Abstract
In this paper, we introduce certain classes of multilinear Calderón–Zygmund operators with kernels of Dini’s type. Applying the sharp method and \(A_{\vec{p}}^{\infty }(\varphi )\) functions, we first establish some weighted norm inequalities for multilinear Calderón–Zygmund operators with kernels of Dini’s type, including pointwise estimates, strong type, and weak endpoint estimates. Furthermore, similar weighted norm inequalities for commutators with \(\mathrm{BMO}_{\theta }(\varphi )\) functions are also obtained, but the weak endpoint estimate is of \(L({\mathrm{log}}L)\) type.
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1 Introduction
In the 1980s, the multilinear Calderón–Zygmund theory was first studied by Coifman and Meyer [1, 2]. The multilinear Calderón–Zygmund operators with standard kernels were then further investigated by many authors, such as [3–8]. Meanwhile, many authors weakened the standard kernel conditions to rough associated kernel conditions; see [9–13]. Particularly, in 1985, Yabuta [10] introduced the Calderón–Zygmund operators of type \(\omega (t)\) (the definition given below) and obtained weighted norm inequalities of the Calderón–Zygmund operators of type \(\omega (t)\) on \(L^{p}\) spaces, here weight functions belong to Muckenhoupt’s class \(A_{p}\). In 2014, Lu and Zhang [12] obtained the weighted boundedness of multilinear Calderón–Zygmund operators of type \(\omega (t)\) and their commutators with BMO functions from weighted \(L^{p}\) spaces to weighted product of \(L^{p}\) spaces. In 2016, Zhang and Sun [13] further considered weighted norm inequalities of iterated commutators that multilinear Calderón–Zygmund operators of type \(\omega (t)\) with BMO functions.
Throughout this paper, \(\omega (t):[0,\infty )\rightarrow [0,\infty )\) is a nondecreasing function with \(0<\omega (1)<\infty \).
For \(a>0\), we say that \(\omega \in {\mathrm{Dini}}(a)\), if
It is worth mentioning that \({\mathrm{Dini}}(a_{1})\subset {\mathrm{Dini}}(a_{2})\) when \(0< a_{1}< a_{2}\).
Definition 1.1
Let \(K(x,y_{1},\ldots,y_{m})\) be a locally integrable function, defined away from the diagonal \(x=y_{1}=\cdots =y_{m}\) in \(( \mathbb{R}^{n})^{m+1}\), it is said to belong a certain class of multilinear Calderón–Zygmund kernel of type \(\omega (t)\), if there exist constant \(A>0\), \(N>0\) such that
for all \((x,y_{1},\ldots,y_{m})\in (\mathbb{R}^{n})^{m+1}\) with \(x\neq y_{j}\) for some \(j=1,2,\ldots,m \) and
whenever \(|x-x'|\leq \frac{1}{2}\max_{1\leq j\leq m}|x-y_{j}|\), and
whenever \(|y_{j}-y'_{j}|\leq \frac{1}{2}\max_{1\leq j \leq m}|x-y_{j}|\).
Let \(T:\mathcal{S}(\mathbb{R}^{n})\times \cdots \times \mathcal{S}( \mathbb{R}^{n})\rightarrow \mathcal{S'}(\mathbb{R}^{n})\) (from the product of Schwarz spaces to the space of tempered distributions) be a multilinear operator with certain classes of multilinear Calderón–Zygmund kernels of type \(\omega (t)\) if there exists a \(K(x,y_{1},\ldots,y_{m})\) that satisfies (1.1)–(1.3), such that
whenever \(x\notin \bigcap_{j=1}^{m}\) \(\operatorname{supp} f_{j}\) and each \(f_{j}\in C_{c}^{\infty }(\mathbb{R}^{n}),j=1,\ldots,m\).
If T can be extended to a bounded multilinear operator:
for some \(1< q_{1},\ldots,q_{m}<\infty \) with \(1/q_{1}+\cdots +1/q_{m}=1/q\), or
for some \(1\leq q_{1},\ldots,q_{m}<\infty \) with \(1/q_{1}+\cdots +1/q_{m}=1/q\), then T is said to belong to the class of multilinear Calderón–Zygmund operators of type \(\omega (t)\).
Remark
When \(N=0\) in Eqs. (1.1)–(1.3), such kernels have a standard kernel of type \(\omega (t)\) as Lu and Zhang [12] and Zhang and Sun [13] considered.
The assumption of (1.6) is reasonable, one may refer to [12, Theorem 1.2].
