The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators

Abstract

We investigate the weighted bounds for multilinear maximal functions and Calderón–Zygmund operators from $L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)$ to $L^{p}(v_{\vec {w}})$, where $1<p_1,\cdots ,p_m<\infty$ with $1/{p_1}+\cdots +1/{p_m}=1/p$ and $\vec {w}$ is a multiple $A_{\vec {P}}$ weight. We prove the sharp bound for the multilinear maximal function for all such $p_1,\ldots , p_m$ and prove the sharp bound for $m$-linear Calderón–Zymund operators when $p\ge 1$.

1 Introduction and Main Results

A weight is a non-negative locally integrable function. Given $p$, $1<p<\infty$, an $A_p$ weight is one that satisfies the following

$$\begin{aligned}{}[w]_{A_p} =\sup _Q \left( \frac{1}{|Q|}\int \limits _Q w\right) \left( \frac{1}{|Q|}\int \limits _Q w^{1-p'}\right) ^{p-1}<\infty . \end{aligned}$$

It is well known that the Hardy–Littlewood maximal operator and Calderón–Zygmund operators are bounded on $L^p(w)$ when $w\in A_p$. The sharp dependence for the Hardy–Littlewood maximal function is given by

$$\begin{aligned} \Vert M\Vert _{L^p(w)\rightarrow L^p(w)}\le C_{n,p}[w]_{A_p}^{\frac{p'}{p}}. \end{aligned}$$

(1.1)

Inequality (1.1) was first proven by Buckley [1]. (We refer the reader to [16] for a beautiful proof of this inequality and a summary of the history.) Later it was proven by Hytönen [9] that if $T$ is a Calderón–Zygmund operator then

$$\begin{aligned} \Vert T\Vert _{L^p(w)\rightarrow L^p(w)}\le C_{n,p,T}[w]_{A_p}^{\max (1,\frac{p'}{p})}. \end{aligned}$$

(1.2)

Again, we refer the reader to [5, 10, 15, 17] for the background material and further references. Later (1.2) was improved in [11–13]. In this article we prove the multilinear analogs of inequalities (1.1) and (1.2). We begin with a few definitions.

First, let us define multiple $A_{\vec {P}}$ weights. Lerner et al. [18] introduced the theory of multiple $A_{\vec {P}}$ weights beginning with the following definition.

Definition 1.1

Let $\vec {P}=(p_1,\cdots ,p_m)$ with $1\le p_1,\cdots ,p_m<\infty$ and $1/{p_1}+\cdots +1/{p_m}=1/p$. Given $\vec {w}=(w_1,\cdots , w_m)$, set

$$\begin{aligned} v_{\vec {w}}=\prod _{i=1}^m w_i^{p/{p_i}}. \end{aligned}$$

We say that $\vec {w}$ satisfies the multilinear $A_{\vec {P}}$ condition if

$$\begin{aligned}{}[\vec {w}]_{A_{\vec {P}}}:=\sup _Q \left( \frac{1}{|Q|}\int \limits _Q v_{\vec {w}}\right) \prod _{i=1}^m\left( \frac{1}{|Q|}\int \limits _Q w_i^{1-p_i'}\right) ^{p/{p_i'}}<\infty , \end{aligned}$$

where $[\vec {w}]_{A_{\vec {P}}}$ is called the $A_{\vec {P}}$ constant of $\vec {w}$. When $p_i=1$, $( \frac{1}{|Q|}\int \limits _Q w_i^{1-p_i'})^{1/{p_i'}}$ is understood as $(\inf _Q w_i)^{-1}$.

It is easy to see that in the linear case (that is, if $m = 1$) $[\vec {w}]_{A_{\vec {P}}}=[w]_{A_p}$ is the usual $A_p$ constant. In [18], it was shown that for $1<p_1,\cdots ,p_m<\infty$, $\vec {w}\in A_{\vec {P}}$ if and only if $w_i^{1-p_i'}\in A_{mp_i'}$ and $v_{\vec {w}}\in A_{mp}$.

Given $\vec {f}=(f_1,\cdots ,f_m)$, we define the multilinear maximal function by

$$\begin{aligned} \mathcal {M}(\vec {f})=\sup _{Q\ni x}\prod _{i=1}^m\frac{1}{|Q|}\int \limits _{Q}|f_i|. \end{aligned}$$

In [18], the authors proved that $\vec {w}\in A_{\vec {P}}$ if and only if

$$\begin{aligned} \Vert \mathcal {M}(\vec {f})\Vert _{L^p(v_{\vec {w}})}\le C \prod _{i=1}^m \Vert f_i\Vert _{L^{p_i}(w_i)}. \end{aligned}$$

Recall that inequality (1.1) is sharp in the sense that the exponent on $[w]_{A_p}$ cannot be improved. The analogous question for the operator $\mathcal {M}$ has remained open. In [6], using mixed estimates involving $A_\infty$, Damián, Lerner and Pérez proved the following result.

Theorem A

[6, Theorem 1.2] Let $1<p_i<\infty$, $i=1,\cdots ,m$ and $1/p=1/{p_1}+\cdots +1/{p_m}$. Denote by $\alpha =\alpha (p_1,\cdots ,p_m)$ the best possible power in

$$\begin{aligned} \Vert \mathcal {M}(\vec {f})\Vert _{L^p(v_{\vec {w}})}\le C_{m,n,\vec {P}} [\vec {w}]_{A_{\vec {P}}}^\alpha \prod _{i=1}^m \Vert f_i\Vert _{L^{p_i}(w_i)}. \end{aligned}$$

(1.3)

Then the following results hold:

• (1) for all $1<p_1,\cdots ,p_m<\infty$, $\frac{m}{mp-1}\le \alpha \le \frac{1}{p}\left( 1+\sum _{i=1}^m\frac{1}{p_i-1}\right)$.

