Abstract
In this paper, we obtain a weighted norm inequality of bilinear Calderón–Zygmund operators in Herz–Morrey spaces with variable exponents and weight in the variable Muckenhoupt class.
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1 Introduction
We denote by \(\mathcal{S}(\mathbb{R}^{n})\) the space of all Schwartz functions on \(\mathbb{R}^{n}\) and by \(\mathcal{S}'(\mathbb{R}^{n})\) the space of all tempered distributions on \(\mathbb{R}^{n}\). Let T be a bilinear operator, which is originally defined on the 2-fold of Schwartz function space \(\mathcal{S}(\mathbb{R}^{n})\), and its value belongs to \(\mathcal{S}'(\mathbb{R}^{n})\):
T is called bilinear Calderón–Zygmund operator, if it extends to a bounded bilinear operator from \(L^{p_{1}} \times L^{p_{2}}\) to \(L^{p}\) with \(1/p_{1}+1/p_{2}=1/p\), and for \(f_{1}\), \(f_{2}\in L^{ \infty }_{C}(\mathbb{R}^{n})\) (the space of compactly supported bounded functions), \(x \notin \mathrm{supp}(f_{1}) \cap \mathrm{supp}(f_{2})\)
where the kernel K is a function in \(\mathbb{R}^{3n}\) off from the diagonal \(x=y_{1}=y_{2}\) and there exist positive constants ε, A such that
and
whenever \(|x-x'| \leq \frac{1}{2}\max \{|x-y_{1}|,|x-y_{2}|\}\), and the two analogous difference estimates with respect to the variables \(y_{1}\) and \(y_{2}\) hold.
Recently, Cruz-Uribe and Guzman proved the boundedness of the bilinear Calderón–Zygmund operator on products of weighted variable Lebesgue spaces in [1]. As a generalization of variable Lebesgue spaces, variable and weighted variable Herz–Morrey (Herz) spaces have been introduced in the last decades; see [2,3,4,5,6,7,8,9,10,11]. Motivated by [1], in this paper, we will prove a weighted norm inequality on products of Herz–Morrey spaces with variable exponents and weight in the variable Muckenhoupt class. We only consider the bilinear Calderón–Zygmund operator for simplicity. The analogs of our result for m-linear Calderón–Zygmund operators also hold for \(m\geq 3\), because our argument and Lemma 8 in Sect. 3 also hold for m-linear Calderón–Zygmund operators with \(m\geq 3\), see Remark 2.7 for [1, Theorem 2.4] in [1]. We mention here that the theory of multilinear Calderón–Zygmund operators started in [12]. After that, the boundedness of multilinear Calderón–Zygmund operators on products of various spaces has been obtained; see [13,14,15,16,17,18,19].
The plan of the paper is as follows. In Sect. 2, we collect some notations and state main result. The proof of the main result will be given in Sect. 3.
2 Notations and main result
In this section, we firstly recall some definitions and notations, then we state our results. Let Ω be a positive measurable subset of \(\mathbb{R}^{n}\), given a measurable function \(p(\cdot ):\varOmega \rightarrow [1,\infty )\), the Lebesgue space with variable exponent \(L^{p(\cdot )}(\varOmega )\) is defined by
The Lebesgue space \(L^{p(\cdot )}(\varOmega )\) becomes a Banach function space equipped with the norm
The space \(L^{p(\cdot )} _{\mathrm{loc}}(\mathbb{R}^{n})\) is defined by \(L^{p(\cdot )} _{\mathrm{loc}}(\mathbb{R}^{n}):=\{f: f\chi _{K} \in L ^{p(\cdot )}(\mathbb{R}^{n})\text{ for all compact subsets} K \subset \mathbb{R}^{n}\}\), where and what follows, \(\chi _{S}\) denotes the characteristic function of a measurable set \(S\subset \mathbb{R}^{n}\). Let \(p(\cdot ):\mathbb{R}^{n}\rightarrow (0,\infty )\), we denote \({p_{-} }: = \mathrm{ess}\inf_{x \in {\mathbb{R}^{n}}} p(x)\), \({p_{+} }: = \mathrm{ess} \sup_{x \in {\mathbb{R}^{n}}} p(x)\). The set \(\mathcal{P}(\mathbb{R}^{n})\) consists of all \(p(\cdot )\) satisfying \(p_{-}>1\) and \(p_{+}<\infty \); \(\mathcal{P}_{0}(\mathbb{R}^{n})\) consists of all \(p(\cdot )\) satisfying \(p_{-}>0\) and \(p_{+}<\infty \). \(L^{p(\cdot )}\) can be similarly defined as above for \(p(\cdot ) \in \mathcal{P}_{0}(\mathbb{R}^{n})\). \(p'(\cdot )\) means that the conjugate exponent of \(p(\cdot ) \), that means \(1/p(\cdot )+1/p'( \cdot )=1\).