Let T be a multilinear operator and \(\vec{b}=(b_{1},\ldots,b_{m})\) be a locally integrable vector function in \({\mathrm{BMO}}^{m}(\mathbb{R}^{n})\), the multilinear commutators of T with b⃗ is defined by
where
In 2003, Pérez and Torres [14] first introduced multilinear commutators of multilinear Calderón–Zygmund operators and established their boundedness from \(L^{q_{1}}(\mathbb{R}^{n})\times \cdots \times L^{q_{m}}(\mathbb{R}^{n})\) to \(L^{q}(\mathbb{R}^{n})\) for \(1< q,q_{1},\ldots,q_{m}<\infty \) with \(1/q_{1}+\cdots +1/q_{m}=1/q\), also, from \(L^{q_{1}}(\mathbb{R}^{n})\times \cdots \times L^{q_{m}}(\mathbb{R}^{n})\) to \(L^{q,\infty }(\mathbb{R}^{n})\) for \(1\leq q,q_{1},\ldots,q_{m}<\infty \) with \(1/q_{1}+\cdots +1/q_{m}=1/q\). In 2009, Lerner et al. [8] obtained some weighted boundedness of multilinear commutators as follows:
and for the weak end-point also it was proved that
To clarify the notation, if T is associated in the usual way with a kernel \(K(x,y_{1},\ldots,y_{m})\) satisfying (1.1)–(1.3), then at a formal level
Lerner [8] obtained weighted norm inequalities of classical multilinear Calderón–Zygmund operators and their commutators with BMO functions through new maximal functions. In 2014, end-point estimates for iterated commutators of multilinear singular integrals were shown by Pérez et al. [15]. Lu and Zhang [12] studied multilinear Calderón–Zygmund operators with type \(\omega (t)\) and multilinear commutators with BMO functions. Simultaneously, they established some weighted norm inequalities, such as strong type and weak end-point estimates. The corresponding result of iterated commutators by Zhang and Sun [13] was shown, where the weights belong to \(A_{\vec{p}}\). In 2015, Pan and Tang [16] and Bui [17], respectively, established weighted norm inequalities for certain classes of multilinear Calderón–Zygmund operators and their commutators with \(\mathrm{BMO}_{\theta }(\varphi )\). The difference is that Pan and Tang also considered weak end-point results. In 2019, Hu and Zhou [18] obtained weighted norm inequalities of Calderón–Zygmund operators of type \(\omega (t)\) and their commutators with \({\mathrm{BMO}}_{\theta }(\varphi )\) functions, here weights belong to \(A_{p}(\varphi )\) functions.
Inspired by the work above, this paper’s primary purpose is to obtain weighted norm inequalities for certain classes of multilinear operators of type \(\omega (t)\) and their commutators, including the pointwise estimate, strong type, and weak end-point estimates.
2 Some preliminaries and notations
In this section, we first recall some notations. For a measure set E, we define \(|E|\) as the Lebesgue measure of E and \(\chi _{E}\) as the characteristic function of E. \(Q(x,r)\) denotes the cube centered at x with the side length r and \(\lambda Q=Q(x,\lambda r)\). \(\vec{q}=(q_{1},q_{2},\ldots,q_{m})\) and θ⃗= \((\theta _{1},\theta _{2},\ldots,\theta _{m})\). For a locally integrable function f, \(f_{Q}\) denotes the average \(f_{Q}=(1/|Q|)\) \(\int _{Q} f(y)\,dy\). In this paper, let \(\varphi _{\theta }(Q)=(1+r)^{\theta }\), where r is the side length of the cube Q.
2.1 The \(A_{\vec{p}}^{\infty }(\varphi )\) weights
According to [16], we say that a weight w belongs to the class \(A_{p}^{\theta }(\varphi )\) for \(1< p<\infty \), if there exists a constant C such that, for all cubes Q,
In particular, when \(p=1\),
Notice that \(A^{\infty }_{p}(\varphi )=\bigcup_{\theta \geq 0}A_{p}^{\theta }( \varphi )\), \(A^{\infty }_{\infty }(\varphi )=\bigcup_{p\geq 1}A_{p}^{\infty }( \varphi )\) and \(A_{p}^{0}(\varphi )\) is equivalent to the Muckenhoupt class of weights \(A_{p}\) in [19] for all \(1\leq p<\infty \). However, in general, the class \(A_{p}^{\infty }(\varphi )\) is strictly larger than the class \(A_{p}\) for all \(1\leq p<\infty \).
Next, we give some necessary properties of \(A_{\vec{p}}^{\theta }(\varphi )\) functions.
Lemma 2.1
([20])
The following statements hold:
-
(i)
\(A_{p}^{\infty }(\varphi )\subset A_{q}^{\infty }(\varphi )\) for \(1\leq p\leq q<\infty \).
-
(ii)
If \(w\in A_{p}^{\infty }(\varphi )\), with \(p>1\) then there exists \(\epsilon >0\) such that \(w\in A_{p-\epsilon }^{\infty }(\varphi )\). Consequently, \(A_{p}^{\infty }(\varphi )=\bigcup_{q< p}A_{q}^{\infty }(\varphi )\).
