L^{\infty}.$ We define the $\mathbb{C}^{2n}$ valued function $\tilde{f}$ by

[TABLE]

and define the $2n\times 2n$ block matrix $\Phi(x)$ by

[TABLE]

so that

[TABLE]

Then we have that

[TABLE]

By assumption, $\Phi^{-1}\tilde{f}$ is bounded with compact support. A direct computation shows that

[TABLE]

Lets plug in $\Phi^{-1}(y)\tilde{f}(y)$ into (3.1) and equate components. Namely

[TABLE]

so that

[TABLE]

However, adding and subtracting $\langle k_{Q}(x,\cdot)f\rangle_{Q}\langle b\rangle_{Q}$ to the first component, we get

[TABLE]

Hence,

[TABLE]

and we are done.

Remark 1*.*

The proof presented above works as well in the case $n=1$, hence providing a new proof for the scalar case that was settled in [22].

4. The reverese Hölder inquality. $A_{1},\,A_{q}$ and $A_{q,\infty}^{sc}$

weights

We recall that if $\rho(x)$ is a norm on ${\mathbb{C}^{n}}$ there exists a positive matrix $A$, that we call reducing operator of $\rho$ such that

[TABLE]

If $1\leq p<\infty$ we will call $\mathcal{W}_{Q,p}$ the reducing operator for

[TABLE]

In the case $1<p<\infty$ we shall call $\mathcal{W}^{\prime}_{Q,p}$ the reducing operator for

[TABLE]

It follows from the proof of Roudenko’s characterization [35, Lemma 1.3] that

[TABLE]

Now we observe that if we call $V=W^{-\frac{1}{p-1}}$, we have that

[TABLE]

This yields that we can take $\mathcal{V}_{Q,p^{\prime}}=\mathcal{W}^{\prime}_{Q,p}$. Analogously

[TABLE]

and we can choose $\mathcal{V}^{\prime}_{Q,p^{\prime}}=\mathcal{W}_{Q,p}$. Consequently we have that

[TABLE]

The preceding discussion can be summarized in the following proposition.

Proposition 1**.**

Let $1<p<\infty$. Then

[TABLE]

In our next result we show that the $A_{1}$ type conditions constants control the corresponding $A_{\infty}$ constants. We include in the statement the case of the $A_{q}$ constant that was already established in [3] for the sake of completeness.

Proposition 2**.**

If $1\leq q<\infty$ and $W\in A_{q}$, then $[W]_{A_{q,\infty}^{sc}}\leq c_{n}[W]_{A_{q}}$.

Proof.

We just settle the case $q=1$. We observe that for every cube $Q$, a.e $y\in Q$, and every $\vec{e}\in\mathbb{C}^{n}$

[TABLE]

or equivalently

[TABLE]

Hence

[TABLE]

and integrating in $y$ over $Q$,

[TABLE]

Consequently

[TABLE]

and since the preceding estimate holds for every cube $Q$ and every $e$ we have that $[W]_{A_{1,\infty}^{sc}}\leq[W]_{A_{1}}$. ∎

Now we recall the quantitative version of the reverse Hölder inequality. This estimate was obtained for first in [12] (see [13] for another proof).

Lemma 1**.**

Let $w\in A_{\infty}$ then if $0<\delta\leq\frac{1}{2^{d+11}[w]_{A_{\infty}}}$ for every cube $Q\subseteq\mathbb{R}^{d}$ we have that

[TABLE]

We would like to end up the section presenting a technical result that will be crucial for the proof of the main results.

Lemma 2**.**

Let $1\leq q<p<\infty$. Assume that $W\in A_{q}$ and let $r=1+\frac{1}{2^{d+11}[W]_{A_{q,\infty}^{sc}}}$. Then we have that a.e $y\in Q$,

[TABLE]

Proof.

