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Boundedness of rough singular integral operators and commutators on Morrey-Herz spaces with variable exponents
Journal of Inequalities and Applications volume 2016, Article number: 222 (2016)
Abstract
By decomposing functions, we establish some boundedness results for some rough singular integrals on the homogeneous Morrey-Herz spaces \(M\dot{K}_{q, p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})\), where the two main indices are variable. The corresponding results as regards their commutators are also considered.
1 Introduction
In recent years, function spaces with variable exponents have been intensively studied; see [1–6] for example. The origin of such spaces is the study of PDE with non-standard growth conditions, fluid dynamics and image restoration; see [7–9]. By virtue of the fundamental work [10] by Kováčik and Rákosník appearing in 1991, the Lebesgue spaces and various other function spaces have been investigated in the variable exponent setting. Meanwhile, the boundedness of some classical operators, such as the Hardy-Littlewood maximal operator, singular integrals and commutators, has been proved on these spaces; see [11–19] and the references therein.
Herz spaces have been playing a central role in harmonic analysis. After they were introduced in [20], the theory of these spaces had a remarkable development in part due to its useful applications. For instance, they are good substitutes of the ordinary Hardy spaces when considering the boundedness of non-translation invariant singular integral operators, they also appear in the characterization of multiplier on Hardy spaces and in the regularity theory for elliptic and parabolic equations in divergence form; see [21–23] for example.
One of the important problems on Herz spaces is the boundedness of sublinear operators satisfying the size condition
for integrable and compactly supported functions f. We mention that condition (1.1) was initially studied by Soria and Weiss [24] and it is satisfied by several classical operators, such as Calderón-Zymund operators, the Carleson maximal operator and the Hardy-Littlewood maximal operator. Hernández, Li, Lu, and Yang [25–27] proved that if a sublinear operator T is bounded on \(L^{p}({\Bbb {R}}^{n})\) and satisfies the size condition (1.1), then T is bounded on the homogeneous Herz spaces \(\dot{K}_{p}^{\alpha,q}({\Bbb{ R}}^{n})\) and on the non-homogeneous Herz spaces \(K_{p}^{\alpha,q}({\Bbb{ R}}^{n})\). This result is extended to the generalized Herz spaces \(\dot{K}_{p(\cdot)}^{\alpha,q}({\Bbb{ R}}^{n})\) and \(K_{p(\cdot)}^{\alpha,q}({\Bbb{ R}}^{n})\) with variable exponent \(p(\cdot)\) by Izuki [28]. In 2012, Almeida and Drihem [11] made a further step and gave boundedness results for T on \(\dot{K}_{p(\cdot)}^{\alpha(\cdot),q}({\Bbb{ R}}^{n})\) and \(K_{p(\cdot)}^{\alpha(\cdot),q}({\Bbb{ R}}^{n})\), where the exponent α is variable as well.
Denote by \(S^{n-1}\) the unit sphere in \({\Bbb {R}}^{n}\) \((n \ge2)\) with normalized Lebesgue measure dσ. Let \(\Omega\in L^{s}(S^{n-1})\) for some \(s\in(1,\infty]\) be homogeneous of degree zero. In this paper, we consider sublinear operators satisfying the size condition
for integrable and compactly supported functions f and commutators defined by
Lu et al. first proved the boundedness for both \(T_{\Omega}\) and \([b,T_{\Omega}]\) on Herz spaces and Morrey-Herz spaces with constant exponent; see [29, 30]. Motivated by the work of Lu [29] and Almeida [11], we shall generalize these results to the case of variable exponent and also consider the boundedness on Morrey-Herz spaces with variable exponent. Our approach is mainly based on some properties of variable exponent and BMO norms obtained by the author [31].
For brevity, we denote \(B:=\{y\in{\Bbb {R}}^{n}:|x-y|< r\}\). \(f_{B}\) stands for the integral average of f on B, i.e. \(f_{B}=\frac{1}{|B|}\int_{B}f(x)\,dx\). \(p'(\cdot)\) means the conjugate exponent \(1/p(\cdot)+1/p'(\cdot)=1\). C denotes a positive constant, which may have different values even in the same line. \(f\lesssim g\) means that \(f\leq Cg\), and \(f\thickapprox g\) means \(f\lesssim g\lesssim f\).
