Abstract
In this paper, we establish some new estimates for commutators of Hausdorff operators on homogeneous Herz and Morrey-Herz spaces, which extend some previous results.
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1 Introduction and results
In 2000, Liflyand and Móricz gave the definition of Hausdorff operator in [1]. Suppose f is a locally integrable function on \(\mathbb{R}\), the Hausdorff operator is defined by
where Φ belongs to \(L^{1}(\mathbb{R})\). The fractional Hausdorff operator in higher dimensional space \(\mathbb {R}^{n}\) is defined in [2] as follows:
where Φ is a radial function defined on \(\mathbb{R}^{n}\), \(0\leq \beta< n\). When \(\beta=0\), \(H_{\Phi,\beta}f=H_{\Phi}f\).
If we set \(\Phi(t)=t^{\beta-n}\chi_{(1,\infty)}(t)\), we get
where \(H_{\beta}\) is the fractional Hardy operator defined by
If we set \(\Phi(t)=\chi_{(0,1)}(t)\), we obtain
where \(H_{\beta}^{*}\) is defined by
The properties of Hausdorff operator in \(L^{p}\), \(H^{p}\), \(h^{p}\) and other spaces can be found in [1, 3–7].
Let T be a classical Calderón-Zygmund operator with its kernel satisfying the standard estimates; the commutator generated by T and a function \(b\in \mathit{BMO}(\mathbb{R}^{n})\) is defined by
In 1976, Coifman et al. [8] proved that \([b,T]\) is bounded on \(L^{p}(\mathbb{R}^{n})\) for any \(1< p<\infty\). In 2012, Gao and Jia studied the boundedness of commutator \([b,H_{\Phi }]\) of Hausdorff operator with central BMO function in Lebesgue space, Morrey-Herz spaces, and Herz spaces in [9]. The theory of commutators of singular integrals has been studied extensively.
In 1981, Cohen [10] considered the following generalized commutators:
where \(\Omega\in L^{1}(S^{n-1})\) satisfies \(\Omega(\lambda x)=\Omega (x)\), \(\lambda>0\), ∀x, \(\int_{S^{n-1}}|x|\Omega(x)\,dx=0\). And he proved when \(\Omega\in \operatorname{Lip}_{1}(S^{n-1})\), \(\nabla A\in \mathit{BMO}(\mathbb {R}^{n})\), \(T_{A}\) is bounded on \(L^{p}(\mathbb{R}^{n})\) for \(1< p<\infty\).
In 2002, Pérez and Trujillo-González [11] established the boundedness for the multilinear commutators of the classical Calderón-Zygmund operators \(T_{\vec{b}}f\), which are defined by \(T_{\vec{b}}f (x)=\int_{\mathbb{R}^{n}}\prod_{j=1}^{m}(b_{j}(x)-b_{j}(y))K(x,y)\,dy\). Later the multilinear commutators were widely studied by many authors.
In this paper, we mainly discuss the properties of generalized commutators and multilinear commutators of Hausdorff operators with central BMO functions in some function spaces.
Let us first of all recall the definition of homogeneous central BMO space.
Definition 1
[12]
Let \(1\leq q< \infty\), \(\mathit{CBMO}_{q}(\mathbb{R}^{n})\) is the space of all functions \(f\in L_{\mathrm{loc}}^{q}(\mathbb{R}^{n})\) satisfying
where \(B(0,r)=\{x\in\mathbb{R}^{n}:|x|< r\}\) and \(f_{B(0,r)}\) is the mean value of f on \(B(0,r)\).
It is easy to see \(\mathit{BMO}(\mathbb{R}^{n})\subsetneqq \mathit{CBMO}_{q}(\mathbb {R}^{n})\), \(1\leq q<\infty\), and \(\mathit{CBMO}_{q}(\mathbb{R}^{n})\subsetneqq \mathit{CBMO}_{p}(\mathbb{R}^{n})\), \(1\leq p< q<\infty\).
Let us give the definitions of the generalized commutators of Hausdorff operator and the multilinear commutators of Hausdorff operators.