• (2) if $p_1 = p_2 = \cdots = p_m = r > 1$, then $\alpha =\frac{m}{r-1}$.

Interestingly, the mixed estimates involving $A_\infty$ do not yield the sharp dependence on the constant $[\vec w]_{A_{\vec {P}}}$. The sharp bound along the diagonal is obtained using similar methods to those found in [16] and these techniques only seem to work along the diagonal. In this paper we find the optimal power on $[\vec {w}]_{A_{\vec {P}}}$ for the full range of exponents, $1<p_1,\ldots ,p_m<\infty$. We emphasize that our results below not only improve those in Theorem A, but are also the best possible.

Theorem 1.2

Suppose $1<p_1,\ldots ,p_m<\infty$, $1/p=1/{p_1}+\cdots +1/{p_m}$, and $\vec {w}\in A_{\vec P}$. Then

$$\begin{aligned} \Vert \mathcal {M}(\vec {f})\Vert _{L^p(v_{\vec {w}})}\le C_{m,n,\vec {P}} [\vec {w}]_{A_{\vec {P}}}^{\max (\frac{p_1'}{p},\cdots , \frac{p_m'}{p})}\prod _{i=1}^m \Vert f_i\Vert _{L^{p_i}(w_i)}. \end{aligned}$$

(1.4)

Moreover the exponent on $[\vec w]_{A_{\vec P}}$ is the best possible. See Fig. 1 for a visualization of the bilinear case.

Fig. 1

The sharp exponents on $[\vec w]_{A_{\vec {P}}}$ for the bilinear maximal function

Next we turn to study weighted bounds of multilinear Calderón–Zygmund operators. The theory of multilinear Calderón–Zygmund operators originated in the works of Coifman and Meyer [3, 4] and was later developed by Christ and Journé [2], Kenig and Stein [14], and Grafakos and Torres [8]. The last work provides a comprehensive account of general multilinear Calderón–Zygmund operators which we follow in this paper.

Definition 1.3

Let $T$ be a multilinear operator initially defined on the $m$-fold product of Schwartz spaces and taking values into the space of tempered distributions,

$$\begin{aligned} T: \fancyscript{S}(\mathbb {R}^n)\times \cdots \times \fancyscript{S}(\mathbb {R}^n)\rightarrow \fancyscript{S}'(\mathbb {R}^n). \end{aligned}$$

we say that $T$ is an $m$-linear Calderón–Zygmund operator if, for some $1\le q_i<\infty$, it extends to a bounded multilinear operator from $L^{q_1} \times \cdots \times L^{q_m}$ to $L^q$ , where $1/{q_1}+\cdots +1/{q_m}=1/q$, and if there exists a function $K$, defined off the diagonal $x=y_1=\cdots =y_m$ in $(\mathbb {R}^n)^{m+1}$, satisfying

$$\begin{aligned} T(f_1,\cdots ,f_m)(x)=\int \limits _{(\mathbb {R}^n)^m}K(x,y_1,\cdots ,y_m)f_1(y_1)\cdots f_m(y_m)dy_1\cdots dy_m \end{aligned}$$

for all $x\notin \bigcap _{j=1}^m\mathrm{supp}\ f_i$ and

$$\begin{aligned} |K(y_0,y_1,\cdots ,y_m)|\le \frac{A}{(\sum _{k,l=0}^m |y_k-y_l|)^{mn}} \end{aligned}$$

and

$$\begin{aligned} |K(y_0,\cdots ,y_i,\cdots ,y_m)-K(y_0,\cdots ,y_i',\cdots ,y_m)|\le \frac{A|y_i-y_i'|^\varepsilon }{(\sum _{k,l=0}^m |y_k-y_l|)^{mn+\varepsilon }} \end{aligned}$$

for some $A,\varepsilon >0$ and all $0\le i\le m$, whenever $|y_i-y_i'|\le \frac{1}{2}\max _{0\le k\le m}|y_i-y_k|$.

It was shown in [8] that if $1/{r_1}+\cdots +1/{r_m}=1/r$, then an $m$-linear Calderón–Zygmund operator $T$ is bounded from $L^{r_1}\times \cdots \times L^{r_m}$ to $L^r$ when $1<r_i<\infty$ for all $i=1,\cdots ,m$; and $T$ is bounded from $L^{r_1}\times \cdots \times L^{r_m}$ to $L^{r,\infty }$ when at least one $r_i=1$. In particular, $T$ is bounded from $L^1\times \cdots \times L^1$ to $L^{1/m,\infty }$.

A weighted theory for $m$-linear Calderón–Zygmund operators was developed in [18], where it was shown that such operators are bounded from $L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)$ to $L^{p}(v_{\vec {w}})$ when $\vec {w}\in A_{\vec {P}}$. In [6], the authors proved a multilinear version of the $A_2$ conjecture. Specifically, for $p_1=\cdots =p_m=m+1$, it was shown that

$$\begin{aligned} \Vert T(\vec {f})\Vert _{L^p(v_{\vec {w}})}\lesssim [\vec {w}]_{A_{\vec {P}}}\prod _{i=1}^m\Vert f_i\Vert _{L^{p_i}(w_i)}, \end{aligned}$$

where the estimate for the power of $[\vec {w}]_{A_{\vec {P}}}$ is the best possible.

