Weighted estimates for commutators associated to singular integral operator satisfying a variant of Hörmander's condition

1.   Introduction and results

In 1952, Calder$\mathrm{\acute{o}}$n and Zygmund ^[1] introduced the concept of singular integral in the study of elliptic partial differential equations and proved the existence of singular integral. Later, Calder$\mathrm{\acute{o}}$n and Zygmund ^[2] studied a class of singular integral of convolution type and proved the $L^{p}$-boundedness $(1 < p < \infty)$.

In the classical Calder$\mathrm{\acute{o}}$n-Zygmund theory, the Hörmander's condition ^[3]

plays a foundational role in Harmonic analysis. As the development of singular integrals, the kernel $K$ which does not satisfy the condition (1.1) has been extensively considered. In ^[4], Grubb and Moore introduced a variant of Hörmander's condition

where $\mathcal{B}_{j}$ and $\phi_{j}$ are appropriate functions and proved the $L^{p}$-boundedness for the singular operator with kernel satisfying (1.2).

Later, Trujillo-Gonz$\acute{a}$lez made the kernel's conditions stronger and established the weighted $L^{p}$-boundedness of singular integral operator as below.

Theorem A. ^[5] Let $K\in L^{2} (\mathbb{R}^{n})$ satisfy

$(K1)$ $\|\hat{K}\|_{\infty}\leq C_{0},$

$(K2)$ $|K(x)|\leq \frac{C_{0}}{|x|^{n}},$

$(K3)$ there exist functions $\mathcal{B}_{1}\cdots \mathcal{B}_{m}$ and $\Phi = \{\phi_{1}, \cdots\phi_{m}\}\subset L^{\infty}(\mathbb{R}^{n})$ such that $|\mathrm{det}[\phi_{j}(y_{i})]|^{2}\in RH_{\infty}(\mathbb{R}^{nm})$,

$(K4)$ for a fixed $\gamma > 0$ and any $|x| > 2|y| > 0,$

For any $f\in C_{c}^{\infty}(\mathbb{R}^{n})$, we define the singular integral operator $T$ related to the kernel $K$ by:

Let $1 < p < \infty$, $\omega\in A_{p}$, then there exists a constant $C > 0$, such that

Remark A. When $m = 1$, $\mathcal{B}_{1}(x) = K(x)$, $\phi_{1}(y) = 1$, then (1.2) is exactly Hörmander's condition and (1.3) is the classical Calder$\mathrm{\acute{o}}$n-Zygmund kernel.

With the development of singular integral operators, their commutators have been well studied. In 1976, Coifman, Rochberg and Weiss ^[6] established the boundedness of commutators on some $L^{p}(\mathbb{R}^{n}) (1 < p < \infty)$. Let $b$ be a locally integrable function on $\mathbb{R}^{n}$ and let $T$ be a Calder$\mathrm{\acute{o}}$n-Zygmund singular integral operator. Consider the commutator $T_{b}$ defined for suitable functions $f$ by

In 2004, Zhou ^[7] proved that the commutators generated by the singular integral operator and a BMO function are bounded on Hardy spaces. In 2009, Kong and Jiang ^[8] got the boundedness of commutators generated by the singular integral operator with a homogeneous kernel and a BMO function on weighted Hardy spaces and weighted Herz spaces. In 2010, Liu ^[9] showed that the commutators generated by the singular integral operator and a BMO function are bounded on Herz-Hardy spaces. For more information about this topic we refer to ^{[10,11,12,13]}.

In 2012, Zhang ^[14] studied the commutators generated by the singular integral operator satisfying a variant of Hörmander's condition and a BMO function are bounded on Hardy spaces. In 2015, Xie ^[15] discussed the boundedness of multilinear operators from Lebesgue spaces to Orlicz spaces when the kernel $K$ satisfies the conditions $(K1)$–$(K4)$. In 2017, Pan ^[16] obtained the boundedness of multilinear operators satisfying a variant of Hörmander's condition on Morrey spaces.