-
(iii)
If \(w\in A_{p}^{\infty }(\varphi )\) with \(p\geq 1\), then exist positive numbers \(\delta,l\) and C so that, for all cubes Q,
$$ \biggl(\frac{1}{ \vert Q \vert } \int _{Q}w^{1+\delta }(x)\,dx \biggr)^{ \frac{1}{1+\delta }} \leq C \biggl(\frac{1}{ \vert Q \vert } \int _{Q}w(x)\,dx \biggr) \varphi ^{l}(Q). $$
Lemma 2.2
([21])
The following statements hold:
-
(i)
\(w\in A_{p}^{\theta }(\varphi )\) if and only if \(w^{-\frac{1}{p-1}}\in A_{p'}^{\theta }(\varphi )\), where \(\frac{1}{p}+\frac{1}{p'}=1\);
-
(ii)
if \(w_{1}\), \(w_{2}\in A_{p}^{\theta }(\varphi )\), \(p\geq 1\), then \(w_{1}^{\alpha }w_{2}^{1-\alpha }\in A_{p}^{\theta }(\varphi )\) for any \(0<\alpha <1\);
-
(iii)
if \(w\in A_{p}^{\theta }(\varphi )\), for \(1\leq p<\infty \), then
$$ \frac{1}{\varphi _{\theta }(Q) \vert Q \vert } \int _{Q} \bigl\vert f(y) \bigr\vert \,dy\leq C \biggl( \frac{1}{w(5Q)} \int _{Q} \bigl\vert f(y) \bigr\vert ^{p}w(y) \,dy \biggr)^{\frac{1}{p}}. $$In particular, let \(f=\chi _{E}\) for any measurable set \(E\subset Q\),
$$ \frac{ \vert E \vert }{\varphi _{\theta }(Q) \vert Q \vert }\leq C \biggl( \frac{w(E)}{w(5Q)} \biggr)^{\frac{1}{p}}. $$
Let \(\vec{p}=(p_{1},\ldots,p_{m})\) and \(1/p=1/p_{1}+\cdots +1/p_{m}\) with \(1\leq p_{1},\ldots,p_{m}<\infty \). Given \(\vec{w}=(w_{1},\ldots,w_{m})\), each \(w_{j}\) being nonnegative measurable, we set
For \(\theta \geq 0\), we say that w⃗ satisfies the \(A^{\theta }_{\vec{p}}(\varphi )\) condition and denote \(\vec{w}\in A^{\theta }_{\vec{p}}(\varphi )\), if
where the supremum is taken over all cubes \(Q\subset \mathbb{R}^{n}\), and the term \((\frac{1}{|Q|}\int _{Q}w_{j}(x)^{1-p'_{j}} )^{1/p'_{j}}\) coincides with \((\inf_{x \in Q}w_{j} )^{-1}\) when \(p_{j}=1\) \(j=1,2,\ldots,m\).
For \(1\leq p_{1},\ldots,p_{m}<\infty \), set \(A_{\vec{p}}^{\infty }(\varphi )=\bigcup_{\theta \geq 0}A_{ \vec{p}}^{\theta }(\varphi )\). When \(\theta =0\), the class \(A_{\vec{p}}^{0}(\varphi )\) coincides with the class of multiple weights \(A_{\vec{p}}\) introduced by [15].
Lemma 2.3
([17])
Let \(1\leq p_{1},\ldots,p_{m}<\infty \) and \(\vec{w}=(w_{1},\ldots,w_{m})\). Then the following statements are equivalent:
-
(i)
\(\vec{w}\in A_{\vec{p}}^{\infty }(\varphi )\);
-
(ii)
\(w_{j}^{1-p'_{j}}\in A_{mp'_{j}}^{\infty },j=1,\ldots,m\), and \(v_{\vec{w}}\in A_{mp}^{\infty }(\varphi )\).
The class \(A_{\vec{p}}^{\infty }(\varphi )\) is not increasing, which means that, for \(\vec{p}=(p_{1},\ldots,p_{m})\) and \(\vec{q}=(q_{1},\ldots,q_{m})\) with \(p_{j}\leq q_{j},j=1,\ldots,m\), the following may not be true \(A_{\vec{p}}^{\infty }(\varphi )\subset A_{\vec{q}}^{\infty }(\varphi )\).
Lemma 2.4
([17])
Let \(1\leq p_{1},\ldots,p_{m}<\infty \) and \(\vec{w}=(w_{1},\ldots,w_{m})\in A_{\vec{p}}^{\infty }(\varphi )\). Then
-
(i)
for any \(r\geq 1, \vec{w}\in A_{r\vec{p}}^{\infty }(\varphi )\);
-
(ii)
if \(1< p_{1},\ldots,p_{m}<\infty \), then there exists \(r>1\) so that \(\vec{w}\in A_{\vec{p}/r}^{\infty }(\varphi )\).
2.2 \({\mathrm{BMO}}_{\infty }(\varphi )\) spaces
Now, recall the definition and properties of the \({\mathrm{BMO}}_{\infty }\) spaces introduced by [20].
A locally integrable function b is in \(\mathrm{BMO}_{\theta }(\varphi )(\theta \geq 0)\) if
When \(\theta =0,{\mathrm{BMO}}_{0}(\varphi )={\mathrm{BMO}}(\mathbb{R}^{n})\). Clearly \({\mathrm{BMO}}(\mathbb{R}^{n})\subset \mathrm{BMO}_{\theta }(\varphi )\) and \(\mathrm{BMO}_{\theta _{1}}(\varphi )\subset {\mathrm{BMO}}_{\theta _{2}}(\varphi )\) for \(\theta _{1}\leq \theta _{2}\). We denote \(\mathrm{BMO}_{\infty }(\varphi )=\bigcup_{\theta \geq 0}\mathrm{BMO}_{\theta }( \varphi )\).