Let A be a function on \(\mathbb{R}^{n}\) having derivatives of order one in \(\mathit{CBMO}_{q}(\mathbb{R}^{n})\). For \(x,t\in\mathbb{R}^{n}\), set
We define the fractional generalized commutators of Hausdorff operators as follows:
where Φ is a radial function. When \(\beta=0\), write \(H_{\Phi ,\beta,A}f(x)=H_{\Phi,A}f(x)\), and \(H_{\Phi,\beta,A}^{1}f(x)=H_{\Phi,A}^{1}f(x)\), \(H_{\Phi,\beta ,A}^{2}f(x)=H_{\Phi,A}^{2}f(x)\).
Set \(\Phi(t)=t^{\beta-n}\chi_{(1,\infty)}(t)\), then
and set \(\Phi(t)=\chi_{(0,1)}(t)\), we have
In 2010, Lu and Zhao [13] considered the properties of generalized commutators of Hardy operators. They got the following results.
Theorem A
Let \(1< p<\infty\), \(u\geq p'\), \(s>n\), and \(\lambda\geq0\). Suppose \(\nabla A \in \mathit{CBMO}_{\max\{s,u\}}\). If \(\alpha< np/p'\), then
Theorem B
Let \(1< p<\infty\), \(1< q<\infty\). Suppose \(u\geq q'\), \(s>n\) and \(\nabla A \in \mathit{CBMO}_{\max\{s,u\}}\). If \(\alpha< n/q'\), then
Theorem C
Let \(1< p<\infty\), \(1< q<\infty\) and \(\lambda\geq0\). Suppose \(u\geq q'\), \(s>n\) and \(\nabla A \in \mathit{CBMO}_{\max\{s,u\}}\). If \(\alpha< n/q'+\lambda\), then
Let \(\vec{b}=(b_{1},\ldots,b_{m})\), \(b_{j}\in \mathit{CBMO}_{q}(\mathbb{R}^{n})\), \(1\leq j\leq m\), \(1\leq q<\infty\). Similarly to [14], we consider the following the multilinear commutator \(H_{\Phi,\beta,\vec{b}}\) generalized by the n dimensional fractional Hausdorff operator \(H_{\Phi,\beta}\) and b⃗:
where m is a nonpositive integer, Φ is a radial function.
We decompose the integral
When \(m=1\), \(\beta=0\), denote
and
Select
and
we have
and
Before we formulate the main theorems, we give some remarks
When \(\beta=0\), denote by \(\mathscr{A}_{\Phi,\beta,r}^{n}=\mathscr {A}_{\Phi,r}^{n}\).
Our main results are the following theorems.
Theorem 1
Suppose \(1< p,q<\infty\), \(0\leq\beta< n\), \(\frac{1}{p}-\frac{1}{q}=\frac{\beta}{n}\), and \(\nabla A\in \mathit{CBMO}_{\max\{s,\frac{pr}{p-r}\}}(\mathbb{R}^{n})\) with \(s>n\), \(r<\min\{p,q'\}\).
-
(a)
If \(\mathscr{A}_{\Phi,\beta,r}^{n}<\infty\), then \(H_{\Phi,\beta,A}^{1}\) is bounded from \(L^{p}(\mathbb{R}^{n})\) to \(L^{q}(\mathbb{R}^{n})\).
-
(b)
If \(\mathscr{B}_{\Phi,r}^{n}<\infty\), then \(H_{\Phi ,\beta,A}^{2}\) is bounded from \(L^{p}(\mathbb{R}^{n})\) to \(L^{q}(\mathbb {R}^{n})\).
-
(c)
If \(\mathscr{A}_{\Phi,\beta,r}^{n}<\infty\), \(\mathscr {B}_{\Phi,r}^{n}<\infty\), then \(H_{\Phi,\beta,A}\) is bounded from \(L^{p}(\mathbb{R}^{n})\) to \(L^{q}(\mathbb{R}^{n})\).
Theorem 2
Suppose \(0 < p < \infty\), \(1< q_{1},q_{2}<\infty\), \(0\leq\beta< n\), \(\frac{1}{q_{1}}-\frac {1}{q_{2}}=\frac{\beta}{n}\), and \(\nabla A\in \mathit{CBMO}_{\max\{s,\frac {q_{1}r}{q_{1}-r}\}}(\mathbb{R}^{n})\) with \(1< r< q_{1}\), \(s>n\).