Choosing ${\vec{e}_{j}}(y)$ an orthonormal basis of eigenvalues $\lambda_{j}(y)$ of $W(y)$, we have by the classical Hölder-McCarthy inequality (see [2, Lemma 2.1]) that

[TABLE]

Hence

[TABLE]

∎

Now, since $r=1+\frac{1}{2^{d+11}[W]_{A_{q,\infty}^{sc}}}$, taking account that $W\in A_{q}\subset A_{q,\infty}^{sc}$, by reverse Hölder inequality we have that choosing any basis $\{{\vec{e}_{i}}\}_{i=1}^{n}$ of ${\mathbb{C}^{n}}$,

[TABLE]

and we are done.

5. Proofs of $A_{1}$ and $A_{q}$ estimates

5.1. Proof of Theorems 1 and 2 for $M^{\prime}_{W,p}$

and $M_{W,p}$

5.1.1. Estimates for $M^{\prime}_{W,p}$

First we deal with (2.2) and (2.7). We proceed as follows. Notice that

[TABLE]

If $q=1$ it suffices to use Lemma 2 and the definition of $A_{1}$ weight to see that

[TABLE]

In that case

[TABLE]

and using the strong type $(p,p)$ for the scalar maximal function we are done.

If $q>1$,

[TABLE]

Now we notice that taking into account Lemma 2,

[TABLE]

To end the estimate, observe that if we call $V=W^{-\frac{1}{q-1}}$

[TABLE]

where the last step is a direct application of Proposition 1. Then

[TABLE]

and using the strong type $(p,p)$ for the scalar operator $M_{q}(f)=M(|f|^{q})^{\frac{1}{q}}$, we are done.

5.1.2. Estimates for $M_{W,p}$

We are going to settle (2.1) and (2.6) at the same time. First we note that by the proof of Lemma 2 we have that

[TABLE]

for any $Q\in\mathscr{D}$. Fix $J\in\mathscr{D}$. Let $\mathcal{J}(J)$ denote the maximal cubes $L\in\mathcal{D}(J)$ (if any exist) where

[TABLE]

By maximality, as usual, we have

[TABLE]

Now let $\mathcal{F}(J)$ be the collection of cubes in $\mathcal{D}(J)$ that are not a subset of any cube $I\in\mathcal{J}(J)$. Furthermore, for ease of notation let $\cup\mathcal{J}(J)=\cup_{L\in\mathcal{J}(J)}L$. Let

[TABLE]

We pointwise dominate $M_{J,W}\vec{f}(x)$ by looking at three cases. First, assume $Q\in\mathcal{F}(J)$ and assume $x\in\cup\mathcal{J}(J)$. Thus, let $x\in Q\in\mathcal{F}(J)$ and $x\in I\in\mathcal{J}(J)$. Then by definition of $\mathcal{F}(J)$ we must have $I\subsetneq Q\subseteq J$ so that in this case, (5.2) and (5.3) gives us

[TABLE]

at this point if $q=1$ we have that

[TABLE]

and if $1<q<\infty$,

[TABLE]

Next, assume $Q\in\mathcal{F}(J)$ and $x\not\in\cup\mathcal{J}(J)$. Pick a sequence $L_{k}^{x}$ of nested dyadic cubes where

[TABLE]

But if

[TABLE]

then for some $k$ we have

[TABLE]

which means that $x\in L_{k}^{x}\subseteq Q$ for some $Q\in\mathcal{J}(J)$. Thus, bearing the computation above in mind,

[TABLE]

in the case $1<q<\infty$ and

[TABLE]

in the case $q=1$.