2 Preliminaries and lemmas
Let \(E\subset{\Bbb{ R}}^{n}\) with the Lebesgue measure \(|E|>0\), \(p(\cdot): E\rightarrow[1,\infty)\) be a measurable function. The Lebesgue space with variable exponent \(L^{p(\cdot)}(E)\) is defined by
This is a Banach space with the Luxemburg-Nakano norm
The space with variable exponent \(L^{p(\cdot)}_{\mathrm{loc}}(E)\) is defined by
We denote
and
where the Hardy-Littlewood maximal operator M is defined by
A function \(\alpha(\cdot):{\Bbb{ R}}^{n} \rightarrow\Bbb {R}\) is called log-Hölder continuous at the origin (or has a log decay at the origin), if there exists a constant \(C_{\log}>0\) such that
for all \(x\in{\Bbb{R}}^{n}\). If, for some \(\alpha_{\infty}\in{\Bbb{ R}}\) and \(C_{\log}>0\), we have
for all \(x\in{\Bbb{ R}}^{n}\), then \(\alpha(\cdot)\) is called log-Hölder continuous at infinity (or has a log decay at infinity).
By \(\mathscr{P}_{0}^{\log}({\Bbb{ R}}^{n})\) and \(\mathscr{P}_{\infty}^{\log}({\Bbb{ R}}^{n})\) we denote the class of all exponents \(p\in\mathscr{P}({\Bbb{ R}}^{n})\) which have a log decay at the origin and at infinity, respectively. It is worth noting that if \(p(\cdot)\in{\mathscr{P}_{0}^{\log}({\Bbb{ R}}^{n})\cap\mathscr{P}_{\infty}^{\log}({\Bbb{ R}}^{n})}\), then we have \(p(\cdot)\in\mathscr{B}({\Bbb{ R}}^{n})\); see [31] or [32].
Let \(B_{k} = \{x \in{\mathbb {R}} ^{n}: |x| \leq2^{k}\}\), \(R_{k} = B_{k} \backslash B_{k-1}\) and \(\chi_{k} = \chi_{R_{k}}\) be the characteristic function of the set \(R_{k}\) for \(k\in\mathbb{Z}\). Almeida and Direhem [11] first introduced the following Herz spaces with variable exponents.
Definition 2.1
Let \(0< q\leq\infty\), \(p(\cdot)\in \mathscr{P}({\Bbb{ R}}^{n})\) and \(\alpha(\cdot):{\Bbb{ R}}^{n} \rightarrow\Bbb {R}\) with \(\alpha\in L^{\infty}({\Bbb{ R}}^{n})\).
-
(1)
The homogeneous Herz space \(\dot{K}_{p(\cdot)}^{\alpha(\cdot),q}({\Bbb{ R}}^{n})\) is defined as the set of all \(f\in L_{\mathrm{loc}}^{p(\cdot)}({{\Bbb{ R}}^{n}}\backslash\{0\})\) such that
$$\|f\|_{{\dot{K}_{p(\cdot)}^{\alpha(\cdot),q}}({\Bbb{ R}}^{n})}:= \biggl(\sum_{k\in\mathbb{Z}} \bigl\| 2^{k\alpha(\cdot)} f\chi_{k}\bigr\| ^{q}_{L^{p(\cdot)}({\Bbb{ R}}^{n})} \biggr)^{1/q}< \infty. $$ -
(2)
The non-homogeneous Herz space \(K_{p(\cdot)}^{\alpha(\cdot),q}({\Bbb{ R}}^{n})\) consists of all \(f\in L_{\mathrm{loc}}^{p(\cdot)}({{\Bbb{ R}}^{n}})\) such that
$$\|f\|_{{K_{p(\cdot)}^{\alpha(\cdot),q}}({\Bbb{ R}}^{n})}:= \|f\chi_{B_{0}}\|_{L^{p(\cdot)}({\Bbb{ R}}^{n})}+ \biggl(\sum _{k\geq 1}\bigl\| 2^{k\alpha(\cdot)} f\chi_{k} \bigr\| ^{q}_{L^{p(\cdot)}({\Bbb{ R}}^{n})} \biggr)^{1/q}< \infty, $$with the usual modification when \(q=\infty\).
