Two matrix weighted inequalities for commutators with fractional integral operators

Abstract

In this paper we prove two matrix weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a matrix symbol. More precisely, we extend the recent results of the second author, Pott, and Treil on two matrix weighted norm inequalities for commutators of Calderon-Zygmund operators and multiplication by a matrix symbol to the fractional integral operator setting. In particular, we completely extend the fractional Bloom theory of Holmes, Rahm, and Spencer to the two matrix weighted setting with a matrix symbol and also provide new two matrix weighted norm inequalities for commutators of fractional integral operators with a matrix symbol and two arbitrary matrix weights. These results are new even in the scalar one weighted setting with a scalar symbol.

Introduction

Let w be a weight on ${\mathbb{R}}^d$ and let $L^p(w)$ be the standard weighted Lebesgue space with respect to the norm

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

Furthermore, let A${}_{p,q}$ for $p,q>1$ be the Muckenhoupt class of weights w satisfying

$$\underset{\begin{array}{c}\hfill Q\subseteq {\mathbb{R}}^d\hfill \\ \hfill Q\text{ is a cube}\hfill \end{array}}{\sup }(\underset{Q}{⨍}w(x)dx){(\underset{Q}{⨍}{w}^{-\frac{p^{\prime }}{q}}(x)dx)}^{\frac{q}{p^{\prime }}}<\infty$$

where ${⨍}_Q$ is the unweighted average over Q (which will also occasionally be denoted by $m_Q$). When $p=q$ we write $A_p:=A_{p,p}$ as usual.

Given a weight ν, we say $b\in {\text{BMO}}_{\nu }$ if

$${\Vert b\Vert }_{{\text{BMO}}_{\nu }}=\underset{\begin{array}{c}\hfill Q\subseteq {\mathbb{R}}^d\hfill \\ \hfill Q\text{ is a cube}\hfill \end{array}}{\sup }\frac{1}{\nu (Q)}\underset{Q}{\int }|b(x)-m_{Q}b|dx<\infty$$

(where $\nu (Q)={\int }_Q\nu$) so that clearly $\text{BMO}={\text{BMO}}_{\nu }$ when $\nu \equiv 1$. Further, given a linear operator T, define the commutator $[M_b,T]=M_{b}T-T{M}_b$ with $M_b$ being multiplication by b. In the papers [10], [11] the authors extended earlier work of S. Bloom [3] and proved that if $u,v\in {\text{A}}_p$ and T is any Calderón-Zygmund operator (CZO) then

$${\Vert [M_b,T]\Vert }_{L^p(u)\to L^p(v)}\lesssim {\Vert b\Vert }_{{\text{BMO}}_{\nu }}$$

where $\nu ={(u{v}^{-1})}^{\frac{1}{p}}$ and it was proved in [11] that if $R_s$ is the $s^{\text{th}}$ Riesz transform then

$${\Vert b\Vert }_{{\text{BMO}}_{\nu }}\lesssim \underset{1\le s\le d}{\max }{\Vert [M_b,R_s]\Vert }_{L^p(u)\to L^p(v)}.$$

Furthermore, let $I_{\alpha }$ be the fractional integral operator defined by the formula

$$I_{\alpha }f(x)=\underset{{\mathbb{R}}^d}{\int }\frac{f(y)}{{|x-y|}^{d-\alpha }}\mathrm{d}y,\text{ for }0<\alpha <d.$$

It was proved in [12] that if $0<\alpha <d$ and $\alpha /d+1/q=1/p$, if $u,v\in A_{p,q}$, and if $\nu =u^{\frac{1}{q}}{v}^{-\frac{1}{q}}$ then

$${\Vert [M_b,I_{\alpha }]\Vert }_{L^p(u^{\frac{p}{q}})\to L^p(v)}\approx {\Vert b\Vert }_{\text{BMO}(\nu )}$$

(see also [1] for an iterated commutator version of (1.3)).

On the other hand, matrix weighted extensions and generalizations of (1.1) and (1.2) that surprisingly hold for two arbitrary matrix weights (and provided new results even in the scalar $p=2$ setting of a single scalar weight) were proved in [15], and it is the purpose of this paper to extend the results of [15] to the fractional setting, providing matrix weighted extensions of (1.3) that hold for two arbitrary matrix weights. Note that for the rest of this paper we will assume that $0<\alpha <d$ and $\alpha /d+1/q=1/p$.