Due to the lack of an appropriate extrapolation theorem for multilinear operators with multiple weights, let alone a version with good constants, the sharp estimate is unknown for any other choices of $p_i$. In this paper, we give a sharp estimate for the case of $p\ge 1$. Specifically, we prove the following Theorem and refer the reader to Fig. 2 for a visualization of the bilinear case.

Theorem 1.4

Let $T$ be a multilnear Calderón–Zygmund operator, $\vec {P}=(p_1,\cdots ,p_m)$ with $1<p_1,\cdots ,p_m<\infty$ and $1/{p_1}+\cdots +1/{p_m}=1/p\le 1$. If $\vec {w}=(w_1,\cdots , w_m)\in A_{\vec {P}}$, then

$$\begin{aligned} \Vert T(f)\Vert _{L^p(v_{\vec {w}})}\le C_{n,m,{\vec {P}},T}[\vec {w}]_{A_{\vec {P}}}^{\max (1,\frac{p_1'}{p},\ldots ,\frac{p_m'}{p})}\prod _{i=1}^m\Vert f_i\Vert _{L^{p_i}(w_i)}. \end{aligned}$$

(1.5)

Moreover, for certain multilinear operators the exponent on $[\vec {w}]_{A_{\vec {P}}}$ is the best possible.

Fig. 2

The sharp exponents on $[\vec w]_{A_{\vec {P}}}$ for bilinear Calderón–Zygmund operators when $p\ge 1$

In order to prove Theorem 1.4 we will approximate multilinear Calderón–Zygmund operators by positive dyadic operators. The result of Damián et al. [6] states the following (see Sect. 2 for pertinent definition).

Theorem 1.5

[6, Theorem 1.4] Let $T$ be a multilinear Calderón–Zygmund operator and let $\mathcal {X}$ be a Banach function space over $\mathbb {R}^n$ equipped with Lebesgue measure. Then, for any appropriate $\vec {f}$,

$$\begin{aligned} \Vert T(\vec {f})\Vert _{\mathcal {X}}\le C_{T,m,n}\sup _{\fancyscript{D},\mathcal {S}}\Vert A_{\fancyscript{D},\mathcal {S}}(|\vec {f}|)\Vert _{\mathcal {X}}. \end{aligned}$$

(1.6)

When $p\ge 1$, $\mathcal {X}=L^p(v)$ is a Banach space. However, for $0<p<1$ it is not. Since the $m$-linear operators map into $L^p$ for $p>1/m$ we are are unable to obtain the full range. It is an interesting question as to whether the same decomposition can be obtained for non Banach spaces such as $L^p$ for $0<p<1$. Moreover, we believe that inequality (1.5) should hold for all $1<p_1,\ldots ,p_m<\infty$.

We will actually prove the estimate in Theorem 1.4 for the sparse operators $A_{\fancyscript{D},\mathcal {S}}$. For these operators, the main techniques are an extension of those found in [19], in which the second author proved the sharp weighted bound for linear Calderón–Zygmund operators without extrapolation.

The rest of this article is devoted to the following. In Sect. 2 we state some brief preliminary material. In Sect. 3 we will prove the main estimates in Theorems 1.2 and 1.4 and in Sect. 4 we will provide examples to show that our results are sharp.

2 Preliminaries

Recall that the standard dyadic grid in $\mathbb {R}^n$ consists of the cubes

$$\begin{aligned} 2^{-k}([0,1)^n+j),\quad k\in \mathbb {Z}, j\in \mathbb {Z}^n. \end{aligned}$$

Denote the standard grid by $\mathcal D$.

By a general dyadic grid $\fancyscript{D}$ we mean a collection of cubes with the following properties: (i) for any $Q\in \fancyscript{D}$ its sidelength, $l_Q$, is of the form $2^k$, $k\in \mathbb {Z}$; (ii) $Q\cap R \in \{Q,R,\varnothing \}$ for any $Q,R\in \fancyscript{D}$; (iii) the cubes of a fixed sidelength $2^k$ form a partition of $\mathbb {R}^n$.

Now we define the dyadic maximal function with respect to arbitrary weight

$$\begin{aligned} M_{w}^{\fancyscript{D}}f(x)=\sup _{Q\ni x, Q\in \fancyscript{D}}\frac{1}{w(Q)}\int \limits _Q |f|w. \end{aligned}$$

It is well-known that

$$\begin{aligned} \Vert M_{w}^{\fancyscript{D}}f\Vert _{L^p(w)}\le p'\Vert f\Vert _{L^p(w)},\quad 1<p<\infty , \end{aligned}$$

(2.1)

we refer the readers to [19] for a proof.

We will also need the notion of a sparse family of cubes. Given a dyadic grid $\fancyscript{D}$ we say that a family $\mathcal {S}$ is sparse if there are disjoint majorizing subsets, that is, for each $Q\in \mathcal {S}$ there exists $E_Q\subset Q$ such that $\{E_Q\}_{Q\in \mathcal {S}}$ is pairwise disjoint and $|E_Q|\ge \frac{1}{2}|Q|$. Sparse families have long played a role in Calderón–Zygmund theory, our definition can be found in [10]. Finally, in [6], the multilinear sparse operators,

$$\begin{aligned} A_{\fancyscript{D},\mathcal {S}}(\vec {f})=\sum _{Q\in \mathcal {S}}\bigg ( \prod _{i=1}^m \frac{1}{|Q|}\int \limits _{Q}f_i\bigg )\chi ^{}_{Q}, \end{aligned}$$

were defined and used to approximate multlinear Calderón–Zygmund operators (see Theorem 1.5).

3 Proof of Theorems 1.2 and 1.4

First, we give a proof for Theorem 1.2.