Motivated by these results, we will study the boundedness of commutators generated by the singular integral operator satisfying a variant of Hörmander's condition and a weighted BMO function on some function spaces in this paper. Now, we state our main results as follows.

Theorem 1.1. Let $T$ be the singular integral operator with the kernel $K$ satisfying $(K1)$–$(K4)$. Let $\gamma$ be as in $K4$. Suppose that $\mu\in A_{1}$, $\frac{n}{n+\gamma} < p\leq1$ and $b\in \rm BMO_{\mu}$. Then $T_{b}$ is bounded from $\mathcal{H}_{\Phi, b}^{p}(\mu)$ to $L^{p}(\mu^{1-p})$.

Remark 1.1. When we take $m = 1, \mathcal{B}_{1}(x) = K(x), \phi_{1}(y) = 1$, then the commutator generated by the classical singular integral operator and a weighted BMO function is bounded from $\mathcal{H}_{b}^{p}(\mu)$ to $L^{p}(\mu^{1-p})$.

Theorem 1.2. Let $T$ be the singular integral operator with the kernel $K$ satisfying $(K1)$–$(K4)$. Suppose that $\mu\in A_{1}$ and $b\in \rm BMO_{\mu}$. Then $T_{b}$ is bounded from $\mathcal{H}_{\Phi}^{1}(\mu)$ to weak $L^{1}(\mathbb{R}^{n})$. That is to say, for any $\lambda > 0$, there exists $C > 0$ such that

Remark 1.2. When we take $m = 1, \mathcal{B}_{1}(x) = K(x), \phi_{1}(y) = 1$, then the commutator generated by the classical singular integral operator and a weighted BMO function is bounded from $\mathcal{H}^{1}(\mu)$ to weak $L^{1}(\mathbb{R}^{n})$.

Theorem 1.3. Let $T$ be the singular integral operator with the kernel $K$ satisfying $(K1)$–$(K4)$. Suppose that $\mu\in A_{1}$, $0 < p\leq\infty, 1 < q < \infty$, $-\frac{n\delta}{q} < \eta < n\delta-\frac{n}{q}$ and $b\in \rm BMO_{\mu}$. Then $T_{b}$ maps $\dot{K}_{q}^{\eta, p}(\mu, \mu)$ continuously into $\dot{K}_{q}^{\eta, p}(\mu, \mu^{1-q})$.

Remark 1.3. When we take $m = 1, \mathcal{B}_{1}(x) = K(x), \phi_{1}(y) = 1$, then the commutator generated by the classical singular integral operator and a weighted BMO function maps $\dot{K}_{q}^{\eta, p}(\mu, \mu)$ continuously into $\dot{K}_{q}^{\eta, p}(\mu, \mu^{1-q})$.

Theorem 1.4. Let $T$ be the singular integral operator with the kernel $K$ satisfying $(K1)$–$(K4)$. Suppose that $\mu\in A_{1}$, $0 < p\leq1$, $1 < q < \infty$, $\eta = n\delta-\frac{n}{q}$, $q\delta\neq1$ and $b\in \rm BMO_{\mu}$. Then $T_{b}$ maps $\dot{K}_{q}^{\eta, p}(\mu, \mu)$ continuously into $W\dot{K}_{q}^{\eta, p}(\mu, \mu^{1-q})$.

Remark 1.4. When we take $m = 1, \mathcal{B}_{1}(x) = K(x), \phi_{1}(y) = 1$, $\eta = n\delta-\frac{n}{q}$, then the commutator generated by the classical singular integral operator and a weighted BMO function maps $\dot{K}_{q}^{\eta, p}(\mu, \mu)$ continuously into $W\dot{K}_{q}^{\eta, p}(\mu, \mu^{1-q})$.

Throughout this paper, we denote by $p'$ the conjugate index of $p$, that is $\frac{1}{p}+\frac{1}{p'} = 1$. The letter $C$, sometimes with additional parameters, will stand for positive constants, not necessarily the same at each occurrence but is independent of the main parameters.

2.   Preliminaries and lemmas

Firstly, let us introduce some important notations that will help us further.