In particular, for any linear operator T acting on scalar valued functions on ${\mathbb{R}}^d$, we can canonically extend T to act on ${\mathbb{C}}^n$ valued functions $\overrightarrow{f}$ by the formula $T\overrightarrow{f}:={\sum }_{j=1}^n\left(T{〈\overrightarrow{f},{\overrightarrow{e}}_j〉}_{{\mathbb{C}}^n}\right){\overrightarrow{e}}_j$ where $\{{\overrightarrow{e}}_j\}$ is any orthonormal basis of ${\mathbb{C}}^n$ (and note that this is easily seen to be independent of the orthonormal basis chosen). Let $W:{\mathbb{R}}^d\to {\mathbb{M}}_{n\times n}$ be an $n\times n$ matrix weight (a positive definite a.e. ${\mathbb{M}}_{n\times n}$ valued function on ${\mathbb{R}}^d$) and let $L^p(W)$ be the space of ${\mathbb{C}}^n$ valued functions $\overrightarrow{f}$ such that

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

Furthermore, for $p,q>1$ we will say that a matrix weight W is a matrix A${}_{p,q}$ weight (see [14]) if it satisfies

$${\Vert W\Vert }_{{\text{A}}_{p,q}}=\underset{\begin{array}{c}\hfill Q\subset {\mathbb{R}}^d\hfill \\ \hfill Q\text{ is a cube}\hfill \end{array}}{\sup }\underset{Q}{⨍}{\left(\underset{Q}{⨍}{\Vert W^{\frac{1}{q}}(x)W^{-\frac{1}{q}}(y)\Vert }^{p^{\prime }}dy\right)}^{\frac{q}{p^{\prime }}}dx<\infty$$

and when $p=q$ we say W is a matrix A_p weight (see [19]).

Now for scalar weights u and v, notice that by multiple uses of the A_q property and Hölder's inequality we have

$$m_Q\nu \approx {(m_{Q}u)}^{\frac{1}{q}}{(m_Q{v}^{-\frac{q^{\prime }}{q}})}^{\frac{1}{q^{\prime }}}\approx {(m_{Q}u)}^{\frac{1}{q}}{(m_{Q}v)}^{-\frac{1}{q}}\approx (m_Q{u}^{\frac{1}{q}}){(m_Q{v}^{\frac{1}{q}})}^{-1}.$$

Thus, $b\in {\text{BMO}}_{\nu }$ when u and v are A_q weights if and only if

$$\underset{\begin{array}{c}\hfill Q\subseteq \mathbb{R}\hfill \\ \hfill Q\text{ is a cube}\hfill \end{array}}{\sup }\underset{Q}{⨍}(m_Q{v}^{\frac{1}{q}}){(m_Q{u}^{\frac{1}{q}})}^{-1}|b(x)-m_{Q}b|dx<\infty ,$$

which is a condition that easily extends to the matrix weighted setting, noting that $A_{p,q}\subset A_q$ when $q>p$, since then $q^{\prime }<p^{\prime }$ and so Hölder's inequality gives us

$$\underset{Q}{⨍}{\left(\underset{Q}{⨍}{\Vert W^{\frac{1}{q}}(x)W^{-\frac{1}{q}}(y)\Vert }^{q^{\prime }}dy\right)}^{\frac{q}{q^{\prime }}}dx\le \underset{Q}{⨍}{\left(\underset{Q}{⨍}{\Vert W^{\frac{1}{q}}(x)W^{-\frac{1}{q}}(y)\Vert }^{p^{\prime }}dy\right)}^{\frac{q}{p^{\prime }}}dx.$$

Namely, if $U,V$ are $n\times n$ matrix A${}_{p,q}$ weights, then we define ${\text{BMO}}_{V,U}^{p,q}$ to be the space of $n\times n$ locally integrable matrix functions B where

$${\Vert B\Vert }_{{\text{BMO}}_{V,U}^{p,q}}=\underset{\begin{array}{c}\hfill Q\subseteq {\mathbb{R}}^d\hfill \\ \hfill Q\text{ is a cube}\hfill \end{array}}{\sup }{(\underset{Q}{⨍}\Vert (m_Q{V}^{\frac{1}{q}})(B(x)-m_{Q}B){(m_Q{U}^{\frac{1}{q}})}^{-1}\Vert dx)}^{\frac{1}{q}}<\infty$$

so that ${\Vert b\Vert }_{{\text{BMO}}_{V,U}^{p,q}}\approx {\Vert b\Vert }_{\text{BMO}\nu }$ if $U,V$ are scalar weights and b is a scalar function. Note that the ${\text{BMO}}_{V,U}^{p,q}$ condition is much more naturally defined in terms of reducing matrices, which will be discussed in Section 3.