-
(a)
If \(\mathscr{A}_{\Phi,\beta,r}^{n}<\infty\), \(\alpha <\frac{n}{r}-\frac{n}{q_{1}}\), then \(H_{\Phi,\beta,A}^{1}\) is bounded from \(\dot{K}^{\alpha,p}_{q_{1}}(\mathbb{R}^{n})\) to \(\dot{K}^{\alpha ,p}_{q_{2}}(\mathbb{R}^{n})\).
-
(b)
If \(\mathscr{B}_{\Phi,r}^{n}<\infty\), \(\alpha>-(\frac {n}{q_{2}}-\frac{n}{r})\), then \(H_{\Phi,\beta,A}^{2}\) is bounded from \(\dot{K}^{\alpha,p}_{q_{1}}(\mathbb{R}^{n})\) to \(\dot{K}^{\alpha ,p}_{q_{2}}(\mathbb{R}^{n})\).
-
(c)
If \(\mathscr{A}_{\Phi,\beta,r}^{n}<\infty\), \(\mathscr {B}_{\Phi,r}^{n}<\infty\), \(-(\frac{n}{q_{2}}-\frac{n}{r})<\alpha<\frac {n}{r}-\frac{n}{q_{1}}\), then \(H_{\Phi,\beta,A}\) is bounded from \(\dot {K}^{\alpha,p}_{q_{1}}(\mathbb{R}^{n})\) to \(\dot{K}^{\alpha ,p}_{q_{2}}(\mathbb{R}^{n})\).
Theorem 3
Suppose \(0 < p < \infty\), \(1< q_{1},q_{2}<\infty\), \(0\leq\beta< n\), \(\lambda\geq0\), \(\frac {1}{q_{1}}-\frac{1}{q_{2}}=\frac{\beta}{n}\), and \(\nabla A\in \mathit{CBMO}_{\max \{s,\frac{q_{1}r}{q_{1}-r}\}}(\mathbb{R}^{n})\) with \(1< r< q_{1}\), \(s>n\).
-
(a)
If \(\mathscr{A}_{\Phi,\beta,r}^{n}<\infty\), \(\alpha <\frac{n}{r}-\frac{n}{q_{1}}+\lambda\), then \(H_{\Phi,\beta,A}^{1}\) is bounded from \(M\dot{K}^{\alpha,\lambda}_{p,q_{1}}(\mathbb{R}^{n})\) to \(M\dot{K}^{\alpha,\lambda}_{p,q_{2}}(\mathbb{R}^{n})\).
-
(b)
If \(\mathscr{B}_{\Phi,r}^{n}<\infty\), \(\alpha>-(\frac {n}{q_{2}}-\frac{n}{r})+\lambda\), then \(H_{\Phi,\beta,A}^{2}\) is bounded from \(M\dot{K}^{\alpha,\lambda}_{p,q_{1}}(\mathbb{R}^{n})\) to \(M\dot{K}^{\alpha,\lambda}_{p,q_{2}}(\mathbb{R}^{n})\).
-
(c)
If \(\mathscr{A}_{\Phi,\beta,r}^{n}<\infty\), \(\mathscr {B}_{\Phi,r}^{n}<\infty\), \(-(\frac{n}{q_{2}}-\frac{n}{r})+\lambda <\alpha<\frac{n}{r}-\frac{n}{q_{1}}+\lambda\), then \(H_{\Phi,\beta ,A}\) is bounded from \(M\dot{K}^{\alpha,\lambda}_{p,q_{1}}(\mathbb {R}^{n})\) to \(M\dot{K}^{\alpha,\lambda}_{p,q_{2}}(\mathbb{R}^{n})\).
Remark 1
Select \(\Phi(t)=t^{\beta -n}\chi_{(1,\infty)}(t)\), it is easy to get \(\mathscr{A}_{\Phi,\beta,r}^{n}<\infty\), and \(H_{\Phi,\beta ,A}f=H_{\Phi,\beta,A}^{1}f=H_{\beta,A}f\). We can get the boundedness of generalized commutator of fractional Hardy operator on Herz and Morrey-Herz spaces.