Lemma 2.5
([20])
Let \(\theta >0,s\geq 1\). If \(b\in {\mathrm{BMO}}_{\theta }(\varphi )\) then for all cubes \(Q=Q(x,r)\)
-
(i)
\((\frac{1}{|Q|}\int _{Q}|b(y)-b_{Q}|^{s}\,dy )^{ \frac{1}{s}}\leq \|b\|_{\mathrm{BMO}_{\theta }(\varphi )}\varphi _{ \theta }(Q)\);
-
(ii)
\((\frac{1}{|3^{k}Q|}\int _{3^{k}Q}|b(y)-b_{Q}|^{s}\,dy )^{\frac{1}{s}}\leq k\|b\|_{\mathrm{BMO}_{\theta }(\varphi )} \varphi _{\theta }(3^{k}Q)\), for all \(k\in N\).
2.3 The norm of Orlicz spaces
For \(\Phi (t)=t(1+{\mathrm{log}}^{+}t)\) and a cube Q in \(\mathbb{R}^{n}\), we will consider the average \(\|f\|_{\Phi,Q}\) of a function f given by the Luxemburg norm
The generalized Hölder inequality in Orlicz spaces together with the corresponding John–Nirenberg inequality in [18, Lemma 2.5] implies that
2.4 Maximal functions and Sharp maximal functions
Maximal functions and sharp maximal functions play an important role in the proof of the main theorem. Next, recall the relevant definition.
For \(0<\eta <\infty \), the maximal operator \(M_{\varphi,\eta }\) is defined by
Definition 2.6
([21])
Let \(0<\eta <\infty \), then the dyadic maximal function \(M_{\varphi,\eta }^{d}\) is defined by
Let Q be a dyadic cube; f is a locally integral function, then the dyadic sharp maximal function \(M_{\varphi,\eta }^{\sharp,d}\) is defined by
where \(f_{Q}=\frac{1}{|Q|}\int _{Q}f(y)\,dy\).
From the above definition, the variants of the dyadic maximal operator and the dyadic sharp maximal operator are as follows:
Lemma 2.7
([21])
Let \(1< p<\infty \), \(w\in A_{p}^{\infty }\), \(0<\eta <\infty \) and \(f\in L^{p}(w)\), then
Lemma 2.8
([21])
Let \(1< p<\infty \), \(\omega \in A_{\infty }^{\infty }\), \(0<\eta <\infty \) and \(\delta >0\) and let \(\psi:(0,\infty )\mapsto (0,\infty )\) be doubling, that is, \(\psi (2a)\leq \psi (a)\) for \(a>0\). Then there exists a constant C depending upon the \(A_{\infty }^{\infty }\) condition of w and the doubling condition of ψ such that
Let \(0<\eta <\infty \), \(\vec{f}=(f_{1},f_{2},\ldots,f_{m})\), then the multilinear maximal operators \(\mathcal{M}_{\varphi,\eta }\) and \(\mathcal{M}_{L(\mathrm{Log}L),\varphi,\eta }\) are defined by
Lemma 2.9
([16])
Let \(1< p_{j}<\infty \), \(j=1,2,\ldots,m\), \(\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}+\cdots +\frac{1}{p_{m}}\) and \(\vec{w}\in A_{\vec{p}}^{\infty }\), then there exists some \(\eta _{0}>0\) depending on \(p,m,p_{j}\) such that
3 Estimates for multilinear operators
Theorem 3.1
Let T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1, assume that \(0<\delta <\frac{1}{m},0<\eta \) and ω is satisfying \(\omega \in {\mathrm{Dini}}(1)\). Then there exists a constant \(C>0\) such that
for all f⃗ in \(L^{p_{1}}(\mathbb{R}^{n})\times \cdots \times L^{p_{m}}(\mathbb{R}^{n})\) with \(1\leq p_{j}<\infty \) for \(j=1,\ldots,m\).
Proof
If \(\omega \in \mathrm{Dini(1)}\), then
For a fixed point \(x\in \mathbb{R}^{n}\) and let \(x\in Q=Q{(x_{0},r)}\), Q is a dyadic cube. To complete the proof, we consider the following two cases of the side length r: \(r\leq 1\) and \(r>1\).
Case 1. When \(r\leq 1\). Since \(0<\delta <\frac{1}{m}<1\), \(\eta >0\) and \(||a|^{t}-|b|^{t}|<|a-b|^{t}\) for \(0< t<1\), for any number C we can estimate
Let \(Q^{*}=8\sqrt{n}Q\), we decompose \(f_{j}=f_{j}^{0}+f_{j}^{\infty }\) for each \(f_{j}\), where \(f_{j}^{0}=f_{j}\chi _{Q^{*}}\). Then
where \(\mathscr{L}=\{(\alpha _{1},\ldots,\alpha _{m}): \mathrm{there~is~at~least~one~}\alpha _{j}\neq 0\}\).
Let \(C=\sum_{\alpha _{1},\ldots,\alpha _{m}\in \mathscr{L}}C_{ \alpha _{1},\ldots,\alpha _{m}}\), from this condition, we can get the following a series of estimates:
Since \(T:L^{1}\times \cdots \times L^{1}\rightarrow L^{\frac{1}{m},\infty }\) and using the Kolmogorov inequality with \(p=\delta \) and \(q=\frac{1}{m}\), we have
To estimate II, we choose \(C_{\alpha _{1},\ldots,\alpha _{m}}=T(f_{1}^{\alpha _{1}},\ldots, f_{m}^{ \alpha _{m}})(x)\), for any \(z\in Q\), the following estimate holds:
We consider first the case when \(\alpha _{1}=\cdots =\alpha _{m}=\infty \). For any \(z\in Q\), we get
where \(\Omega _{k}=(2^{k+3}\sqrt{n}Q)\setminus (2^{k+2}\sqrt{n}Q)\) for \(k=1,2,\ldots \) .