Definition 2.2
Let \(0< q\leq\infty\), \(p(\cdot)\in \mathscr{P}({\Bbb{ R}}^{n})\), \(0\leq\lambda<\infty\) and \(\alpha(\cdot):{\Bbb{ R}}^{n} \rightarrow\Bbb {R}\) with \(\alpha\in L^{\infty}({\Bbb{ R}}^{n})\). The homogeneous Morrey-Herz space \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})\) is defined as the set of all \(f\in L_{\mathrm{loc}}^{p(\cdot)}({{\Bbb{ R}}^{n}}\backslash\{0\})\) such that
with the usual modification when \(q=\infty\).
Remark 2.3
It is easy to see that \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),0}({\Bbb{ R}}^{n})=\dot{K}_{p(\cdot)}^{\alpha(\cdot),q}({\Bbb{ R}}^{n})\). If \(\alpha(\cdot)\) is constant, then \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})= M\dot{K}_{q,p(\cdot)}^{\alpha,\lambda}({\Bbb{ R}}^{n})\), which is first defined by Izuki [33]. If both \(\alpha(\cdot)\) and \(p(\cdot)\) are constant, then \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),0}({\Bbb{ R}}^{n})=\dot{K}_{q}^{\alpha,p}({\Bbb{ R}}^{n})\) are classical Herz spaces in [22], and \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})=M\dot{K}_{p,q}^{\alpha,\lambda}({\Bbb{ R}}^{n})\) are classical Morrey-Herz spaces in [30].
In [32, 34], Lu and Zhu obtained the following result:
Proposition 2.4
Let \(0< q\leq\infty\), \(p(\cdot)\in \mathscr{P}({\Bbb{ R}}^{n})\), \(0\leq\lambda<\infty\) and \(\alpha\in L^{\infty}({\Bbb{ R}}^{n})\). If α is log-Hölder continuous both at origin and at infinity, then
Before stating our main results, we introduce some key lemmas which will be used later.
Lemma 2.5
([10]) (Generalized Hölder inequality)
Let \(p(\cdot)\in\mathscr{P}({\Bbb{ R}}^{n})\), if \(f\in L^{p(\cdot)}{({\Bbb {R}}^{n})}\) and \(g\in L^{p'(\cdot)}{({\Bbb {R}}^{n})}\), then
where \(r_{p}=1+1/p_{-}-1/p_{+}\).
Lemma 2.6
([35])
Let \(\Omega\in L^{s}(S^{n-1})\), \(s\in [1,\infty]\). If \(a>0\), \(d\in(0, s]\) and \(-n+\frac{(n-1)d}{s}<\nu<\infty\), then
Lemma 2.7
([36])
Let \(p(\cdot)\in\mathscr{P}({\Bbb{ R}}^{n})\). If \(q\in(p_{+}, \infty)\) and \(\frac{1}{p(x)}=\frac{1}{\widetilde{q}(x)}+\frac{1}{q} \) (\(x\in {\Bbb{ R}}^{n}\)), then we have
for all measurable functions f and g.
We remark that Lemmas 2.8-2.10 were shown in Izuki [14, 31], and Lemma 2.11 was considered by Almeida and Drihem in [11].
Lemma 2.8
Let \(p(\cdot)\in\mathscr{B} ({\Bbb{ R}}^{n})\), then we see that for all balls B in \({\Bbb{ R}}^{n}\),
Lemma 2.9
Let \(p(\cdot)\in\mathscr{B} ({\Bbb{ R}}^{n})\), then we see that for all balls B in \({\Bbb{ R}}^{n}\) and all measurable subsets \(S\subset B\),
where \(\delta_{1},\delta_{2}\) are constants with \(0<\delta_{1}, \delta_{2}<1\).
Lemma 2.10
Let \(b\in BMO({\Bbb{ R}}^{n})\), \(k>j\) (\(k,j\in \mathbb{N}\)), then we have
and
Lemma 2.11
Let \(\alpha\in L^{\infty}({\Bbb{ R}}^{n})\) and \(r_{1}>0\). If α is log-Hölder continuous both at origin and at infinity, then
for any \(x\in B(0,r_{1})\backslash B(0,r_{1}/2)\) and \(y\in B(0,r_{2})\backslash B(0,r_{2}/2)\), with the implicit constant not depending on \(x,y,r_{1}\), and \(r_{2}\).