Remark 2
If we select \(\Phi (t)=t^{\beta-n}\chi_{(1,\infty)}(t)\), it is easy to get \(\mathscr{B}_{\Phi,r}^{n}<\infty\), and \(H_{\Phi,\beta,A}f=H_{\Phi ,\beta,A}^{2}f=H_{\beta,A}^{*}f\), by Theorems 2 and 3, we can get the boundedness of \(H_{\beta,A}^{*}\) on Herz and Morrey-Herz spaces.
Remark 3
When \(\beta=0\), by Remark 1 and the definitions of Herz and Morrey-Herz spaces, we can easily get Theorem A, B, and C.
Theorem 4
Let \(0 < p< \infty\), \(1< q_{1},q_{2}<\infty\), \(0\leq\beta< n\), \(\frac{1}{q_{1}}-\frac {1}{q_{2}}=\frac{\beta}{n}\), \(1< r< q_{1}\), \(s>n\), and \(b_{j}\in \mathit{CBMO}_{\max\{r_{j}q_{2},\frac{q_{1}r}{q_{1}-r}r_{j}\}}(\mathbb {R}^{n})\), \(r_{j}>1\) (\(1\leq j\leq m\)), \(\frac{1}{r_{1}}+\cdots+\frac {1}{r_{m}}=1\).
-
(a)
If \(A_{\Phi,\beta,r}^{n}<\infty\), \(\alpha<\frac {n}{r}-\frac{n}{q_{1}}\), then \(H_{\Phi,\beta,\vec{b}}^{1}\) is bounded from \(\dot{K}^{\alpha,p}_{q_{1}}(\mathbb{R}^{n})\) to \(\dot{K}^{\alpha ,p}_{q_{2}}(\mathbb{R}^{n})\).
-
(b)
If \(\mathscr{B}_{\Phi,r}^{n}<\infty\), \(\alpha>-(\frac {n}{q_{2}}-\frac{n}{r})\), then \(H_{\Phi,\beta,\vec{b}}^{2}\) is bounded from \(\dot{K}^{\alpha,p}_{q_{1}}(\mathbb{R}^{n})\) to \(\dot{K}^{\alpha ,p}_{q_{2}}(\mathbb{R}^{n})\).
-
(c)
If \(\mathscr{A}_{\Phi,\beta,r}^{n}<\infty\), \(\mathscr {B}_{\Phi,r}^{n}<\infty\), \(-(\frac{n}{q_{2}}-\frac{n}{r})<\alpha<\frac {n}{r}-\frac{n}{q_{1}}\), then \(H_{\Phi,\beta,\vec{b}}\) is bounded from \(\dot{K}^{\alpha,p}_{q_{1}}(\mathbb{R}^{n})\) to \(\dot{K}^{\alpha ,p}_{q_{2}}(\mathbb{R}^{n})\).
Theorem 5
Let \(\lambda\geq0\), \(0 < p< \infty\), \(1< q_{1},q_{2}<\infty\), \(0\leq\beta< n\), \(\frac{1}{q_{1}}-\frac {1}{q_{2}}=\frac{\beta}{n}\), \(1< r< q_{1}\), \(s>n\), and \(b_{j}\in \mathit{CBMO}_{\max\{r_{j}q_{2},\frac {q_{1}r}{q_{1}-r}r_{j}\}}(\mathbb{R}^{n})\), \(r_{j}>1\) (\(1\leq j\leq m\)), \(\frac {1}{r_{1}}+\cdots+\frac{1}{r_{m}}=1\).
-
(a)
If \(\mathscr{A}_{\Phi,\beta,r}^{n}<\infty\), \(\alpha <\frac{n}{r}-\frac{n}{q_{1}}+\lambda\), then \(H_{\Phi,\beta,\vec {b}}^{1}\) is bounded from \(M\dot{K}^{\alpha,\lambda}_{p,q_{1}}(\mathbb {R}^{n})\) to \(M\dot{K}^{\alpha,\lambda}_{p,q_{2}}(\mathbb{R}^{n})\).