We will need a definition before we state our first result. We say that a linear operator R acting on scalar functions is a fractional lower bound operator if for any $n\in \mathbb{N}$ and any $n\times n$ matrix weight W we have

$${\Vert W\Vert }_{{\text{A}}_{p,q}}^{\frac{1}{q}}\lesssim {\Vert T\Vert }_{L^p(W^{\frac{p}{q}})\to L^q(W)}$$

with the bound independent of W (but not necessarily independent of n), and ${\Vert T\Vert }_{L^p(W^{\frac{p}{q}})\to L^q(W)}<\infty$ if W is a matrix A${}_{p,q}$ weight.

Theorem 1.1

Let T be any linear operator acting on scalar valued functions where its canonical ${\mathbb{C}}^n$ valued extension is bounded from $L^p(W^{\frac{p}{q}})$ to $L^q(W)$ for all $n\times n$ matrix A${}_{p,q}$ weights W and all $n\in \mathbb{N}$ with bound depending on $T,n,d,p$, and ${\Vert W\Vert }_{{\mathit{\text{A}}}_{p,q}}$ (which is known to be true for fractional integral operators, see [14, Theorem 1.4]). If $U,V$ are $m\times m$ matrix A_p weights and B is an $m\times m$ locally integrable matrix function for some $m\in \mathbb{N}$, then

$${\Vert [M_B,T]\Vert }_{L^p(U^{\frac{p}{q}})\to L^q(V)}\lesssim {\Vert B\Vert }_{{\mathit{\text{BMO}}}_{V,U}^{p,q}}$$

with bounds depending on $T,m,d,p,{\Vert U\Vert }_{{\mathit{\text{A}}}_{p,q}}$ and ${\Vert V\Vert }_{{\mathit{\text{A}}}_{p,q}}$.

Furthermore, for any fractional lower bound operator T we have the lower bound estimate

$${\Vert B\Vert }_{{\mathit{\text{BMO}}}_{V,U}^{p,q}}\lesssim {\Vert [M_B,T]\Vert }_{L^p(U^{\frac{p}{q}})\to L^q(V)}$$

Like in [15], we will use matrix weighted arguments inspired by [9] in the next section to prove Theorem 1.1 in terms of a weighted BMO quantity ${\Vert B\Vert }_{{\widetilde{\text{BMO}}}_{V,U}^{p,q}}$ that is equivalent to ${\Vert B\Vert }_{{\text{BMO}}_{V,U}^{p,q}}$ when U and V are matrix A${}_{p,q}$ weights (see Theorem 3.1) but is much more natural for more arbitrary matrix weights U and V. More precisely, define

$${\Vert B\Vert }_{{\widetilde{\text{BMO}}}_{V,U}^{p,q}}^q=\underset{\begin{array}{c}\hfill Q\subseteq {\mathbb{R}}^d\hfill \\ \hfill Q\text{ is a cube}\hfill \end{array}}{\sup }\underset{Q}{⨍}{\left(\underset{Q}{⨍}{\Vert V^{\frac{1}{q}}(x)(B(x)-B(y))U^{-\frac{1}{q}}(y)\Vert }^{p^{\prime }}dy\right)}^{\frac{q}{p^{\prime }}}dx.$$

We will then give relatively short proofs of the following two results in Section 2.

Lemma 1.2

Let T be any linear operator defined on scalar valued functions where its canonical ${\mathbb{C}}^n$ valued extension T for any $n\in \mathbb{N}$ satisfies

$${\Vert T\Vert }_{L^p(W^{\frac{p}{q}})\to L^q(W)}\le \varphi ({\Vert W\Vert }_{{\mathit{\text{A}}}_{p,q}})$$

for some positive increasing function ϕ (possibly depending on $T,d,n,p,q$). If $U,V$ are $m\times m$ matrix A${}_{p,q}$ weights and B is a locally integrable $m\times m$ matrix valued function for some $m\in \mathbb{N}$, then

$${\Vert [M_B,T]\Vert }_{L^p(U^{\frac{p}{q}})\to L^q(V)}\le {\Vert B\Vert }_{{\widetilde{\mathit{\text{BMO}}}}_{V,U}^{p,q}}\varphi (3^{\frac{q}{p^{\prime }}}({\Vert U\Vert }_{{\mathit{\text{A}}}_{p,q}}+{\Vert V\Vert }_{{\mathit{\text{A}}}_{p,q}})+1)$$

Lemma 1.3

If T is any fractional lower bound operator then for any $m\times m$ matrix A${}_{p,q}$ weights $U,V$ and an $m\times m$ matrix symbol B we have

$${\Vert B\Vert }_{{\widetilde{\mathit{\text{BMO}}}}_{V,U}^{p,q}}\lesssim {\Vert [M_B,T]\Vert }_{L^p(U^{\frac{p}{q}})\to L^q(V)}$$

where the bound depends possibly on $n,p,d$ and T but is independent of U and V.