Note that, for \(x,z\in Q\) and any \((y_{1},\ldots,y_{m})\in (\Omega _{k})^{m}\),
since ω is nondecreasing, and through the kernel condition (1.2), we have
Then, taking \(N\geq m\eta \),
We are now to consider \(\alpha _{j_{1}}=\cdots =\alpha _{j_{l}}=0\) for \(1\leq l< m\). Let \(\mathscr{J}:=\{j_{1},\ldots,j_{l}\}\) then \(\alpha _{j}=\infty \) for \(j\notin \mathscr{J}\). Thus
Similar to the above discussion, taking \(N\geq m\eta \), we have
Case 2. When \(r>1\), since \(0<\delta <\frac{1}{m}<1\), and \(\eta >0\), it follows that
For I, by the Kolmogorov inequality and \(T:L^{1}\times \cdots \times L^{1}\rightarrow L^{\frac{1}{m},\infty }\), we have
To estimate II, note that, for \(z\in Q\) and any \((y_{1},\ldots,y_{m})\in (\Omega _{k})^{m},|z-y_{j}|\geq 2^{k} \sqrt{n}r\). Consider now \(\alpha _{1}=\cdots =\alpha _{m}=\infty \), taking \(N\geq m\eta +1\), the following estimate holds:
When \(\alpha _{j_{1}}=\cdots =\alpha _{j_{l}}=0\) for \(1\leq l< m\). Let \(\mathscr{J}:=\{j_{1},\ldots,j_{l}\}\) then \(\alpha _{j}=\infty \) for \(j\notin \mathscr{J}\), taking \(N\geq m\eta +1\), then
Pan and Tang in [16, Lemma 2.7] proved the result in our framework, which is similar to the classical Fefferman–Stein inequalities. Next, using Lemma 2.7 of our paper, we obtain the result as follows. □
Corollary 3.2
Let T be a multilinear operator satisfying (1.1)–(1.5), and suppose that ω is satisfying \(\omega \in {\mathrm{Dini}}(1)\), \(w\in A_{\infty }^{\infty }\), \(\eta >0\) and \(p>0\). Then there exist constants \(C>0\), such that
and
Proof
From Lemma 2.7 and Theorem 3.1, we get
□
Similarly, with the help of Lemma 2.8, the weak-type estimate is obtained.
Theorem 3.3
Let T be a multilinear operator satisfying (1.1)–(1.5), \(\vec{w}\in A_{\vec{p}}^{\infty }(\varphi )\) and \(1/p=1/p_{1}+\cdots +1/p_{m}\). If ω is satisfying \(\omega \in {\mathrm{Dini}}(1)\), then there exists a constant \(C>0\), such that:
-
(i)
If \(1< p_{j}<\infty \), \(j=1,\ldots,m\), then
$$ \bigl\Vert T(\vec{f}) \bigr\Vert _{L^{p}(v_{\vec{w}})}\leq C\prod_{j=1}^{m} \Vert f_{j} \Vert _{L^{p_{j}}(w_{j})}. $$ -
(ii)
If \(1\leq p_{j}<\infty \), \(j=1,\ldots,m\), and at least one of \(p_{j}=1\), then
$$ \bigl\Vert T(\vec{f}) \bigr\Vert _{L^{p,\infty }(v_{\vec{w}})}\leq C\prod_{j=1}^{m} \Vert f_{j} \Vert _{L^{p_{j}}(w_{j})}. $$
Proof
The desired result directly is obtained from Theorem 3.1, Corollary 3.2, Lemma 2.4 and Lemma 2.9. The proof is completed. □
4 Estimates for multilinear commutators
To ensure the fluency of the demonstration in this section, we need first to explain the meaning of some notations. We write
We always take \(\sigma (i)\leq \sigma (j)\) if \(i\leq j\).
For any \(\sigma '\in C_{j}^{m}\), we have \(\sigma '=\{\sigma (1),\sigma (2),\ldots,\sigma (m)\}\setminus \sigma \) and \(\sigma '\in C_{m-j}^{m}\).
Let b⃗ be m-tuple functions and \(\sigma \in C_{j}^{m}\), we have the j-tuple function \(\vec{b}=(b_{\sigma (1)},b_{\sigma (2)}, \ldots, b_{\sigma (j)})\). For all \(b_{\sigma (j)}\in {\mathrm{BMO}}_{\theta }(\varphi )\), \(1\leq j\leq m\), we have \(\vec{b}=(b_{\sigma (1)},b_{\sigma (2)},\ldots,b_{\sigma (m)})\in { \mathrm{BMO}}_{\vec{\theta }}^{m}(\varphi )\). See [15, 16].