Proof of Theorem 1.2

In [6], the authors proved that there exists $2^n$ families of dyadic grids $\fancyscript{D}_\beta$ such that

$$\begin{aligned} \mathcal {M}(\vec {f})(x)\le 6^{mn}\sum _{\beta =1}^{2^n}\mathcal {M}^{\fancyscript{D}_\beta }(\vec {f})(x), \end{aligned}$$

where

$$\begin{aligned} \mathcal {M}^{\fancyscript{D}_\beta }(\vec {f})(x)=\sup _{Q\ni x, Q\in \fancyscript{D}_\beta }\prod _{i=1}^m\frac{1}{|Q|}\int \limits _{Q}|f_i|. \end{aligned}$$

Without loss of generality, it suffices to prove that

$$\begin{aligned} \Vert \mathcal {M}^{\fancyscript{D}}(\vec {f\sigma })\Vert _{L^p(v_{\vec {w}})}\le C_{m,n,\vec {P}} [\vec {w}]_{A_{\vec {P}}}^{\max _i(\frac{p_i^\prime }{p})}\prod _{i=1}^m \Vert f_i\Vert _{L^{p_i}(\sigma _i)}. \end{aligned}$$

for a general dyadic grid $\fancyscript{D}$, and $\mathcal {M}^\fancyscript{D}(\vec {f\sigma })=\mathcal {M}^\fancyscript{D}(f_1\sigma _1,\ldots ,f_m\sigma _m)$. Moreover, it was shown in [6, Lemma 2.2] that there exists a sparse subset $\mathcal {S}\subset \fancyscript{D}$ such that

$$\begin{aligned} \mathcal {M}^{\fancyscript{D}}(\vec {f\sigma })\lesssim \sum _{Q\in \mathcal {S}}\prod _{i=1}^m\left( \frac{1}{|Q|}\int \limits _Q |f_i|\sigma _i\,\right) \chi _{E_Q}. \end{aligned}$$

Without loss of generality, assume that $p_1=\min \{p_1,\cdots ,p_m\}$. We have

$$\begin{aligned} \int \limits _{\mathbb {R}^n}\mathcal {M}^{\fancyscript{D}}(\vec {f\sigma })^pv_{\vec {w}}&\lesssim \sum _{Q\in \mathcal {S}} \prod _{i=1}^m\left( \frac{1}{|Q|}\int \limits _Q |f_i|\sigma _i\,\right) ^pv_{\vec {w}}(Q)\\&= \sum _{Q\in \mathcal {S}}\frac{v_{\vec {w}}(Q)^{p_1'}\prod _{i=1}^m\sigma _i(Q)^{pp_1'/{p_i'}}}{|Q|^{mpp_1'}}\bigg ( \prod _{i=1}^m \int \limits _{Q}|f_i|\sigma _i \bigg )^p\nonumber \\&\quad \times \frac{|Q|^{mp(p_1'-1)}}{v_{\vec {w}}(Q)^{p_1'-1}\prod _{i=1}^m\sigma _i(Q)^{pp_1'/{p_i'}}}\nonumber \\&\le [\vec {w}]_{A_{\vec {P}}}^{p_1'}\sum _{Q\in \mathcal {S}}\frac{2^{mp(p_1'-1)}|E_Q|^{mp(p_1'-1)}}{v_{\vec {w}}(Q)^{p_1'-1}\prod _{i=1}^m\sigma _i(Q)^{pp_1'/{p_i'}}}\cdot \bigg ( \prod _{i=1}^m \int \limits _{Q}|f_i|\sigma _i \bigg )^p.\nonumber \end{aligned}$$

By Hölder’s inequality, we have

$$\begin{aligned} |E_Q|&= \int \limits _{E_Q}v_{\vec {w}}^{\frac{1}{mp}}\sigma _1^{\frac{1}{mp_1'}}\cdots \sigma _m^{\frac{1}{mp_m'}} \\&\le v_{\vec {w}}(E_Q)^{\frac{1}{mp}}\sigma _1(E_Q)^{\frac{1}{mp_1'}}\cdots \sigma _m(E_Q)^{\frac{1}{mp_m'}}\nonumber . \end{aligned}$$

(3.1)

Therefore,

$$\begin{aligned} |E_Q|^{mp(p_1'-1)}\le v_{\vec {w}}(E_Q)^{p_1'-1}\sigma _1(E_Q)^{\frac{p(p_1'-1)}{p_1'}}\cdots \sigma _m(E_Q)^{\frac{p(p_1'-1)}{p_m'}} \end{aligned}$$

and

$$\begin{aligned} \frac{p(p_1'-1)}{p_i'}-\frac{p}{p_i}=\frac{pp_1'}{p_i'}-p\ge 0. \end{aligned}$$

Since $E_Q\subset Q$, we have

$$\begin{aligned} v_{\vec {w}}(E_Q)^{p_1'-1}\le v_{\vec {w}}(Q)^{p_1'-1} \end{aligned}$$

and hence

$$\begin{aligned} \sigma _i(E_Q)^{\frac{p(p_1'-1)}{p_i'}-\frac{p}{p_i}}\le \sigma _i(Q)^{\frac{pp_1'}{p_i'}-p},\quad i=1,\cdots ,m. \end{aligned}$$