As in [15], we will prove that the fractional integral operator is a fractional lower bound operator in Section 4 by utilizing the Schur multiplier/Wiener algebra ideas from [17], and thus recover (1.7). These arguments will in fact prove the following (see [15] for an analogous result with respect to the Riesz transforms). Here, for ease of notation, we set $U^{\prime }=U^{-\frac{p^{\prime }}{q}}$ and $V^{\prime }=V^{-\frac{p^{\prime }}{q}}$.

Theorem 1.4

Let U and V be any (not necessarily A_p) matrix weights. If B is any locally integrable $m\times m$ matrix valued function then

$$\max \{{\Vert B\Vert }_{{\widetilde{\mathit{\text{BMO}}}}_{V,U}^{p,q}},{\Vert B\Vert }_{{\widetilde{\mathit{\text{BMO}}}}_{U^{\prime },V^{\prime }}^{q^{\prime },p^{\prime }}}\}\lesssim {\Vert [M_B,I_{\alpha }]\Vert }_{L^p(U^{\frac{p}{q}})\to L^q(V)}.$$

Note that the two quantities ${\Vert B\Vert }_{{\widetilde{\text{BMO}}}_{V,U}^{p,q}}$ and ${\Vert B\Vert }_{{\widetilde{\text{BMO}}}_{U^{\prime },V^{\prime }}^{q^{\prime },p^{\prime }}}$ are equivalent when $U,V\in {\text{A}}_{p,q}$ (which will be proved in Section 4) and in general should be thought of as “dual” matrix weighted BMO quantities. Finally, we will show that an Orlicz “bumped” version of these conditions is sufficient for the general two matrix weighted boundedness of fractional integral operators. In particular, we will prove the following result in Section 5 (see [15] for an analogous result for Calderon-Zygmund operators):

Proposition 1.5

Let U and V be any $m\times m$ matrix weights, and suppose that C and D are Young functions with $\overline{D}\in B_{p,q}$ and $\overline{C}\in B_{q^{\prime }}$.

Then

$${\Vert [M_B,I_{\alpha }]\Vert }_{L^p(U^{\frac{p}{q}})\to L^q(V)}\lesssim \min \{{\kappa }_1,{\kappa }_2\}$$

where

$$\begin{gathered} {\kappa }_1=\underset{Q}{\sup }{\Vert {\Vert V^{\frac{1}{q}}(x)(B(x)-B(y))U^{-\frac{1}{q}}(y)\Vert }_{C_x,Q}\Vert }_{D_y,Q} \\ {\kappa }_2=\underset{Q}{\sup }{\Vert {\Vert V^{\frac{1}{q}}(x)(B(x)-B(y))U^{-\frac{1}{q}}(y)\Vert }_{D_y,Q}\Vert }_{C_x,Q} \end{gathered}$$

We refer the reader to Section 5.2 in [4] for the standard Orlicz space related definitions used in the statement of Proposition 1.5.

It is important to emphasize that Theorem 1.4 and Proposition 1.5 are new, even in the scalar setting of a single weight (and a scalar symbol). In fact, the upper bound in Proposition 1.5 is the first such scalar weighted upper bound type result for commutators of fractional integral operators with a scalar symbol and two arbitrary scalar weights (see [20] for closely related but different upper bound results for commutators of fractional integral operators with a scalar symbol and two arbitrary scalar weights). On the other hand, Theorem 1.4 is (to the authors' knowledge) the only scalar weighted lower bound result known for commutators of fractional integral operators with a scalar symbol and two arbitrary scalar weights (see however [7] for closely related two weighted lower bound results for iterated commutators of singular integral operators with a scalar symbol and two arbitrary scalar weights).

Finally, let us mention that results similar to Theorem 1.4 and Proposition 1.5 for iterated commutators of singular integral operators with (not necessarily equal) matrix symbols have been proved by the author in a preprint with S. Pott and I. Rivera-Rios. It is likely that versions of Theorem 1.4 and Proposition 1.5 for iterated commutators of fractional integral operators with (again not necessarily equal) matrix symbols should also be true, and will be explored in a forthcoming paper.