3 Main results and their proofs
In this section, we prove the boundedness of sublinear operators \(T_{\Omega}\) satisfying the size condition (1.2) and the commutators \([b,T_{\Omega}]\) defined as in (1.3) on the homogeneous Morrey-Herz space \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})\), respectively.
Our main results in this paper can be stated as follows.
Theorem 3.1
Suppose that \(p(\cdot)\in{\mathscr{P}_{0}^{\log}({\Bbb{ R}}^{n})\cap\mathscr{P}_{\infty}^{\log}({\Bbb{ R}}^{n})}\) and \(\Omega\in L^{s}(S^{n-1})\), \(s>(p')_{+}\). Let \(0<\lambda<n\), \(0< q\leq\infty\), \(-1/s<\nu<\infty\) and let \(\alpha(\cdot)\in L^{\infty}({\Bbb{ R}}^{n})\) be log-Hölder continuous both at the origin and at infinity, such that
where \(0<\delta_{1}, \delta_{2}<1\) are the constants appearing in Lemma 2.9. Then every sublinear operator \(T_{\Omega}\) satisfying (1.2) which is bounded on \(L^{p(\cdot)}({\Bbb{ R}}^{n})\) is also bounded on \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})\).
Theorem 3.2
Suppose that \(b\in BMO({\Bbb{ R}}^{n})\) and let \([b,T_{\Omega}]\) be defined as in (1.3). Under the assumptions in Theorem 3.1, if \([b,T_{\Omega}]\) is bounded on \(L^{p(\cdot)}({\Bbb{ R}}^{n})\), then \([b,T_{\Omega}]\) is also bounded on \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})\).
Remark 3.3
According to Remark 2.3, Theorem 3.1 and Theorem 3.2 extend the corresponding results in [29] to a more generalized function space. Moreover, comparing with [11, 28], our main results generalize the integral kernel to the case of \(\Omega\in L^{s}(S^{n-1})\).
Our proofs use partially some decomposition techniques already used in [30] where the constant exponent case was studied. We consider only \(0< q<\infty\), the arguments are similar in the case \(q=\infty\).
Proof of Theorem 3.1
Let \(f\in M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})\). We decompose
Then we have
First we estimate \(U_{1}\). Noting that if \(x\in R_{k}, y\in R_{j}\) and \(j\leq{k-2}\), then \(|x-y|\approx|x|\approx2^{k}\), by Lemma 2.11, we get
which combining with Lemma 2.5 yields
We define a variable exponent \(\widetilde{p}(\cdot)\) by \(\frac{1}{p'(x)}=\frac{1}{\widetilde{p}'(x)}+\frac{1}{s}\), since \(s>(p')_{+}\), by Lemma 2.7 and Lemma 2.6, we obtain
where the last inequality is based on the fact that \(\|\chi_{B_{j}}(\cdot) \|_{L^{\widetilde{p}'(\cdot)}({\Bbb{R}}^{n})}\approx\|\chi_{B_{j}}(\cdot) \|_{L^{p'(\cdot)}({\Bbb{R}}^{n})}|B_{j}|^{-1/s}\); see [37], p.258, for the details.