Lastly, if $Q\not\in\mathcal{F}(J)$ then $Q\subseteq L$ for some $L\in\mathcal{J}(J)$ so obviously if $x\in Q$ then $x\in\cup\mathcal{J}(J)$. Combining all this gives

[TABLE]

Thus, for $1\leq q<\infty,$

[TABLE]

If as usual $\mathcal{J}_{k}(J)=\{L\in\mathcal{J}(Q):Q\in\mathcal{J}_{k-1}(J)\}$ with $\mathcal{J}_{0}(J)=\{J\}$ and $\mathcal{S}=\cup_{k}\mathcal{J}_{k}(J)$ then $\mathcal{S}$ is sparse and iteration gives us

[TABLE]

5.2. Proof of Theorems 1 and 2 for singular

operators

5.2.1. A reduction to bump conditions

We recall that $A:[0,\infty)\rightarrow[0,\infty)$ is Young function if $A(0)=0$ and it is a convex and increasing function. Given a function $f$ and a measurable set $E$ with finite measure, we can define the average on $E$ of $f$ associated to $A$ by

[TABLE]

From that definition it readily follows that if

[TABLE]

then

[TABLE]

Given a Young function $A$ it is natural to define a maximal operator $M_{A}$ hinging upon the preceding definition of average as follows

[TABLE]

The boundedness of those operators on $L^{p}$ spaces was thoroughly studied by C. Pérez [30], under the aditional condition that $A$ is doubling, assumption that was proved to be superfluous by Liu and Luque [24]. The condition is the following

[TABLE]

Associated to each Young function we can define the so called associated Young function $\overline{A}$ by

[TABLE]

That function has some interesting properties. The first of them is that

[TABLE]

The second one, that will be very interesting for us, is the following generalized Hölder inequality

[TABLE]

For more details about Young functions we remit the reader to [28, 33].

Let $T$ is a Calderón-Zygmund operator and $W,V$ be matrix weights. If we call

[TABLE]

and

[TABLE]

then Theorems 3 and 4 immediately give us that

[TABLE]

and

[TABLE]

Armed with the preceding definitions and results and arguing in the spirit of [3] we can prove a lemma that will be fundamental for our purposes.

Lemma 3**.**

Let $A,B$ be Young functions. Then

[TABLE]

where $\kappa_{1}=\sup_{Q}\|\|\|W^{\frac{1}{p}}(x)V^{-\frac{1}{p}}(y)\|\|_{A_{x},Q}\|_{B_{y},Q}$ and $\kappa_{2}=\sup_{Q}\|\|\|W^{\frac{1}{p}}(x)V^{-\frac{1}{p}}(y)\|\|_{B_{y},Q}\|_{A_{x},Q}$.

Proof.

Without loss of generality we may assume that $f,g\geq 0$. Then taking into account generalized Hölder inequality

$\begin{aligned} &\sum_{Q\in\mathcal{S}}\frac{1}{|Q|}\int_{Q}\int_{Q}\|W^{\frac{1}{p}}(x)V^{-\frac{1}{p}}(y)\|f(y)g(x)dydx\\ &=\sum_{Q\in\mathcal{S}}\frac{1}{|Q|}\int_{Q}\int_{Q}\|W^{\frac{1}{p}}(x)V^{-\frac{1}{p}}(y)\|g(x)f(y)dxdy\\ &\leq 2\sum_{Q\in\mathcal{S}}\|g\|_{\overline{A},Q}\int_{Q}\|\|W^{\frac{1}{p}}(x)V^{-\frac{1}{p}}(y)\|\|_{A_{x},Q}f(y)dy\leq\\ &\leq c\sum_{Q\in\mathcal{S}}\|f\|_{\overline{B},Q}\|g\|_{\overline{A},Q}|E_{Q}|\|\|\|W^{\frac{1}{p}}(x)V^{-\frac{1}{p}}(y)\|\|_{A_{x},Q}\|_{B_{y},Q}\\ &\leq c\sup_{S}\|\|\|W^{\frac{1}{p}}(x)V^{-\frac{1}{p}}(y)\|\|_{A_{x},Q}\|_{B_{y},S}\sum_{Q\in\mathcal{S}}\|f\|_{\overline{B},Q}\|g\|_{\overline{A},Q}|E_{Q}|\\ &\leq c\|M_{\overline{A}}\|_{L^{p^{\prime}}}\|M_{\overline{B}}\|_{L^{p}}\sup_{Q}\|\|\|W^{\frac{1}{p}}(x)V^{-\frac{1}{p}}(y)\|\|_{A_{x},Q}\|_{B_{y},Q}\|f\|_{{L^{p}(\mathbb{R}^{d})}}\|g\|_{{L^{p^{\prime}}(\mathbb{R}^{d})}}.\end{aligned}$