It follows that

$$\begin{aligned}&\sum _{Q\in \mathcal {S}}\frac{|E_Q|^{mp(p_1'-1)}}{v_{\vec {w}}(Q)^{p_1'-1}\prod _{i=1}^m\sigma _i(Q)^{pp_1'/{p_i'}}} \cdot \bigg ( \prod _{i=1}^m \int \limits _{Q}|f_i|\sigma _i\bigg )^p\\&\quad \le \sum _{Q\in \mathcal {S}}\prod _{i=1}^m \bigg ( \frac{1}{\sigma _i(Q)}\int \limits _{Q}|f_i|\sigma _i\bigg )^p \sigma _i(E_Q)^{p/{p_i}}\nonumber \\&\quad \le \prod _{i=1}^m\left( \sum _{Q\in \mathcal {S}}\bigg (\frac{1}{\sigma _i(Q)}\int \limits _{Q}|f_i|\sigma _i \bigg )^{p_i} \sigma _i(E_{Q})\right) ^{p/{p_i}}\nonumber \\&\quad \le \prod _{i=1}^m\Vert M_{\sigma _i}^{\fancyscript{D}}(f_i)\Vert _{L^{p_i}(\sigma _i)}^p\nonumber \\&\quad \lesssim \prod _{i=1}^m\Vert f_i\Vert _{L^{p_i}(\sigma _i)}^p.\nonumber \end{aligned}$$

Hence

$$\begin{aligned} \Vert \mathcal {M}^{\fancyscript{D}}(\vec {f})\Vert _{L^p(v_{\vec {w}})}\le C_{m,n,\vec {P}} [\vec {w}]_{A_{\vec {P}}}^{\max _i(\frac{p_i'}{p})}\prod _{i=1}^m \Vert f_i\Vert _{L^{p_i}(w_i)}. \end{aligned}$$

This completes the proof. $\square$

We now turn our attention to the proof of Theorem 1.4. First, we note the following symmetry of $A_{\vec {P}}$ weights.

Lemma 3.1

Suppose that $\vec {w}=(w_1,\cdots ,w_m)\in A_{\vec {P}}$ and that $1<p$, $p_1$, $\cdots$, $p_m<\infty$ with $1/{p_1}+\cdots +1/{p_m}=1/p$. Then $\vec {w}^i:=(w_1$, $\cdots$, $w_{i-1}$, $v_{\vec {w}}^{1-p'}$, $w_{i+1}$, $\cdots$, $w_m)\in A_{\vec {P}^i}$ with $\vec {P}^i=(p_1,\cdots , p_{i-1}, p', p_{i+1},\cdots ,p_m)$ and

$$\begin{aligned}{}[\vec {w}^i]_{A_{\vec {P}^i}}=[\vec {w}]_{A_{\vec {P}}}^{p_i'/p}. \end{aligned}$$

Proof

We will prove the conclusion for $i=1$; the other cases are similar. Notice that

$$\begin{aligned} 1/{p'}+1/{p_{2}}+\cdots +1/{p_m}=1/{p_1'} \end{aligned}$$

and

$$\begin{aligned} v_{\vec {w}}^{(1-p')p_1'/{p'}}\cdot w_{2}^{p_1'/{p_{2}}}\cdots w_m^{p_1'/p_m}=w_1^{1-p_1'}. \end{aligned}$$

By the definition of multiple $A_{\vec {P}}$ constant, we have

$$\begin{aligned}{}[\vec {w}^1]_{A_{\vec {P}^1}}&= \sup _Q \left( \frac{1}{|Q|}\int \limits _Q w_1^{1-p_1'}\right) \cdot \left( \frac{1}{|Q|}\int \limits _Q (v_{\vec {w}}^{1-p'})^{1-p}\right) ^{p_1'/p}\\&\quad \times \prod _{i=2}^m\left( \frac{1}{|Q|}\int \limits _Q w_i^{1-p_i'}\right) ^{p_1'/{p_i'}}\\&= [\vec {w}]_{A_{\vec {P}}}^{p_1'/p}. \end{aligned}$$

$\square$

By Theorem 1.5 we reduce our problem to consider the behavior of the operator $A_{\fancyscript{D},\mathcal {S}}$. For these operators we have the following Theorem, which holds for all $1<p_1,\ldots ,p_m<\infty$.

Theorem 3.2

Suppose that $1<p_1,\cdots ,p_m$ $<\infty$ with $1/{p_1}+\cdots +1/{p_m}=1/p$ and $\vec {w}\in A_{\vec {P}}$. Then

$$\begin{aligned} \Vert A_{\fancyscript{D},\mathcal {S}}(\vec {f})\Vert _{L^p(v_{\vec {w}})}\lesssim [\vec {w}]^{\max (1,\frac{p_1'}{p},\ldots ,\frac{p_m'}{p})}_{A_{\vec {P}}}\prod _{i=1}^m\Vert f_i\Vert _{L^{p_i}(w_i)}. \end{aligned}$$

Proof

We first consider the case when $\frac{1}{m}<p\le 1$. In this case

$$\begin{aligned} \int \limits _{\mathbb {R}^n}A_{\fancyscript{D},\mathcal {S}}(\vec {f})^pv_{\vec {w}}\le \sum _{Q\in \mathcal {S}}\left( \prod _{i=1}^m\frac{1}{|Q|}\int \limits _Q f_i\right) ^pv_{\vec {w}}(Q), \end{aligned}$$

which can be handled in exactly the same manner as the estimates in proof of Theorem 1.2.