Let \(p(\cdot ) \in \mathcal{P}(\mathbb{R}^{n})\) and w be a weight which is a non-negative measurable function on \(\mathbb{R}^{n}\). Then the weighted variable exponent Lebesgue space \(L^{p(\cdot )}( w )\) is the set of all complex-valued measurable function f such that \(fw \in L^{p(\cdot )}\). The space \(L^{p(\cdot )}(w )\) is a Banach space equipped with the norm
Let \(f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\). Then the standard Hardy–Littlewood maximal function of f is defined by
where the supremum is taken over all balls containing x in \(\mathbb{R}^{n}\). In general, the Hardy–Littlewood maximal operator is not bounded on weighted variable Lebesgue spaces. But if \(p(\cdot ) \in \mathcal{P}(\mathbb{R}^{n})\) and satisfies the following global log-Hölder continuous and \(w \in A_{p(\cdot )}\), then M is bounded on \(L^{p(\cdot )}(w)\).
Definition 1
Let \(\alpha (\cdot )\) be a real-valued measurable function on \(\mathbb{R}^{n}\).
-
(i)
The function \(\alpha (\cdot )\) is locally log-Hölder continuous if there exists a constant \(C_{1}\) such that
$$ \bigl\vert \alpha (x) - \alpha (y) \bigr\vert \leq \frac{C_{1}}{ {\log ( {e + 1/ \vert {x - y} \vert } )}},\quad x,y \in {\mathbb{R}^{n}}, \vert x - y \vert < \frac{1}{2}. $$ -
(ii)
The function \(\alpha (\cdot )\) is log-Hölder continuous at the origin if there exists a constant \(C_{2}\) such that
$$ \bigl\vert \alpha (x) - \alpha (0) \bigr\vert \leq \frac{C_{2}}{ {\log ( {e + 1/ \vert x \vert } )}}, \quad\forall x \in {\mathbb{R}^{n}}. $$Denote by \(\mathcal{P}^{\log }_{0}(\mathbb{R}^{n})\) the set of all log-Hölder continuous functions at the origin.
-
(iii)
The function \(\alpha (\cdot )\) is log-Hölder continuous at infinity if there exist \(\alpha _{\infty }\in \mathbb{R}\) and a constant \(C_{3}\) such that
$$ \bigl\vert \alpha (x) - \alpha _{\infty } \bigr\vert \leq \frac{C _{3}}{{\log ( {e + \vert x \vert } )}},\quad \forall x \in {\mathbb{R}^{n}}. $$Denote by \(\mathcal{P}^{\log }_{\infty }(\mathbb{R}^{n})\) the set of all log-Hölder continuous functions at infinity.
-
(iv)
The function \(\alpha (\cdot )\) is global log-Hölder continuous if \(\alpha (\cdot )\) are both locally log-Hölder continuous and log-Hölder continuous at infinity. Denote by \(\mathcal{P}^{\log }(\mathbb{R}^{n})\) the set of all global log-Hölder continuous functions.