Corresponding to the classical form, can define the following form of the iterated commutators:
Theorem 4.1
Let T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1, \(T_{\Pi _{\vec{b}}}\) be a multilinear commutator with \(\vec{b}\in {\mathrm{BMO}}^{m}_{\vec{\theta }}(\varphi )\). We have \(0<\delta <\epsilon <1/m\) and \(\eta >(\theta _{1},\ldots,\theta _{m})/ (1/\delta -1/\epsilon )\), assume that ω is satisfying
Then there exists a constant \(C>0\) such that
for all m-tuples \(\vec{f}=(f_{1},\ldots,f_{m})\) of bounded measurable functions with compact support.
Proof
For simplicity, we only prove the case \(m=2\) and \(\theta _{1}=\theta _{2}=\theta \).
If ω is satisfying (4.1), then \(\omega \in \mathrm{Dini(1)}\) and
For \(b_{1},b_{2}\in \mathrm{BMO}_{\theta }(\varphi )\), it suffices to prove that
For any constants \(\lambda _{1},\lambda _{2}\), it follows that
where
and
Now, we fix \(x\in \mathbb{R}^{n}\), a dyadic cube \(Q\ni x\) and a constant c, then, since \(0<\delta <\frac{1}{2}\), we only need to consider the two cases \(r\leq 1\) and \(r>1\).
Case 1: When \(r\leq 1\), the following estimate holds:
Let \(Q^{*}=8\sqrt{n}Q\) and let \(\lambda _{j}=(b_{j})_{Q^{*}}\) be the average of \(b_{j}\) on \(Q^{*},j=1,2\). For any \(1< r_{1},r_{2},r_{3}<\infty \) with \(1/r_{1}+1/r_{2}+1/r_{3}=1\), choosing a δ to make \(\delta r_{i}<1,i=1,2\) and \(r_{3}<\epsilon /\delta \).
By Hölder’s inequality, we have
For II, let \(1< t_{1},t_{2}<\infty \) with \(1=1/t_{1}+1/t_{2}\) and \(t_{2}<\epsilon /\delta \). By Hölder’s inequality,
Similarly, we obtain
Now for the last term IV. We split each \(f_{j}\) as \(f_{j}=f_{j}^{0}+f_{j}^{\infty }\) where \(f_{i}^{0}=f_{j}\chi _{Q^{*}}\) and \(f^{\infty }_{j}=f_{j}-f_{j}^{0}\).
Let \(c=c_{1}+c_{2}+c_{3}\), where
Then
For \(\mathit{IV}_{1}\), choosing \(1< p<\frac{1}{2\delta }\) and applying Kolmogorov’s inequality with \(p=\delta <\frac{1}{2}\), \(q=\frac{1}{2}\),
Next to estimate \(\mathit{IV}_{2}\). For any \(z\in Q\), let \(c_{1}=T((b_{1}-\lambda _{1})f_{1}^{0},(b_{2}-\lambda _{2})f_{2}^{ \infty })(x)\), we have
For any \(z\in Q,y_{1}\in Q^{*}\) and \(y_{2}\in \Omega _{k}\),
then we have
Note that, for the constant \(\lambda _{j}=(b_{j})_{Q^{*}}\), the following holds:
Taking \(N\geq 2\eta \), then
Similarly to \(\mathit{IV}_{2}\), we can estimate
Now for the term \(\mathit{IV}_{4}\). For any \(z\in Q\) and \(y_{1},y_{2}\in \Omega _{k}\),
Note \(c_{3}=T((b_{1}-\lambda _{1})f_{1}^{\infty },(b_{2}-\lambda _{2})f_{2}^{ \infty })(x)\). Then
Case 2: When \(r>1\). Let \(0<\delta <\epsilon <1\), the following holds:
Let \(Q^{*}=8\sqrt{n}Q\) and let \(\lambda _{j}=(b_{j})_{Q^{*}}\) be the average of \(b_{j}\) on \(Q^{*},j=1,2\). For any \(1< r_{1},r_{2},r_{3}<\infty \) with \(1/r_{1}+1/r_{2}+1/r_{3}=1\), we choose a δ small enough to make \(\delta r_{i}<1,i=1,2\) and \(r_{3}<\epsilon /\delta \).
Using Hölder’s inequality, choosing η so that \(\eta (\frac{1}{\delta }-\frac{1}{\epsilon })>2\theta \), then
By the Hölder inequality, and Lemma 2.5, we have
Now to estimate \(\mathit{II}_{1}\), using the Kolmogorov inequality and the boundedness of operators, we have
The way of estimate \(\mathit{II}_{2}\) is the same as \(\mathit{II}_{3}\), we only prove \(\mathit{II}_{2}\):
Choosing η such that \(\eta /\delta -1>0\), \(N\geq \theta +2\eta +1\), we get
Let \(N\geq \theta +2\eta +1\) and \(\eta /\delta -1>0\), then
Now estimate IV. We first split any function
Similar to the estimate of \(\mathit{II}_{1}\), taking \(\eta (\frac{1}{\delta }-2)>2\theta \), then
\(\mathit{IV}_{2}\) and \(\mathit{IV}_{3}\) are symmetric, here, only to estimate of \(\mathit{IV}_{2}\), similar to \(\mathit{II}_{2}\), taking \(N\geq 2\theta +2\eta +1\), we get
Finally, similarly we estimate \(\mathit{IV}_{4}\),
Thus, we completed the proof of Theorem 4.1. □
Theorem 4.2
Let T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1, \(T_{\Pi _{\vec{b}}}\) be a multilinear commutator with \(\vec{b}\in {\mathrm{BMO}}^{m}_{\vec{\theta }}(\varphi )\) and \(\vec{w}\in A_{\vec{p}}^{\infty }(\varphi )\) with \(1/p=1/p_{1}+\cdots +1/p_{m}\) and \(1< p_{j}<\infty,j=1,\ldots,m\). If ω satisfies
then there exists a constant \(C>0\) such that
Proof
It is sufficient to prove that, \(p>0\), \(w\in A_{\infty }^{\infty }\),
By the related Fefferman–Stein inequality (Lemma 2.7) and Theorem 4.1, we get
Here \(\epsilon,\eta \) are the same as in Theorem 4.1.