Now consider the case $p\ge \max _i p_i'$. It is sufficient to prove that

$$\begin{aligned} \Vert A_{\fancyscript{D},\mathcal {S}}(\vec {f\sigma })\Vert _{L^p(v_{\vec {w}})}\lesssim [\vec {w}]_{A_{\vec {P}}}\prod _{i=1}^m\Vert f_i\Vert _{L^{p_i}(\sigma _i)}, \end{aligned}$$

where $\sigma _i=w_i^{1-p_i'}$, $A_{\fancyscript{D},\mathcal {S}}(\vec {f\sigma })= A_{\fancyscript{D},\mathcal {S}}(f_1\sigma _1,\cdots , f_m\sigma _m)$, and $f_i\ge 0$. By duality, it suffices to estimate the $(m+1)$-linear form

$$\begin{aligned} \int \limits _{\mathbb {R}^n} A_{\fancyscript{D},\mathcal {S}}(\vec {f\sigma })g v_{\vec {w}}= \sum _{Q\in \mathcal {S}} \int \limits _{Q} g v_{\vec {w}}\cdot \prod _{i=1}^m \frac{1}{|Q|}\int \limits _{Q}f_i\sigma _i \end{aligned}$$

for $g\ge 0$ belonging to $L^{p'}(v_{\vec {w}})$. We have

$$\begin{aligned}&{\sum _{Q\in \mathcal {S}} \int \limits _{Q}g v_{\vec {w}}\cdot \prod _{i=1}^m \frac{1}{|Q|}\int \limits _{Q}f_i\sigma _i }\\&\quad =\sum _{Q\in \mathcal {S}} \frac{v_{\vec {w}}(Q)\prod _{i=1}^m \sigma _i(Q)^{p/{p_i'}}}{|Q|^{mp}}\cdot \frac{|Q|^{m(p-1)}}{v_{\vec {w}}(Q)\prod _{i=1}^m \sigma _i(Q)^{p/{p_i'}}}\cdot \int \limits _{Q}g v_{\vec {w}}\cdot \prod _{i=1}^m \int \limits _{Q}f_i\sigma _i \\&\quad \le [\vec {w}]_{A_{\vec {P}}}\sum _{Q\in \mathcal {S}}\frac{|Q|^{m(p-1)}}{v_{\vec {w}}(Q)\prod _{i=1}^m \sigma _i(Q)^{p/{p_i'}}} \cdot \int \limits _{Q}g v_{\vec {w}}\cdot \prod _{i=1}^m \int \limits _{Q}f_i\sigma _i \\&\quad \le [\vec {w}]_{A_{\vec {P}}}\sum _{Q\in \mathcal {S}}\frac{2^{m(p-1)}|E_Q|^{m(p-1)}}{v_{\vec {w}}(Q)\prod _{i=1}^m \sigma _i(Q)^{p/{p_i'}}} \cdot \int \limits _{Q}g v_{\vec {w}}\cdot \prod _{i=1}^m \int \limits _{Q}f_i\sigma _i . \end{aligned}$$

By (3.1),

$$\begin{aligned} |E_Q|\le v_{\vec {w}}(E_Q)^{\frac{1}{mp}}\sigma _1(E_Q)^{\frac{1}{mp_1'}}\cdots \sigma _m(E_Q)^{\frac{1}{mp_m'}}. \end{aligned}$$

Since $p\ge \max _i\{p_i'\}$ and $E_Q\subset Q$, we have $\sigma _i(Q)^{1-\frac{p}{p_i'}}\le \sigma _i(E_Q)^{1-\frac{p}{p_i'}}$ for any $i=1,\cdots ,m$. Therefore,

$$\begin{aligned}&\sum _{Q\in \mathcal {S}} \int \limits _{Q}g v_{\vec {w}}\cdot \prod _{i=1}^m \frac{1}{|Q|}\int \limits _{Q}f_i\sigma _i \\&\le 2^{m(p-1)}[\vec {w}]_{A_{\vec {P}}}\sum _{j,k}v_{\vec {w}}(E_Q)^{\frac{1}{p'}}\prod _{i=1}^m \sigma _i(E_Q)^{\frac{p-1}{p_i'}}\sigma _i(Q)^{1-\frac{p}{p_i^{\prime }}}\\&\quad \times \frac{1}{v_{\vec {w}}(Q)}\int \limits _{Q}g v_{\vec {w}}\cdot \prod _{i=1}^m \frac{1}{\sigma _i(Q)}\int \limits _{Q}f_i\sigma _i \\&\le 2^{m(p-1)}[\vec {w}]_{A_{\vec {P}}}\sum _{Q\in \mathcal {S}}v_{\vec {w}}(E_Q)^{\frac{1}{p'}}\prod _{i=1}^m \sigma _i(E_Q)^{\frac{1}{p_i}} \frac{1}{v_{\vec {w}}(Q)}\int \limits _{Q}g v_{\vec {w}}\\&\quad \times \prod _{i=1}^m \frac{1}{\sigma _i(Q)}\int \limits _{Q}f_i\sigma _i \\&\le 2^{m(p-1)}[\vec {w}]_{A_{\vec {P}}}\left( \sum _{Q\in \mathcal {S}}\bigg (\frac{1}{v_{\vec {w}}(Q)}\int \limits _{Q}g v_{\vec {w}}\bigg )^{p'}v_{\vec {w}}(E_Q)\right) ^{1/{p'}}\\&\quad \times \prod _{i=1}^m\left( \sum _{Q\in \mathcal {S}}\bigg (\frac{1}{\sigma _i(Q)}\int \limits _{Q}f_i \sigma _i \bigg )^{p_i}\sigma _i(E_Q)\right) ^{1/{p_i}}\\&\le 2^{m(p-1)}[\vec {w}]_{A_{\vec {P}}}\Vert M_{v_{\vec {w}}}^{\fancyscript{D}}(g)\Vert _{L^{p'}(v_{\vec {w}})}\prod _{i=1}^m \Vert M_{\sigma _i}^{\fancyscript{D}}(f_i)\Vert _{L^{p_i}(\sigma _i)}\\&\lesssim 2^{m(p-1)}[\vec {w}]_{A_{\vec {P}}}\Vert g\Vert _{L^{p'}(v_{\vec {w}})}\prod _{i=1}^m \Vert f_i\Vert _{L^{p_i}(\sigma _i)}, \end{aligned}$$