-
(b)
If \(\mathscr{B}_{\Phi,r}^{n}<\infty\), \(\alpha>-(\frac {n}{q_{2}}-\frac{n}{r})+\lambda\), then \(H_{\Phi,\beta,\vec{b}}^{2}\) is bounded from \(M\dot{K}^{\alpha,\lambda}_{p,q_{1}}(\mathbb{R}^{n})\) to \(M\dot{K}^{\alpha,\lambda}_{p,q_{2}}(\mathbb{R}^{n})\).
-
(c)
If \(\mathscr{A}_{\Phi,\beta,r}^{n}<\infty\), \(\mathscr {B}_{\Phi,r}^{n}<\infty\), \(-(\frac{n}{q_{2}}-\frac{n}{r})+\lambda <\alpha<\frac{n}{r}-\frac{n}{q_{1}}+\lambda\), then \(H_{\Phi,\beta ,\vec{b}}\) is bounded from \(M\dot{K}^{\alpha,\lambda }_{p,q_{1}}(\mathbb{R}^{n})\) to \(M\dot{K}^{\alpha,\lambda }_{p,q_{2}}(\mathbb{R}^{n})\).
Remark 4
If we select \(\Phi(t)=t^{\beta -n}\chi_{(1,\infty)}(t)\), it is obvious that \(\mathscr{A}_{\Phi ,\beta,r}^{n}<\infty\), we may obtain the boundedness of \(H_{\beta,\vec{b}}\) on Herz and Morrey-Herz spaces. If we select \(\Phi(t)=\chi_{(0,1)}(t)\), \(\mathscr{B}_{\Phi ,r}^{n}<\infty\), we may obtain the boundedness of \(H^{\ast}_{\beta ,\vec{b}}\) on Herz and Morrey-Herz spaces.
Corollary 1
Let \(1< p,q<\infty\), \(s>n\), \(0\leq\beta< n\), \(\frac{1}{p}-\frac{1}{q}=\frac{\beta}{n}\), \(r<\min\{p,q'\}\), and \(b_{j}\in \mathit{CBMO}_{\max\{r_{j}q,\frac{pr}{p-r}r_{j}\}}(\mathbb{R}^{n})\), \(r_{j}>1\) (\(1\leq j\leq m\)), \(\frac{1}{r_{1}}+\cdots+\frac{1}{r_{m}}=1\).
-
(a)
If \(\mathscr{A}_{\Phi,\beta,r}^{n}<\infty\), then \(H_{\Phi,\beta,\vec{b}}^{1}\) is bounded from \(L^{p}(\mathbb{R}^{n})\) to \(L^{q}(\mathbb{R}^{n})\).
-
(b)
If \(\mathscr{B}_{\Phi,r}^{n}<\infty\), then \(H_{\Phi ,\beta,\vec{b}}^{2}\) is bounded from \(L^{p}(\mathbb{R}^{n})\) to \(L^{q}(\mathbb{R}^{n})\).
-
(c)
If \(\mathscr{A}_{\Phi,\beta,r}^{n}<\infty\), \(\mathscr {B}_{\Phi,r}^{n}<\infty\), then \(H_{\Phi,\beta,\vec{b}}\) is bounded from \(L^{p}(\mathbb{R}^{n})\) to \(L^{q}(\mathbb{R}^{n})\).
Remark 5
When \(\beta=0\), \(m=1\), the boundedness of commutator \(H_{\Phi,b}\) can be obtained on Lebesgue, Herz, and Morrey-Herz spaces.
The rest of this paper is organized as follows. After recalling some preliminary notations and lemmas in Section 2, we will prove our results in Section 3. We would like to remark that the main methods employed in this paper are a combination of ideas and arguments from [8, 9] and [13].
Throughout this paper, we let \(p'\) satisfy \(1/p+1/p'=1\). The letter C, sometimes with additional parameters, will stand for positive constants, not necessarily the same at each occurrence, but C is independent of the essential variables.
2 Preliminaries and lemmas
In order to prove the theorems, we will formulate some lemmas and preliminaries. For the multi-indices \(\gamma=(\gamma_{1},\ldots,\gamma_{n})\), we will always use notations \(|\gamma|=\gamma_{1}+\cdots+\gamma_{n}\), \(\gamma_{j}\) (\(1\leq j\leq n\)) being nonnegative integers, \(x^{\gamma}=x_{1}^{\gamma_{1}}\cdots x_{n}^{\gamma_{n}}\). Let \(\nabla A=(D_{1}A,D_{2}A,\ldots,D_{n}A)\) where \(D_{j}A=\frac{\partial A}{\partial x_{j}}\), \(j=1,\ldots,n\).