The other estimate is obtained arguing analogously. ∎

In the case of commutators we can provide the following counterpart

Lemma 4**.**

Let $A,B,C,D$ be Young functions. Then

[TABLE]

where $\Lambda_{1}=\|M_{\overline{A}}\|_{L^{p^{\prime}}}\|M_{\overline{B}}\|_{L^{p}}\min\left\{\kappa_{1},\kappa_{2}\right\}$ with

[TABLE]

and $\Lambda_{2}=\|M_{\overline{C}}\|_{L^{p^{\prime}}}\|M_{\overline{D}}\|_{L^{p}}\min\left\{\kappa_{3},\kappa_{4}\right\}$ with

[TABLE]

Proof.

We recall that

[TABLE]

Without loss of generality we may assume that $f,g\geq 0$. For the first term we can argue as follows

$\begin{aligned} &\sum_{Q\in\mathcal{S}}\frac{1}{|Q|}\int_{Q}|b(x)-\langle b\rangle_{Q}|\int_{Q}\|W^{\frac{1}{p}}(x)V^{-\frac{1}{p}}(y)\|f(y)g(x)dydx\\ &=\sum_{Q\in\mathcal{S}}\frac{1}{|Q|}\int_{Q}\int_{Q}|b(x)-\langle b\rangle_{Q}|\|W^{\frac{1}{p}}(x)V^{-\frac{1}{p}}(y)\|g(x)f(y)dxdy\\ &\leq 2\sum_{Q\in\mathcal{S}}\|g\|_{\overline{A},Q}\int_{Q}\||b(x)-\langle b\rangle_{Q}|\|W^{\frac{1}{p}}(x)V^{-\frac{1}{p}}(y)\|\|_{A_{x},Q}f(y)dy\leq\\ &\leq c\sum_{Q\in\mathcal{S}}\|f\|_{\overline{B},Q}\|g\|_{\overline{A},Q}|E_{Q}|\|\||b(x)-\langle b\rangle_{Q}|\|W^{\frac{1}{p}}(x)V^{-\frac{1}{p}}(y)\|\|_{A_{x},Q}\|_{B_{y},Q}\\ &\leq c\sup_{S}\|\||b(x)-b_{Q}|\|W^{\frac{1}{p}}(x)V^{-\frac{1}{p}}(y)\|\|_{A_{x},Q}\|_{B_{y},S}\sum_{Q\in\mathcal{S}}\|f\|_{\overline{B},Q}\|g\|_{\overline{A},Q}|E_{Q}|\\ &\leq c\|M_{\overline{A}}\|_{L^{p^{\prime}}}\|M_{\overline{B}}\|_{L^{p}}\sup_{Q}\|\||b(x)-\langle b\rangle_{Q}|\|W^{\frac{1}{p}}(x)V^{-\frac{1}{p}}(y)\|\|_{A_{x},Q}\|_{B_{y},Q}\|f\|_{{L^{p}(\mathbb{R}^{d})}\|g\|_{L^{p^{\prime}}(\mathbb{R}^{d})}}.\end{aligned}$