where (2.1) is used in the last step. For the other cases we use duality. Notice that the operator $A_{\fancyscript{D},\mathcal {S}}$ is self adjoint as a multilinear operator, in the sense that for any $i$, $i=1,\ldots ,m$, we have

$$\begin{aligned} \int \limits _{\mathbb {R}^n} A_{\fancyscript{D},\mathcal {S}}(f_1,\ldots ,f_m)g= \int \limits _{\mathbb {R}^n} A_{\fancyscript{D},\mathcal {S}}(f_1,\ldots ,f_{i-1},g,f_{i+1},\ldots f_m)f_i. \end{aligned}$$

Without loss of generality suppose $p_1'\ge \max (p,p_2',\ldots ,p_m')$. Hence, by duality and self adjiontness we have

$$\begin{aligned} \Vert A_{\fancyscript{D},\mathcal {S}}\Vert _{L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)\rightarrow L^p(v_{\vec {w}})}&= \Vert A_{\fancyscript{D},\mathcal {S}}\Vert _{L^{p'}(v_{\vec {w}}^{1-p'})\times \cdots \times L^{p_m}(w_m)\rightarrow L^{p_1'}(w_1^{1-p_1'})}\\&\lesssim [\vec {w}^1]_{\vec {P}^1}= [\vec {w}]_{A_{\vec {P}}}^{\frac{p_1'}{p}}. \end{aligned}$$

$\square$

4 Examples

Finally, we end with some examples to show that our bounds are sharp. First we show that Theorem 1.2 is sharp. Consider the case $m=2$ (we leave it to the reader to modify the example for $m>2$) and suppose that we had a better exponent than the one in inequality (1.4), that is, suppose

$$\begin{aligned} \Vert \mathcal {M}\Vert _{L^{p_1}(w_1)\times L^{p_2}(w_2) \rightarrow L^p(v_{\vec {w}})}\lesssim [\vec {w}]_{A_{\vec {P}}}^{r\max (\frac{p_1'}{p},\frac{p_2'}{p})} \end{aligned}$$

(4.1)

for some $r<1$. Further suppose that $p_1'\ge p_2'$. For $0<\varepsilon <1$, let $f_1(x)=|x|^{\varepsilon -n}\chi ^{}_{B(0,1)}(x)$, $f_2(x)=|x|^{\frac{\varepsilon -n}{p_2}}\chi _{B(0,1)}(x)$, $w_1(x)=|x|^{(n-\varepsilon )(p_1-1)}$ and $w_2(x)=1$. Calculations show that

$$\begin{aligned} \Vert f_1\Vert _{L^{p_1}(w_1)}\simeq \varepsilon ^{-{1}/{p_1}}, \Vert f_2\Vert _{L^{p_2}(w_2)}\simeq \varepsilon ^{-1/p_2}, v_{\vec {w}}(x)=|x|^{(n-\varepsilon )\frac{p}{p_1^{\prime }}} \end{aligned}$$

and

$$\begin{aligned}{}[\vec {w}]_{A_{\vec {P}}}\simeq \varepsilon ^{-{p}/{p_1'}}. \end{aligned}$$

For $x\in B(0,1)$ we have

$$\begin{aligned} \mathcal {M}(f_1,f_2)(x)&\gtrsim \frac{1}{|x|^n}\int \limits _{B(0,|x|)} |y_1|^{\varepsilon -n}\,dy_1 \cdot \frac{1}{|x|^n}\int \limits _{B(0,|x|)} |y_2|^{\frac{\varepsilon -n}{p_2}}\,dy_2\\&\gtrsim \frac{f_1(x)f_2(x)}{\varepsilon \cdot (\frac{\varepsilon -n}{p_2}+n)}\gtrsim \frac{f_1(x)f_2(x)}{\varepsilon }. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert \mathcal {M}(f_1,f_2)\Vert _{L^p(v_{\vec {w}})}&\gtrsim \frac{1}{\varepsilon }\left( \int \limits _{B(0,1)}|x|^{(\varepsilon -n)(p+\frac{p}{p_2}-\frac{p}{p_1'})}\,dx\right) ^{1/p}\\&\simeq \frac{1}{\varepsilon }\left( \int \limits _0^1x^{\varepsilon -1}\,dx\right) ^{1/p}\\&= \frac{1}{\varepsilon }\left( \frac{1}{\varepsilon }\right) ^{1/p}. \end{aligned}$$

Combining this with inequality (4.1) we see for some $r<1$,

$$\begin{aligned} \left( \frac{1}{\varepsilon }\right) ^{1+\frac{1}{p}}\lesssim \left( \frac{1}{\varepsilon }\right) ^{r+\frac{1}{p}}, \end{aligned}$$

which is impossible as $\varepsilon \rightarrow 0$.

Next we show that Theorem 1.4 is sharp again when $m=2$. Recall that the first the bilinear Riesz transform is defined by

$$\begin{aligned} R_1(f_1,f_2)(x)=p.v.\int \limits _{\mathbb {R}^{2n}}\frac{\sum _{j=1}^2(x_1-(y_j)_1)}{(\sum _{j=1}^2|x-y_j|^2)^{(n2+1)/2}}f_1(y_1) f_2(y_2)dy_1 dy_2, \end{aligned}$$

where $(y_j)_1$ denotes the first coordinate of $y_j$.