For any positive integer m and j (\(1\leq j\leq m\)), we denote by \(\mathscr{C}_{j}^{m}\) the family of all finite subsets \(\sigma=\{\sigma(1),\sigma(2),\ldots,\sigma(j)\}\) of \(\{1,2,\ldots,m\}\) of j different elements. For any \(\sigma\in \mathscr{C}_{j}^{m}\), we associate the complementary sequence \(\sigma '\in\mathscr{C}_{m-j}^{j}\) given by \(\sigma'=\{1,2,\ldots,m\}\setminus\sigma\). We also denote by \(|\sigma|\) the number of elements in σ, and
Let \(b_{j}\in \mathit{CBMO}_{qr_{j}}(\mathbb{R}^{n})\) (\(1\leq j\leq m\)), \(\vec{b}=(b_{1},b_{2},\ldots,b_{m})\), for \(1< r_{i},q<\infty\),
denote by \(\|\vec{b}\|_{\mathit{CBMO}_{qr_{i}}(\mathbb{R}^{n})}=\|b_{1}\| _{\mathit{CBMO}_{qr_{1}}(\mathbb{R}^{n})}\|b_{2}\|_{\mathit{CBMO}_{qr_{2}}(\mathbb{R}^{n})}\cdots \|b_{m}\|_{\mathit{CBMO}_{qr_{m}}(\mathbb{R}^{n})}\).
For all \(1\leq j\leq m\) and \(\sigma=\{\sigma(1),\sigma(2),\ldots,\sigma(j)\}\in\mathscr{C}_{j}^{m}\). Denote \(\vec{b}_{\sigma}=(b_{\sigma(1)},b_{\sigma(2)},\ldots,b_{\sigma(j)})\), \(\vec{b}_{\sigma'}=(b_{\sigma(j+1)},\ldots,b_{\sigma(m)})\). Denote
and
where B is any ball in \(\mathbb{R}^{n}\), \((b_{\sigma(j)})_{B}\) is the average of \(b_{\sigma(j)}\) over ball B. Denote by
where \(1< q<\infty\), \(r_{\sigma_{(j)}}\in\{r_{1},\ldots,r_{m}\}\), \(1\leq j\leq m\),
and
For all \(\sigma\in\mathscr{C}_{j}^{m}\), denote
when \(\sigma=\{1,2,\ldots,m\}\), \(\sigma'=\emptyset\), write \(\vec {b}_{\sigma}=\vec{b}\), \(H_{\Phi,\beta,\vec{b}_{\sigma}}=H_{\Phi ,\beta,\vec{b}}\), \(H_{\Phi,\beta,\vec{b}_{\sigma'}}=H_{\Phi,\beta}\).
For \(k\in\mathbb{Z}\), let \(B_{k}=\{x\in\mathbb{R}^{n}:|x|\leq2^{k}\}\), \(C_{k}=B_{k}\setminus B_{k-1}\), and \(\chi_{k}\) (\(k\in\mathbb{Z}\)) denote the characteristic function of the set \(C_{k}\).
Definition 2
[15]
Let \(\alpha\in \mathbb{R}\), \(0< p\leq\infty\), \(0< q<\infty\). The homogeneous Herz space \(\dot{K}^{\alpha,p}_{q}(\mathbb{R}^{n})\) is defined by
where
with the usual modification made when \(p=\infty\).
Obviously, \(\dot{K}^{0,q}_{q}(\mathbb{R}^{n})=L^{q}(\mathbb{R}^{n})\), \(\dot {K}^{\frac{\alpha}{q},q}_{q}(\mathbb{R}^{n})=L^{q}(\mathbb {R}^{n},|x|^{\alpha})\), so the Herz space is the natural generalization of the Lebesgue spaces with power weight \(|x|^{\alpha}\).