Arguing analogously we obtain the rest of the estimates. ∎

5.2.2. Proof of estimates (2.3) and (2.5)

Again we deal first with (2.3). We will use Lemma 3. Let us choose $\overline{B}(t)=t^{\frac{p+1}{2}}$ and $A(t)=t^{rp}$ with $r=1+\frac{1}{2^{d+11}[W]_{A_{1,\infty}^{sc}}}$. We observe that $B(t)\simeq t^{\frac{p+1}{p-1}}$ . It’s not hard to check that ${\|M_{\overline{B}}\|_{{L^{p}\rightarrow L^{p}}}\leq c_{d}(p^{\prime})^{\frac{1}{p}}}$ and that ${\|M_{\overline{A}}\|_{L^{p^{\prime}}\rightarrow L^{p^{\prime}}}\leq c_{d}p^{\frac{1}{p^{\prime}}}[W]_{A_{1,\infty}^{sc}}^{\frac{1}{p^{\prime}}}}$ . We observe that using Lemma 2 and the definition of $A_{1}$ weight,

[TABLE]

and we are done.

Now we turn our attention to (2.5). We use Lemma 4. First we choose $\overline{B}(t)=t^{\frac{p+1}{2}}$ and $A(t)=t^{sp}$ with $s=\frac{r+1}{2}$ and $r=1+\frac{1}{2^{d+11}[W]_{A_{1,\infty}^{sc}}}$. For that choice of $s$ we have that $\left(\frac{r}{s}\right)^{\prime}{=}2r^{\prime}$. Notice that again $B(t)\simeq t^{\frac{p+1}{p-1}},\ {\|M_{\overline{B}}\|_{L^{p}\rightarrow L^{p}}\leq c(p^{\prime})^{\frac{1}{p}}},$ and that $\|M_{\overline{A}}\|_{L^{p^{\prime}}\rightarrow L^{p^{\prime}}}\leq c_{d}p^{\frac{1}{p^{\prime}}}[W]_{A_{1,\infty}^{sc}}^{\frac{1}{p^{\prime}}}$ . On the other hand,

[TABLE]

From this point arguing as in (5.6) we have that

[TABLE]

For the other term, we choose $\overline{D}(t)=t^{\frac{p+1}{2}}$ and $C(t)=t^{rp}$ with $r=1+\frac{1}{2^{d+11}[W]_{A_{1,\infty}^{sc}}}$. Then

[TABLE]

Arguing as above,

[TABLE]

Hence

[TABLE]

Consequently gathering all the preceding estimates we obtain (2.5).

5.2.3. Proof of estimates (2.8) and (2.10)

We deal first with (2.8). We rely again upon Lemma 3. We note that choosing $A(t)=t^{rp}$ with $r=1+\frac{1}{2^{d+11}[W]_{A_{q,\infty}^{sc}}}$ and $B(t)=t^{q^{\prime}}$ we have that ${\|M_{\overline{A}}\|_{L^{p^{\prime}}\rightarrow L^{p^{\prime}}}\leq c_{d}(r^{\prime})^{\frac{1}{p^{\prime}}}\leq c_{d}[W]_{A_{q,\infty}^{sc}}^{\frac{1}{p^{\prime}}}}$ and $\|M_{\overline{B}}\|_{L^{p}}\leq c_{d,p,q}$. On the other hand, notice that

[TABLE]

By Lemma 2

[TABLE]

Then

[TABLE]

by (5.1) we have that

[TABLE]

Gathering the preceding estimates and taking into account that $r=1+\frac{1}{2^{d+11}[W]_{A_{q,\infty}^{sc}}}$ we obtain that (2.8) holds.