Definition 2.1. ^[17] A non-negative locally integrable function is called a weight function. Let $\omega$ be a weight function, $1 < p < \infty$. If there is a constant $C > 0,$ such that for any ball $B\subset\mathbb{R}^n$,

then we say $\omega\in A_p.$ We say $\omega\in A_1$, if there is a constant $C > 0$, such that for any ball $B\subset\mathbb{R}^n$,

A weight function $\omega\in A_{\infty}$ if it satisfies the $A_p$ condition for some $1 \leq p < \infty$. The smallest constant satisfying the fomulas above is called $A_p$ constant of $\omega$ and we denote it by $[\omega]_{A_p}$.

For $1\leq p < q < \infty$, we have $A_{1}\subset A_{p}\subset A_{q}$ and $A_{\infty} = \cup_{p\geq1}A_{p}$.

Definition 2.2. ^[5] Given a positive and locally integrable function $g$ in $\mathbb{R}^{n}$, we say that $g$ satisfies the reverse Hölder $RH_{\infty}$ condition, in short, $g\in RH_{\infty}(\mathbb{R}^{n})$, if for any cube Q centered at the origin we have

with $C > 0$ being an absolute constant independent of $Q$.

Definition 2.3. ^[18] Suppose $\mu\in A_{\infty}$. We will say that a locally integrable function $b(x)$ belongs to the weighted $\mathrm{BMO}_{\mu}$, that is

where the supremum is taken over all balls $B\subset\mathbb{R}^{n}$, $\mu(B) = \int_{B}\mu(x)\mathrm{d}x$, the smallest constant $C$ is denoted as $\|b\|_{\ast, \mu}$.

For $1\leq p, q \leq\infty$, we have

Similar to the definition of the Hardy space related to $\Phi$ in ^[19], we define the weighted Hardy space related to $\Phi$.

Definition 2.4. For $0 < p\leq1$, $\omega\in A_{\infty}$ and $\Phi = \{\phi_{1}, \cdots, \phi_{m}\}\subset L^{\infty}(\mathbb{R}^{n})$, a function $a(x)$ is called an $\omega-(p, \infty, \Phi)$ atom centered at $x_0$, if

${\rm(1)} \mathrm{supp}a\subset B(x_{0}, r)$ for some $x_{0}\in \mathbb{R}^{n}$ and $r > 0$,

${\rm(2)} \|a\|_{L^{\infty}} \leq \omega(B(x_{0}, r))^{-\frac{1}{p}}$,

${\rm(3)}\int_{\mathbb{R}^{n}}a(x)\mathrm{d}x = \int_{\mathbb{R}^{n}}a(x)\phi_{j}(x-x_{0})\mathrm{d}x = 0, j = 1, 2, \cdots, m.$

Definition 2.5. For $0 < p\leq1$ and $\omega\in A_{\infty}$, we say that a distribution $f$ on $\mathbb{R}^{n}$ belongs to the weighted Hardy space $\mathcal{H}_{\Phi}^{p}(\omega)$, if in distributional sense, it can be written as $f = \mathop {\sum}\limits_{j = -N}^{N}\lambda_{j}a_{j}$, where each $a_{j}$ is an $\omega-(p, \infty, \Phi)$ atom, $N\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and $\mathop {\sum}\limits_{j = -N}^{N}|\lambda_{j}|^{p} < \infty$. Moreover,

with the infimum taken over all decompositions of $f$.

Similar to the Definition 2.5, we define the weighted Hardy space related to $\Phi$ and $b$.