From (3.2), (3.3), Lemma 2.8, and Lemma 2.9, we deduce that
For convenience below we put \(\sigma=n\delta_{2}-\alpha_{+}-\nu-n/s\). It follows from the condition (3.1) that \(\sigma>0\). Now we can distinguish two cases as follows:
Case 1∘: If \(0< q\leq1\), using the well-known inequality
we have
Case 2∘: If \(1< q<\infty\), by Hölder’s inequality, we have
Next we estimate \(U_{2}\). By Proposition 2.4 and the hypothesis \(T_{\Omega}\) is bounded on \(L^{p(\cdot)}({\Bbb{ R}}^{n})\), so that
For \(U_{3}\), once again by Proposition 2.4, we have
To estimate I, observe that if \(x\in R_{k}, y\in R_{j}\) and \(j\geq {k+2}\), then \(|x-y|\approx|y|\approx2^{j}\). We apply Lemma 2.5 and obtain
An application of (3.7), (3.3), and Lemma 2.9 gives
In the sequel, we put \(\eta=n\delta_{1}+\nu+n/s\) for short. If \(0< q\leq1\), in view of \(\eta>0\), from (3.6), (3.8), and (3.5), we conclude that
For \(I_{1}\), noting that \(\eta+\alpha(0)>\eta+\alpha_{-}>0\), hence we have
For \(I_{2}\), since \(\lambda-\eta-\alpha(0)<\lambda-\eta-\alpha_{-}<0\), we get
If \(1< q<\infty\), we have
For \(I_{a}\), by Hölder’s inequality, we get
For \(I_{b}\), as argued in \(I_{2}\), we obtain
Hence, we arrive at the desired inequality,
We omit the estimation of J since it is essentially similar to that of I. Consequently, the proof of Theorem 3.1 is complete. □
Proof of Theorem 3.2
Let \(f\in M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}({\Bbb{ R}}^{n})\). We decompose
Then we have
First we estimate \(V_{1}\). Noting that \(|x-y|\approx|x|\approx2^{k}\) for \(x\in R_{k}, y\in R_{j}\) and \(j\leq{k-2}\), then, from Lemma 2.11 and Lemma 2.5, it follows that
Similar to (3.3), we define a variable exponent \(\widetilde{p}(\cdot)\) by \(\frac{1}{p'(x)}=\frac{1}{\widetilde{p}'(x)}+\frac{1}{s}\), since \(s>(p')_{+}\), by Lemma 2.7, Lemma 2.6, and Lemma 2.10, we get
Then by (3.9), Lemma 2.5, (3.3), (3.10), Lemma 2.10, and Lemma 2.9, we obtain
By comparing (3.11) with (3.4), after applying the same arguments used in the estimation of \(U_{1}\) in Theorem 3.1, we can easily get, for all \(0< q<\infty\),
Next we estimate \(V_{2}\). Using Proposition 2.4 and the boundedness of \([b,T_{\Omega}]\) on \(L^{p(\cdot)}({\Bbb{ R}}^{n})\), we derive the estimate
For \(V_{3}\), we apply Proposition 2.4 again and obtain
To estimate G, we note that if \(x\in R_{k}, y\in R_{j}\) and \(j\geq {k+2}\), then \(|x-y|\approx|y|\approx2^{j}\). By Lemma 2.5, we have
As argued for (3.10), we get actually
From (3.13) and (3.14), it follows that
Hence we have
By comparing (3.16) with (3.8), as long as we repeat the same procedure as the estimation of I in Theorem 3.1, we can immediately get for all \(0< q<\infty\),
We omit the estimation of H since it is essentially similar to that of G. Consequently, the proof of Theorem 3.2 is complete. □
References
Almeida, A, Hästö, P: Besov spaces with variable smoothness and integrability. J. Funct. Anal. 258, 1628-1655 (2010)
Cruz-Uribe, D, Fiorenza, A: Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis. Birkhäuser, Basel (2013)
Diening, L, Hästö, P, Roudenko, S: Function spaces of variable smoothness and integrability. J. Funct. Anal. 256, 1731-1768 (2009)
Yan, X, Yang, D, Yuan, W, Zhuo, C: Variable weak Hardy spaces and their applications. J. Funct. Anal. (2016). doi:10.1016/j.jfa.2016.07.006
Yang, D, Zhuo, C, Yuan, W: Triebel-Lizorkin type spaces with variable exponents. Banach J. Math. Anal. 9, 146-202 (2015)
Yang, D, Zhuo, C, Yuan, W: Besov-type spaces with variable smoothness and integrability. J. Funct. Anal. 269, 1840-1898 (2015)
Chen, Y, Levine, S, Rao, R: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383-1406 (2006)
Diening, L, Ru̇žička, M: Calderón-Zygmund operators on generalized Lebesgue spaces \(L^{p(\cdot)}\) and problems related to fluid dynamics. J. Reine Angew. Math. 563, 197-220 (2003)
Harjulehto, P, Hästö, P, Lê, ÚV, Nuortio, M: Overview of differential equations with non-standard growth. Nonlinear Anal. 72, 4551-4574 (2010)