Suppose that $m=2$, $p_1'\ge p_2'$ and $p_1'\ge p$. Let

$$\begin{aligned} U&= \{x\in \mathbb {R}^n: |x|\le 1,0<x_i\le x_1,i=2,\cdots ,n\}, \\ V&= \{x\in \mathbb {R}^n:\, |x|\le 1, x_i\le 0, i=1,\cdots ,n \}. \end{aligned}$$

For $0<\varepsilon <1$, let $f_1(x)=|x|^{\varepsilon -n}\chi ^{}_V(x)$, $f_2(x)=|x|^{\frac{\varepsilon -n}{p_2}}\chi ^{}_V(x)$, $w_1(x)=|x|^{(n-\varepsilon )(p_1-1)}$ and $w_2(x)=1$. For $x\in U$ and $y_j\in V$ with $|y_j|\le |x|$, we have

$$\begin{aligned} \frac{\sum _{j=1}^2(x_1-(y_j)_1)}{(\sum _{j=1}^2|x-y_j|^2)^{1/2}} \ge \frac{2\frac{|x|}{\sqrt{n}}}{4|x|}\gtrsim 1. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{\sum _{j=1}^2(x_1-(y_j)_1)}{(\sum _{j=1}^2|x-y_j|^2)^{(2n+1)/2}}\gtrsim \frac{1}{|x|^{2n}}. \end{aligned}$$

It follows that

$$\begin{aligned} R_1(\vec {f})(x)&= p.v.\int \limits _{(\mathbb {R}^n)^2}\frac{\sum _{j=1}^2(x_i-(y_j)_i)}{(\sum _{j=1}^2|x-y_j|^2)^{(2n+1)/2}}f_1(y_1) f_2(y_2)dy_1dy_2\\&\gtrsim \int \limits _{|y_1|\le |x|}\int \limits _{|y_2|\le |x|}\frac{1}{|x|^{2n}} |y_1|^{\varepsilon -n}\cdot |y_2|^{\frac{\varepsilon -n}{p_2}}dy_1dy_2\\&\gtrsim \frac{1}{\varepsilon }f_1(x)f_2(x). \end{aligned}$$

Hence

$$\begin{aligned} \Vert R_1(\vec {f})\Vert _{L^p(v_{\vec {w}})}&\gtrsim \frac{1}{\varepsilon }\left( \int \limits _U |x|^{(\varepsilon -n)(p+p/{p_2}-p/{p_1'})}dx\right) ^{1/p}\\&= \frac{1}{\varepsilon }\left( \int \limits _U |x|^{\varepsilon -n}dx\right) ^{1/p}\\&\gtrsim \frac{1}{\varepsilon }\left( \int \limits _{\{|x|\le 1\}} |x|^{\varepsilon -n}dx\right) ^{1/p}\quad \text{(by } \text{ symmetry) }\\&\gtrsim (\frac{1}{\varepsilon })^{1+1/p}. \end{aligned}$$

Then by similar arguments as the above we can show that the exponent is sharp when $\max ( p_1',p_2')\ge p\ge 1$. When $p>\max (p_1',p_2')$. Again, suppose that $p_1'\ge p_2'$. We consider the adjoint in the first variable, $(R_1)^{1,*}$. Notice that

$$\begin{aligned} (R_1)^{1,*}(f_1,f_2)(x)=\int \limits _{(\mathbb {R}^n)^2}\frac{2(y_1)_1-x_1-(y_2)_1}{(|x-y_1|^2+|y_1-y_2|^2)^{(2n+1)/2}} f(y_1)f(y_2)dy_1dy_2. \end{aligned}$$

Let

$$\begin{aligned} U_1&= \{x\in \mathbb {R}^n: |x|\le 1,x_1\le x_i< 0 ,i=2,\cdots ,n\}, \\ V_1&= \{x\in \mathbb {R}^n:\, |x|\le 1, x_i\ge 0, i=1,\cdots ,n \}. \end{aligned}$$

For $0<\varepsilon <1$, let $f_1(x)=|x|^{\varepsilon -n}\chi ^{}_{V_1}(x)$, $f_2(x)=|x|^{\frac{\varepsilon -n}{p_2}}\chi ^{}_{V}(x)$, $w_1(x)=|x|^{(\varepsilon -n)p_1/p}$ and $w_2(x)=1$. Then $v_{\vec {w}}(x)=|x|^{\varepsilon -n}$, $v_{\vec {w}}^{1-p'}=|x|^{(n-\varepsilon )(p'-1)}$ and $w_1^{1-p_1'}=|x|^{(n-\varepsilon )p_1'/p}$. Similar arguments as the above show that

$$\begin{aligned} \Vert (R_1)^{1,*}\Vert _{L^{p'}(v_{\vec w}^{1-p'})\times L^{p_2}(w_2)\rightarrow L^{p_1'}(w_1^{1-p_1'})} \gtrsim \frac{1}{\varepsilon }=[\vec {w}^1]_{A_{\vec {P}^1}}^{p/{p_1'}}=[\vec {w}]_{A_{\vec {P}}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert R_1 \Vert _{L^{p_1}(w_1)\times L^{p_2}(w_2)\rightarrow L^p(v_{\vec {w}})}=\Vert (R_1)^{1,*}\Vert _{L^{p'}(v_{\vec w}^{1-p'})\times L^{p_2}(w_2)\rightarrow L^{p_1'}(w_1^{1-p_1'})}\gtrsim [\vec {w}]_{A_{\vec {P}}}. \end{aligned}$$

This shows the sharpness of the exponent when $p>\max _i \{p_i'\}$, which completes the proof.