Definition 2
Let \(p(\cdot ) \in \mathcal{P}(\mathbb{R}^{n})\), a positive measurable function w is said to be in \(A_{p(\cdot )}\), if exists a positive constant C for all balls B in \(\mathbb{R}^{n}\) such that
Remark 1
In [20], Cruz-Uribe, Fiorenza and Neugebauer found that if \(p(\cdot ) \in \mathcal{P}(\mathbb{R}^{n})\) and \(w \in A_{p(\cdot )}\), then \(w ^{-1} \in A_{p'(\cdot )}\).
The Muckenhoupt \(A_{p}\) class with constant exponent \(p \in (1,\infty )\) firstly proposed by Muckenhoupt in [21]. The variable Muckenhoupt \(A_{p(\cdot )}\) was considered in [20, 22,23,24,25].
Lemma 1
(see [20, Theorem 1.5])
If \(p(\cdot )\in \mathcal{P}^{\log }(\mathbb{R}^{n})\cap \mathcal{P}( \mathbb{R}^{n})\) and \(w \in A_{p(\cdot )}\), then there is a positive constant C such that, for each \(f\in L^{p(\cdot )}(w )\),
To give the definitions of the Herz space and the Herz–Morrey space with variable exponents, we use the following notations. For each \(k \in \mathbb{Z}\) we define
Definition 3
Let \(q \in (0, \infty ]\), \(p(\cdot )\in \mathcal{P}_{0} (\mathbb{R} ^{n})\), and \(\alpha (\cdot ):\mathbb{R}^{n}\rightarrow \mathbb{R}\) with \(\alpha \in L^{\infty }(\mathbb{R}^{n}) \).
-
(1)
The homogeneous weighted Herz space \(\dot{K}_{p( \cdot )}^{\alpha ( \cdot ),q}(w )\) is defined by
$$ \dot{K}_{p( \cdot )}^{\alpha ( \cdot ),q}(w ):= \bigl\{ {f \in L_{\mathrm{loc}}^{p( \cdot )}\bigl({\mathbb{R}^{n}}\backslash \{ 0 \}, w \bigr):{{ \Vert f \Vert }_{\dot{K}_{p( \cdot )}^{\alpha ( \cdot ),q}(w )}} < \infty } \bigr\} , $$where
$$ \Vert f \Vert _{\dot{K}_{p( \cdot )}^{\alpha ( \cdot ),q}(w )}:= { \Biggl\{ {\sum _{k = - \infty }^{\infty }{ \bigl\Vert {{2^{k \alpha ( \cdot )}}f{\chi _{k}}} \bigr\Vert _{L^{p( \cdot )}(w )} ^{q}} } \Biggr\} ^{1/q}}. $$ -
(2)
The inhomogeneous weighted Herz space \(K_{p( \cdot )}^{\alpha ( \cdot ),q}(w )\) is defined by
$$ K_{p( \cdot )}^{\alpha ( \cdot ),q}(w ):= \bigl\{ {f \in L_{ \mathrm{loc}}^{p( \cdot )}( w ):{{ \Vert f \Vert }_{K_{p( \cdot )}^{\alpha ( \cdot ),q}(w )}} < \infty } \bigr\} , $$where
$$ { \Vert f \Vert _{K_{p( \cdot )}^{\alpha ( \cdot ),q}(w )}} := { \Biggl\{ {\sum _{m = 0}^{\infty }{ \bigl\Vert {{2^{m\alpha ( \cdot )}}f{ \widetilde{\chi } _{m}}} \bigr\Vert _{L^{p( \cdot )}(w )}^{q}} } \Biggr\} ^{1/q}}. $$
Remark 2
If \(0 < q_{1} \leq q_{2} \leq \infty \) and \(w \equiv 1\), then \(\dot{K}^{\alpha (\cdot ),q_{1}}_{p(\cdot )}(\mathbb{R}^{n}) \subset \dot{K}^{\alpha (\cdot ),q_{2}}_{p(\cdot )}(\mathbb{R}^{n})\). If \(w \equiv 1\), \(\alpha (\cdot )\) and \(p(\cdot )\) are constants, then \(\dot{K}^{\alpha (\cdot ),q}_{p(\cdot )}(\mathbb{R}^{n})= \dot{K}^{ \alpha,q}_{p}(\mathbb{R}^{n})\) is the classical Herz spaces in [26, 27].