By Theorem 3.1, then
For simplicity, we only prove the case \(m=2\) in the following:
Similar to the estimate of \(\|M_{\epsilon,\varphi,\eta }^{\sharp,d}(T(\vec{f}))\|_{L^{p}(w)}\), and Eqs. (4.2) and (4.3),
To sum up, we obtain
By Lemma 2.3, we get
If \(\mu >1\), and since \(\Phi (t)=t(1+{\mathrm{log}}^{+}(t))\leq t^{\mu } (t>1)\), we easily get
By Lemma 2.9, then
The inequality \(\|\mathcal{M}_{\mu,\varphi,\eta }(\vec{f})\|_{L^{p}_{(v(\vec{w}))}} \leq C \prod_{j=1}^{m}\|f_{j}\|_{L^{p}_{(w_{j})}}\) is equivalent to \(\|\mathcal{M}_{\varphi,\eta }(\vec{f})\|_{L^{p/u}(v_{\vec{w}})}\leq C \prod_{j=1}^{m}\|f_{j}\|_{L^{p}_{(w_{j})}}\), which was proved in [16]. For some \(\mu >1\), using Lemma 2.4, we get
Thus, this proof is finished. By the proof of Theorem 4.2, the following results are obtained. □
Theorem 4.3
Let \(p>0\) and w be a weight in \(A_{\infty }^{\infty }(\varphi )\), and suppose that \(T_{\Pi _{\vec{b}}}\) is a multilinear iterated commutator with \(\vec{b}\in {\mathrm{BMO}}_{\vec{\theta }}^{m}(\varphi )\). Let \(\eta >0\) and ω is satisfying (4.1). Then there exist constants \(C>0\) depending on the \(A_{\infty }^{\infty }(\varphi )\) constant of w such that
and
Proof
The first result is proved in Theorem 4.2, the proof of second result is similar to the first, also refer to the [8, Theorem 3.19], we need to use Lemma 2.8, Theorem 3.1 and Theorem 4.1. We omit the details here. □
Lemma 4.4
([21])
Let \(w\in A_{\vec{1}}^{\theta }(\varphi )\) and \(\eta >2\theta \). Then there exists a constant \(C>0\) such that
where \(\Phi (t)=t(1+{\mathrm{log}}^{+}t)\) and \(\Phi ^{(m)}=\underbrace{\Phi \circ \cdots \circ \Phi }\).
Theorem 4.5
Let T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1, \(T_{\Pi _{\vec{b}}}\) be a multilinear commutator with \(\vec{b}\in {\mathrm{BMO}}^{m}_{\vec{\theta }}(\varphi )\) and \(\vec{w}\in A_{\vec{1}}^{\infty }(\varphi )\), assume that ω is satisfying (4.1). Then there exists a constant \(C>0\) such that
Proof
Now by Theorem 4.1, Theorem 4.3 and Lemma 4.4, we can get the above result. Since this argument is the same as the proof of [16, Theorem 4.2], here, we omit the proof. □
Corollary 4.6
Let T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1, \(T_{\Sigma _{\vec{b}}}\) be a multilinear commutator with \(\vec{b}\in {\mathrm{BMO}}^{m}_{\vec{\theta }}(\varphi )\), assume that ω is satisfying (4.1), \(0<\delta <\epsilon <1/m\) and \(\eta \geq 2(\theta _{1},\ldots,\theta _{m})/(1/\delta -1/\epsilon )\). Then there exists a constant \(C>0\) such that
For all m-tuples \(\vec{f}=(f_{1},\ldots,f_{m})\) of bounded measurable functions with compact support.
Proof
In fact, the multilinear commutator is a special case of iterate commutator, so we can directly obtain this result through Theorem 4.1. □
Corollary 4.7
Let \(T_{\Sigma _{\vec{b}}}\) be a multilinear commutator with \(\vec{b}\in {\mathrm{BMO}}^{m}_{\vec{\theta }}(\varphi )\), T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1and \(\vec{w}\in A_{\vec{p}}^{\infty }(\varphi )\) with \(1/p=1/p_{1}+\cdots +1/p_{m}\) and \(1< p_{j}<\infty \), \(j=1,\ldots,m\). If ω is satisfying
then there exists a constant \(C>0\) such that
Proof
Obviously, the multilinear commutator is a special case of the iterate commutator, then, through Theorem 4.2 we can directly obtain this result. □
Corollary 4.8
Let \(p>0\) and w be a weight in \(A_{\infty }^{\infty }(\varphi )\), and suppose that \(T_{\Sigma _{\vec{b}}}\) be a multilinear commutator with \(\vec{b}\in {\mathrm{BMO}}^{m}_{\vec{\theta }}(\varphi )\), T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1. Let \(\eta >0\) and ω is satisfying (4.1). Then there exist constants \(C>0\) depending on the \(A_{\infty }^{\infty }(\varphi )\) constant of w such that
and
Proof
Similar to the proof of [8, Theorem 3.19], we need to use Lemma 2.7, Lemma 2.8, Theorem 3.1, and Corollary 4.6. We omit the details here. □
Lemma 4.9
([16])
Let \(w\in A_{\vec{1}}^{\theta }(\varphi )\) and \(\eta >2\theta \). Then there exists a constant C such that
where \(\Phi (t)=t(1+{\mathrm{log}}^{+}t)\).