Definition 2.6. For $0 < p\leq1$, $\omega\in A_{\infty}$, $\Phi = \{\phi_{1}, \cdots, \phi_{m}\}\subset L^{\infty}(\mathbb{R}^{n})$ and $b$ is a locally integrable function, a function $a(x)$ is called an $\omega-(p, \infty, \Phi, b)$ atom centered at $x_0$, if

${\rm(1)} \mathrm{supp}a\subset B(x_{0}, r)$ for some $x_{0}\in \mathbb{R}^{n}$ and $r > 0$,

${\rm(2)} \|a\|_{L^{\infty}} \leq \omega(B(x_{0}, r))^{-\frac{1}{p}}$,

${\rm(3)} \int_{\mathbb{R}^{n}}a(x)\mathrm{d}x = \int_{\mathbb{R}^{n}}a(x)\phi_{j}(x-x_{0})\mathrm{d}x = \int_{\mathbb{R}^{n}}a(x)b(x)\phi_{j}(x-x_{0})\mathrm{d}x = 0, j = 1, 2, \cdots, m.$

Definition 2.7. For $0 < p\leq1$ and $\omega\in A_{\infty}$, we say that a distribution $f$ on $\mathbb{R}^{n}$ belongs to the weighted Hardy space $\mathcal{H}_{\Phi, b}^{p}(\omega)$, if in distributional sense, it can be written as $f = \mathop {\sum}\limits_{j = -N}^{N}\lambda_{j}a_{j}$, where each $a_{j}$ is an $\omega-(p, \infty, \Phi, b)$ atom, $N\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and $\mathop {\sum}\limits_{j = -N}^{N}|\lambda_{j}|^{p} < \infty$. Moreover,

with the infimum taken over all decompositions of $f$.

For $k\in\mathbb{Z}$, $B_{k} = B(0, 2^{k})$ and $E_{k} = B_{k}\backslash B_{k-1}$. Let $\chi_{k} = \chi_{E_{k}}$ denote the characteristic function of the set $E_{k}$.

Definition 2.8. ^[20] Let $\eta\in \mathbb{R}$, $0 < p < \infty$, $1 < q < \infty$, $\omega_{1}$ and $\omega_{2}$ be non-negative weighted functions. The homogeneous Herz space $\dot{K}_{q}^{\eta, p}(\omega_{1}, \omega_{2})$ is defined by

where $\|f\|_{ \dot{K}_{q}^{\eta, p}(\omega_{1}, \omega_{2})} = \big\{ \mathop {\sum}\limits_{k = -\infty}^{\infty} \omega_{1}(B_{k})^{\frac{\eta p}{n}} \|f\chi_{k}\|_{L_{\omega_{2}}^{q}}^{p} \big\}^{\frac{1}{p}}$ with the usual modification made when $p = \infty$.

Definition 2.9. ^[20] Let $\eta\in \mathbb{R}$, $0 < p < \infty$, $1 < q < \infty$, $\omega_{1}$ and $\omega_{2}$ be non-negative weighted functions. The measurable function $f(x)$ belongs to the weighted weak Herz space $W\dot{K}_{q}^{\eta, p}(\omega_{1}, \omega_{2})$, if

where $m_{k}(\lambda, f) = \omega_{2}(\{x\in E_{k}:|f(x)| > \lambda\})$ with the usual modification made when $p = \infty$.

Lemma 2.1. ^[18] Suppose that $\omega\in A_{1}$. Then there exist constant $C_{1}, C_{2} > 0$ and $0 < \delta < 1$ for each measurable subset of $B\subset \mathbb{R}^{n}$ such that

Lemma 2.2. ^[8] Suppose that $\mu\in A_{1}$ and $b\in \rm BMO_{\mu}(\mathbb{R}^{n})$. Then there exists a constant $C > 0$ for each measurable subset of $B\subset \mathbb{R}^{n}$ such that

Lemma 2.3. ^[21] Suppose that $\mu\in A_{1}$ and $1 < q < \infty$. Then there exists a constant $C > 0$ for each measurable subset of $B\subset \mathbb{R}^{n}$ such that

According to ^[22], we have the following lemma :

Lemma 2.4. Let $T$ be the singular integral operator with the kernel $K$ satisfying $(K1)$–$(K4)$. Suppose that $1 < p < \infty$, $\mu\in A_{1}$ and $b\in \rm BMO_{\mu}$. Then $T_{b}$ is bounded from $L^{p}(\mu)$ to $L^{p}(\mu^{1-p})$.