Kováčik, O, Rákosník, J: On spaces \(L^{p(x)}\) and \(W^{k,p(x)}\). Czechoslov. Math. J. 41, 592-618 (1991)
Almeida, A, Drihem, D: Maximal, potential and singular type operators on Herz spaces with variable exponents. J. Math. Anal. Appl. 394, 781-795 (2012)
Cruz-Uribe, D, SFO, Fiorenza, A, Martell, J, Pérez, C: The boundedness of classical operators on variable \(L^{p}\) spaces. Ann. Acad. Sci. Fenn., Math. 31, 239-264 (2006)
Diening, L, Harjulehto, P, Hästö, P, Ru̇žička, M: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Heidelberg (2011)
Izuki, M: Herz and amalgam spaces with variable exponent, the Haar wavelets and greediness of the wavelet system. East J. Approx. 15, 87-109 (2009)
Kokilashvili, V, Meskhi, A, Sarwar, M: Potential operators in variable exponent Lebesgue spaces: two-weight estimates. J. Inequal. Appl. 2010, Article ID 329571 (2010)
Qu, M, Wang, J: A note on fractional integral operators on Herz spaces with variable exponent. J. Inequal. Appl. 2016, Article ID 2 (2016)
Samko, S: Variable exponent Herz spaces. Mediterr. J. Math. 10, 2007-2025 (2013)
Xu, J: Variable Besov and Triebel-Lizorkin spaces. Ann. Acad. Sci. Fenn., Math. 33, 511-522 (2008)
Zhang, P, Wu, J: Commutators of the fractional maximal function on variable exponent Lebesgue spaces. Czechoslov. Math. J. 64, 183-197 (2014)
Herz, C: Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms. J. Math. Mech. 18, 283-324 (1968)
Baernstein, A, Sawyer, E: Embedding and multiplier theorems for \(H^{p}({\Bbb{R}}^{n})\). Mem. Am. Math. Soc. 53, 318 (1985)
Lu, S, Yang, D, Hu, G: Herz Type Spaces and Their Applications. Science Press, Beijing (2008)
Ragusa, M: Homogeneous Herz spaces and regularity results. Nonlinear Anal. 71, 1909-1914 (2009)
Soria, F, Weiss, G: A remark on singular integrals and power weights. Indiana Univ. Math. J. 43, 187-204 (1994)
Hernández, E, Yang, D: Interpolation of Herz spaces and applications. Math. Nachr. 205, 69-87 (1999)
Li, X, Yang, D: Boundedness of some sublinear operators on Herz spaces. Ill. J. Math. 40, 484-501 (1996)
Lu, S, Yang, D: The decomposition of the weighted Herz spaces and its application. Sci. China Ser. A 38, 147-158 (1995)
Izuki, M: Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization. Anal. Math. 36, 33-50 (2010)
Lu, S, Tang, L, Yang, D: Boundedness of commutators on homogeneous Herz spaces. Sci. China Ser. A 41, 1023-1033 (1998)
Lu, S, Xu, L: Boundedness of rough singular integral operatorson the homogeneous Morrey-Herz spaces. Hokkaido Math. J. 34, 299-314 (2005)
Izuki, M: Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent. Rend. Circ. Mat. Palermo 59, 461-472 (2010)
Lu, Y, Zhu, Y: Boundedness of some sublinear operators and commutators on Morrey-Herz spaces with variable exponents. Czechoslov. Math. J. 64, 969-987 (2014)
Izuki, M: Fractional integrals on Herz-Morrey spaces with variable exponent. Hiroshima Math. J. 40, 343-355 (2010)
Lu, Y, Zhu, Y: Boundedness of multilinear Calderón-Zygmund singular operators on Morrey-Herz spaces with variable exponents. Acta Math. Sin. Engl. Ser. 30, 1180-1194 (2014)
Muckenhoupt, B, Wheeden, R: Weighted norm inequalities for singular and fractional integrals. Trans. Am. Math. Soc. 161, 249-258 (1971)
Nakai, E, Sawano, Y: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262, 3665-3748 (2012)
Wang, H: Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent. Czechoslov. Math. J. 66, 251-269 (2016)
Acknowledgements
The authors would like to thank the referee for his very valuable comments and suggestions. Meng Qu is supported by National Natual Science Foundation of China (11471033) and University NSR Project of Anhui Province (KJ2014A087).
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Wang, L., Qu, M. & Shu, L. Boundedness of rough singular integral operators and commutators on Morrey-Herz spaces with variable exponents. J Inequal Appl 2016, 222 (2016). https://doi.org/10.1186/s13660-016-1161-6
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DOI: https://doi.org/10.1186/s13660-016-1161-6