Definition 3
[16]
Let \(\alpha\in \mathbb{R}\), \(0< p\leq\infty\), \(0< q<\infty\), and \(\lambda\geq0\). The homogeneous Morrey-Herz space \(M\dot{K}^{\alpha,\lambda }_{p,q}(\mathbb{R}^{n})\) is defined by
where
with the usual modification made when \(p=\infty\).
Obviously, \(M\dot{K}^{\alpha,0}_{p,q}(\mathbb{R}^{n})=\dot{K}^{\alpha ,p}_{q}(\mathbb{R}^{n})\).
Similarly to the discussion of Lemma 3.1 in [9], it is easy to get the following results.
Lemma 1
Let \(\beta\geq0\), \(1< r\leq p<\infty\), Φ is a radial function, then
Lemma 2
[14]
Let A be a function on \(\mathbb{R}^{n}\) with derivatives of order one in \(L^{q}(\mathbb {R}^{n})\) for some \(q>n\). Then
where \(I_{x}^{y}\) is the cube centered at x with sides parallel to the axes and whose side length is \(2\sqrt{n}|x-y|\).
Lemma 3
[10]
Suppose that \(f\in \mathit{CBMO}_{q}(\mathbb{R}^{n})\), \(1\leq q<\infty\), and \(r_{1},r_{2}>0\). Then
3 Proofs of main theorems
It is easily to see that Theorem 1 can be immediately deduced from Theorem 2 by letting \(\alpha=0\), \(1< p=q_{1}<\infty\), \(1< q_{2}=q<\infty\). Thus it is sufficient to prove Theorems 2 and 3.
Proof of Theorem 2
We only consider the case \(1< p<\infty\), while the case \(p=\infty\) follows after slight modifications.
For simplicity, we denote \(q=\frac{q_{1}r}{q_{1}-r}\).
(a) When \(\mathscr{A}_{\Phi,\beta,r}^{n}<\infty\), we get
For fixed k, set
where \(m_{B_{k}}(\nabla A)\) is the mean value of ∇A on \(B_{k}\). By a simple calculation, we get \(R(A;x,y)=R(A_{k};x,y)\). (More details may be found in [17].)
Since \(y\in C_{i}\) and \(x\in C_{k}\), \(i\leq k-2\), then \(|x-y|\sim|x|\sim 2^{k}\). It is easy to deduce that \(\nabla A_{k}(y)=\nabla A(y)-m_{B_{k}}(\nabla A)\). Then by Lemma 2 for \(s>n\), similarly as the discussion in [15] we have
Then we can split I into two parts
Now we estimate the \(I_{1}\). Using Lemma 1(i) and noting that \(\frac {1}{q_{2}}=\frac{1}{q_{1}}-\frac{\beta}{n}\), we get
For \(y\in C_{i}\), we have
Then we can decompose \(I_{2}\) into two parts
For \(I_{2}'\), by Lemma 1(i) for \(r< q_{1}\) and Hölder’s inequality, we have
For \(I_{2}''\), using Lemma 1(i) and \(\frac{1}{q_{2}}=\frac{1}{q_{1}}-\frac {\beta}{n}\), we get
To estimate II, take a \(\phi\in C_{0}^{\infty}\), such that \(\operatorname{supp}\phi \subset B(0,2)\) and \(\phi=1\) in \(B(0,1)\). Set \(M=\max\{\|\phi\|_{\infty},\|\nabla\phi\|_{\infty}\}\). Take \(y_{0}\in C_{k+4}\), and let
For \(x\in C_{k}\), \(y\in C_{i}\), \(k-1\leq i\leq k\), and \(\phi=1\) in \(B(0,1)\), we may have \(\phi(2^{-k}x)=1\), \(\phi(2^{-k}y)=1\). Since
we have
Then we have
Now we consider \(\mathit{II}_{1}\). Since \(k-1\leq i\leq k\), \(y\in C_{i}\), \(y_{0}\in C_{k+4}\), \(|y-y_{0}|\sim2^{k}\), by Lemma 2, we get
Then we see
Since \(\nabla A_{k}(y)=\nabla A(y)-m_{B_{k}}(\nabla A)\), we get
Employing the same idea for estimating \(I_{2}\), we have
We now estimate \(\mathit{II}_{2}''\). Applying Lemma 1(i), we get
Next we turn to estimate \(\mathit{II}_{1}\). By Lemma 1(i), we get the following estimate:
Now we estimate \(|A_{k}^{\phi}(x)-A_{k}^{\phi}(y)|\). By Lemma 2 and Lemma 3, we obtain
Then by \(\frac{1}{q_{2}}=\frac{1}{q_{1}}-\frac{\beta}{n}\), we get
Then we have
Case 1. \(0< p\leq1\). By the well-known inequality
we have the following:
Similarly to the estimate of L, we obtain