3.   Proofs of theorems

Proof of Theorem 1.1. It suffices to prove that, for any $\mu-(p, \infty, \Phi, b)$ atom $a$, the following inequality holds

In fact, if (3.1) holds, then for any $f = \mathop {\sum}\limits_{j = -N}^{N}\lambda_{j}a_{j}\in\mathcal{H}_{\Phi, b}^{p}(\mu)$, where each $a_{j}$ is a $\mu-(p, \infty, \Phi, b)$ atom, since $\frac{n}{n+\gamma} < p\leq1$, we have

Then the proof can be completed by taking the infimum for all atomic decompositionns of $f$.

Now, we need to prove (3.1), suppose that $\mathrm{supp}a\subset B = B(x_{B}, r)$. Write

By the Hölder inequality for $q > 1$ and Lemma 2.4, we have

Let $C_{k} = 2^{k+1}B\backslash 2^{k}B$

Write

For $I_{21}$, when $y\in B$, $x\in 2^{k+1}B\backslash 2^{k}B$, we have $|y-x|\thicksim |x-x_{B}|\geq2|y-x_{B}|$, by the vanishing condition of $a$, we obtain

For $I_{22}$, we have

Since $\frac{n}{n+\gamma} < p\leq1$,

Combining the estimates for $I_{1}$ and $I_{2}$, we finish the proof. □

Proof of Theorem 1.2. We can write $f = \mathop{\sum}\limits_{j = -N}^{N}\lambda_{j}a_{j}\in\mathcal{H}_{\Phi}^{1}(\mu)$ with each $a_{j}$ being a $\mu-(1, \infty, \Phi)$ atom and $\mathop {\sum}\limits_{j = -N}^{N} |\lambda_{j}| < \infty$. Suppose $\mathrm{supp} a_{j}\subset B_{j} = B(x_{j}, r_{j})$. Write

By the Hölder inequality and the Theorem A, we obtain

and

Thus, we have

Noting that

By the weak $(1, 1)$ boundedness of $T$ in ^[4], we obtain

Therefore,

Taking the infimum over all decompositions of $f$, the proof is finished. □

Proof of Theorem 1.3. We only consider the case $0 < p < \infty$. When $p = \infty$, we can make appropriate modifications.

Firstly, we estimate $L_{2}$. By Lemma 2.4

Obviously

Now, let us estimate $L_{1}$, for $j\geq k+2$, by Lemmas 2.1–2.3, we have

and

When $\eta > 0$, we have

Thus

When $\eta < 0$, we have

Thus

Lastly, we estimate $L_{3}$, for $k\geq j+2$, by Lemmas 2.1–2.3, we have

and

When $\eta > 0$

Thus

Similarly, when $\eta < 0$, we have

Then we finish the proof. □

Proof of Theorem 1.4. For $k\in \mathbb{Z}$, we can write $f$ as

Thus

and

By Lemma 2.4 and $0 < p\leq1$, we obtain

For $G_{1}$, by the $\mathrm{H\ddot{o}lder}$ inequality and Lemmas 2.1–2.3, we have

For $G_{2}$, by Lemmas 2.1–2.3, we have

Then we finish the proof. □

4.   Conclusions

Some new weighted inequalities for commutators related to singular integral operator satisfying a variant of Hörmander's condition are proved. Our results further generalize the corresponding results in ^[8] and include the unweighted case.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author would like to thank the referee for his/her invaluable comments and suggestions.

This work was supported by a Research Project of the Basic Scientific Research Expenses for Heilongjiang Provincial Higher Institutions (No.1355ZD010), the Reform and Development Foundation for Local Colleges and Universities of the Central Government (Excellent Young Talents Project of Heilongjiang Province) (No.2020YQ07), the project of Mudanjiang Normal University (No.GP2019006), the Ideology and Politics Project of Postgraduate course (No.KCSZKC-2022026, KCSZAL-2022013) and the project of Scientific Research Team of Education Department of Heilongjiang Province (No.1354MSYTD006; 2019-KYYWF-0909).

Conflict of interest

All of authors in this article declare no conflict of interest.