Let us deal now with (2.10). Arguing analogously as above, we will use Lemma 4. First we choose $B(t)=t^{q^{\prime}}$ and $A(t)=t^{sp}$ with $s=\frac{r+1}{2}$ and $r=1+\frac{1}{2^{d+11}[W]_{A_{q,\infty}^{sc}}}$. For that choice of $s$ we have that $\left(\frac{r}{s}\right)^{\prime}{=}2r^{\prime}$. It is also straightforward that $\|M_{\overline{B}}\|_{L^{p}}\leq c_{n,p,q}$ and that $\|M_{\overline{A}}\|_{L^{p^{\prime}}}\leq c_{n,d}p\left(s^{\prime}\right)^{\frac{1}{p^{\prime}}}\leq c_{n,d}p[W]_{A_{q,\infty}^{sc}}^{\frac{1}{p^{\prime}}}$ . Then we have that

[TABLE]

Arguing as above,

[TABLE]

For the other term we note that choosing $C(t)=t^{rp}$ with $r=1+\frac{1}{2^{d+11}[W]_{A_{q,\infty}^{sc}}}$ and $D(t)=t^{q^{\prime}}$ we have that $\|M_{\overline{C}}\|_{L^{p^{\prime}}}\leq c_{d}(r^{\prime})^{\frac{1}{p^{\prime}}}\leq c_{n,d}{[W]_{A_{q,\infty}^{sc}}^{\frac{1}{p^{\prime}}}}$ and $\|M_{\overline{D}}\|_{L^{p}}\leq c_{d,p,q}$. We observe that

[TABLE]

Taking into account (5.7), and arguing as above

[TABLE]

Gathering all the choices and estimates above, a direct application of Lemma 4 yields the desired estimate.

5.3. Proof of the estimates for Maximal Rough Singular Integrals

Arguing as in [4], we have that

[TABLE]

where we interpret the product in second term as the right endpoint of the Minkowski product

[TABLE]

which, in the case of ${A,B\subset\mathbb{R}^{d}}$ being convex symmetric sets is a closed symmetric interval. The estimate for the first term is (2.1) in the case $q=1$ and (2.6) in the case $q>1$, so we are left with settling the estimate for the second term. We proceed as follows.

First we notice that if $a\in\left\langle\left\langle W^{-\frac{1}{p}}\vec{f}\right\rangle\right\rangle_{1+\varepsilon,Q}$ and $b\in\left\langle\left\langle W^{\frac{1}{p}}\vec{g}\right\rangle\right\rangle_{1+\varepsilon,Q}$ then

[TABLE]

where $\varphi_{a,Q},\psi_{b,Q}\in L^{(1+\varepsilon)^{\prime}}(Q)$. Since $W^{\frac{1}{p}}$ is positive definite and symmetric a.e. we have that

[TABLE]

Then

[TABLE]

We are going to obtain a suitable control for this term providing an argument analogous to the one we gave for Calderón-Zygmund operators. Let $\tau=8\cdot 2^{d+11}$,

[TABLE]

Then we have that

[TABLE]

It is not hard to check that

[TABLE]

Indeed, taking into account (5.5) it suffices to prove that

[TABLE]

First we note that

[TABLE]

Working on the denominator we have that

[TABLE]

It is clear that $(p-1)\tau\varepsilon\leq 1$. Combining this estimate with the identities above,

[TABLE]

Now we focus on the proof of (2.4). In that case, by Lemma 2, since $(1+\varepsilon)s\leq 1+\frac{1}{2^{d+11}[W]_{A_{q,\infty}^{sc}}}$,

[TABLE]

This yields that

[TABLE]

by (5.8) and taking into account that from (5.5) it follows that

[TABLE]

In the case $q>1$, namely, to settle (2.9), we argue as follows. By Lemma 2, since $(1+\varepsilon)s\leq 1+\frac{1}{2^{d+11}[W]_{A_{q,\infty}^{sc}}}$

[TABLE]

If we call $W=V^{1-q}$, then

[TABLE]

where the last estimate follows from Proposition 1. Taking that estimate into account,

[TABLE]

by (5.8) and taking into account that from (5.5) it is not hard to derive that

[TABLE]

This ends the proof of (2.9).

Acknowledgements

We would like to thank Francesco Di Plinio for suggesting us to address the problem of the maximal rough singular integral.

The second author would like to express his gratitude to the Department of Mathematics of Lund University for the hospitality shown during his visit between September and November 2017.