For K, we have
Case 2. \(1< p<\infty\). Using Hölder’s inequality, we get
Similarly to the estimate of L, we obtain
For K, we have
This completes the proofs of (a). The proofs of (b) are similar to that for (a), thus we omit the details. By combining the estimate (a) and (b), we get (c). □
Proof of Theorem 3
(a) We only give the proof when \(\lambda>0\). By a similar method to the proof of I, II in Theorem 2, we have
For \(E_{1}\), noting that \(\alpha<\frac{n}{r}-\frac{n}{q_{1}}+\lambda\), we have
Similarly to the proof of \(E_{1}\), we have
For \(E_{3}\), we have the following estimate:
This finishes the proof of (a). The proof of Theorem 3(b) is similar to that for (a). We omit the details. By combining the estimates of (a) and (b), we can easily get (c). □
Proof of Theorem 4
We prove (a) first. When \(\mathscr{A}_{\Phi,\beta,r}^{n}<\infty\), we get
To estimate I, by Lemma 1(i) and \(\frac{1}{q_{2}}=\frac{1}{q_{1}}-\frac{\beta}{n}\), \(\frac{1}{r_{1}}+\cdots+\frac{1}{r_{m}}=1\), \(1< r_{j}<\infty\), for \(1< r< q_{1}\), we get
For II, we have
By Lemma 1(i), we have
And by Hölder’s inequality for \(\frac{1}{r_{\sigma(1)}}+\frac {1}{r_{\sigma(2)}}+\cdots+\frac{1}{r_{\sigma(j)}}+\frac {1}{r_{\sigma'}}=1\), and noting that \(\frac{1}{q_{2}}=\frac{1}{q_{1}}-\frac{\beta}{n}\), we get
By Lemma 3, we have
Now we estimate III. Since
and by Lemma 3 and Minkowski’s inequality, we get
Then by Hölder’s inequality, Lemma 1, and Lemma 3, we have
Also
Then
For \(0< p\leq1\), by the well-known inequality
We have the following:
For \(1< p<\infty\), using Hölder’s inequality, we get
This finishes the proof of (a). The proof of Theorem 4(b) is similar to that for (a). We omit the details. By combining the estimates of (a) and (b), we can easily get (c). □
Proof of Theorem 5
We only give the proof when \(\lambda>0\). By the definition of Morrey-Herz spaces and the estimates for I, II, III in the proof of Theorem 4, we have
For K, noting that \(\alpha<\frac{n}{r}-\frac{n}{q_{1}}+\lambda\), we have
This finishes the proof of (a). The proof of Theorem 5(b) is similar to that for (a). We omit the details. By combining the estimates of (a) and (b), we can easily get (c). □
4 Conclusions
This paper proves the boundedness of the generalized commutators of Hausdorff operators \(H_{\Phi,\beta,A}\) and the multilinear commutators of Hausdorff operators \(H_{\Phi,\beta ,\vec{b}}\) with central BMO function, not only in Herz spaces, but also in Morrey-Herz spaces, which promotes some results of Hardy operators or the multilinear commutators of Hausdorff operators \(H_{\Phi,\beta,b}^{m}\).
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Acknowledgements
We would like to appreciate the reviewers for reading the paper and make helpful comments that improved the original paper. This work was supported by National Natural Science Foundation of China (Grant Nos. 11071065, 11171306, and 11571160). This work was also supported by the Foundation of Mudanjiang Normal University (Grant No. sy201325).
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Sun, J., Shi, X. Some estimates for commutators of Hausdorff operators. J Inequal Appl 2015, 409 (2015). https://doi.org/10.1186/s13660-015-0933-8
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DOI: https://doi.org/10.1186/s13660-015-0933-8