Corollary 4.10
Let \(T_{\Sigma _{\vec{b}}}\) be a multilinear commutator with \(\vec{b}\in {\mathrm{BMO}}^{m}_{\vec{\theta }}(\varphi )\), T be a multilinear Calderón–Zygmund operator of type \(\omega (t)\) as in Definition 1.1and \(\vec{w}\in A_{\vec{1}}^{\infty }(\varphi )\), assume that ω is satisfying (4.1). Then there exists a constant C such that
Proof
Now by Corollary 4.8 and Lemma 4.9. We can get Corollary 4.10. Since this argument is the same as the proof of [16, Theorem 4.2], here, we omit the proof. □
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References
Coifman, R.R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212, 315–331 (1975)
Coifman, R.R., Meyer, Y.: Commutateurs d’intégrales singulières et opérateurs multilinéaires. Ann. Inst. Fourier 28(3), 177–202 (1978)
Grafakos, L., Torres, R.H.: Multilinear Calderón–Zygmund theory. Adv. Math. 165(1), 124–164 (2002)
Kenig, C., Stein, E.M.: Multilinear estimates and fractional integration. Math. Res. Lett. 6(1), 1–15 (1999)
Grafakos, L., Torres, R.H.: Maximal operator and weighted norm inequalities for multilinear singular integrals. Indiana Univ. Math. J. 51(5), 1261–1276 (2000)
Hu, G., Meng, Y., Yang, D.: Weighted norm inequalities for multilinear Calderón–Zygmund operators on non-homogeneous metric measure spaces. Forum Math. 26(5), 1289–1322 (2014)
Grafakos, L., He, D.: Multilinear Calderón–Zygmund operators on Hardy spaces. J. Math. Anal. Appl. 416(2), 511–521 (2014)
Lerner, A.K., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo-González, R.: New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory. Adv. Math. 220(4), 1222–1264 (2009)
David, G., Journé, J.L.: A boundedness criterion for generalized Calderón–Zygmund operators. Ann. Math. 120(2), 371–397 (1984)
Yabuta, K.: Generalizations of Calderón–Zygmund operators. Stud. Math. 1, 17–31 (1985)
Hu, G.: Estimates for the maximal bilinear singular integral operators. Acta Math. Sin. 31, 847–862 (2015)
Lu, G., Zhang, P.: Multilinear Calderón–Zygmund operators with kernels of Dini’s type and applications. Nonlinear Anal. 107, 92–117 (2014)
Zhang, P., Sun, J.: Commutators of multilinear Calderón–Zygmund operators with kernels of Dini’s type and applications (2016) arXiv:1605.07449
Pérez, C., Torres, R.H.: Sharp maximal function estimates for multilinear singular integrals. Contemp. Math. 320, 323–331 (2003)
Pérez, C., Pradolini, G., Torres, R.H., Trujillo-González, R.: End-point estimates for iterated commutators of multilinear singular integrals. Bull. Lond. Math. Soc. 46, 26–42 (2014)
Pan, G., Tang, L.: New weighted norm inequalities for certain classes of multilinear operators and their iterated commutators. Potential Anal. 43, 371–398 (2015)
Bui, T.A.: New class of multiple weights and new weighted inequalities for multilinear operators. Forum Math. 27(2), 995–1023 (2015)
Hu, X., Zhou, J.: New weighted norm inequalities for Calderón–Zygmund operators with kernels of Dini’s type and their commutators. Proc. Indian Acad. Sci. Math. Sci. 129, 56–71 (2019)
García-Cuerva, J., Rubio de Francia, J.: Weighted Norm Inequalities and Related Topics. Amsterdam, New York (1985)
Bongioanni, B., Haboure, E., Salinas, O.: Classes of weights related to Schrodinger operators. J. Math. Anal. Appl. 373, 563–579 (2011)
Tang, L.: Weighted norm inequalities for pseudo-differential operators with smooth symbols and their commutators. J. Funct. Anal. 262, 1603–1629 (2012)
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The research was supported by Natural Science Foundation of China (Grant Nos. 12061069).
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Zhao, Y., Zhou, J. New weighted norm inequalities for multilinear Calderón–Zygmund operators with kernels of Dini’s type and their commutators. J Inequal Appl 2021, 29 (2021). https://doi.org/10.1186/s13660-021-02560-8
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DOI: https://doi.org/10.1186/s13660-021-02560-8