To generalize the above spaces to variable exponent \(q(\cdot )\), we need the notation of the variable mixed sequence space \(\ell ^{q(\cdot )}(L ^{p(\cdot )})\), which is firstly defined by Almeida and Hästö in [28]. Let w be a non-negative measurable function. Given a sequence of functions \(\{f_{j}\}_{j\in \mathbb{Z}}\), define the modular
where \(\lambda ^{1/\infty }=1\). If \(q^{+}<\infty \) or \(q(\cdot )\le p( \cdot )\), the above can be written as
The norm is
Now, spaces \(\dot{K}^{\alpha (\cdot ),q(\cdot )}_{p(\cdot )}(w )\) and \(K^{\alpha (\cdot ),q(\cdot )}_{p(\cdot )}(w )\) are defined, respectively, by
where
and
For any quantities A and B, we shall write \(A \lesssim B\) to indicate that there exists a constant \(C>0\) such that \(A\leq CB\). If \(A\lesssim B\) and \(B \lesssim A\), we write \(A\approx B\).
The following lemma is a corollary of [29, Theorem 3].
Lemma 2
Let \(\alpha (\cdot )\in L^{\infty }(\mathbb{R}^{n})\), \(p(\cdot )\), \(q(\cdot )\in \mathcal{P}_{0}(\mathbb{R}^{n})\) and w be a weight. If \(\alpha (\cdot )\) and \(q(\cdot )\) are log-Hölder continuous at infinity, then
Additionally, if \(\alpha (\cdot )\) and \(q(\cdot )\) are log-Hölder continuous at the origin, then
Definition 4
Let \(p(\cdot )\), \(q(\cdot )\in \mathcal{P}_{0}(\mathbb{R}^{n})\), \(\lambda \in [0, \infty )\). Let \(\alpha (\cdot )\) be a bounded real-valued measurable function on \(\mathbb{R}^{n}\). The homogeneous weighted Herz–Morrey space \(M\dot{K}^{\alpha (\cdot ),q(\cdot )}_{p( \cdot ),\lambda }(w )\) and non-homogeneous weighted Herz–Morrey space \(MK^{\alpha (\cdot ),q(\cdot )}_{p(\cdot ),\lambda }(w )\) are defined, respectively, by
and
where
and
Proposition 1
Let \(p(\cdot )\), \(q(\cdot )\in \mathcal{P}_{0}(\mathbb{R}^{n})\), w be a weight, \(\lambda \in [0,\infty )\), and \(\alpha (\cdot )\in L^{\infty }(\mathbb{R}^{n})\).
-
(i)
If \(\alpha (\cdot )\), \(q(\cdot )\in \mathcal{P}^{\log } _{0}(\mathbb{R}^{n})\cap \mathcal{P}^{\log }_{\infty }(\mathbb{R}^{n})\), then, for any \(f\in L^{p(\cdot )}_{\mathrm{loc}}(\mathbb{R}^{n}\backslash \{0\},w )\),
$$\begin{aligned} & \Vert f \Vert _{M\dot{K}^{\alpha (\cdot ),q(\cdot )}_{p(\cdot ),\lambda }( w )}\\ &\quad\approx\max \Bigl\{ \sup _{L\leqslant 0,L\in \mathbb{Z}} 2^{-L \lambda } \bigl\Vert \bigl(2^{k\alpha (0)}f \chi _{k}\bigr)_{k \leq L} \bigr\Vert _{\ell ^{q_{0}}({L^{p(\cdot )}( w )})}, \\ & \qquad\sup_{L>0,L\in \mathbb{Z}} \bigl[2^{-L\lambda } \bigl\Vert \bigl(2^{k\alpha (0)}f\chi _{k}\bigr)_{k< 0} \bigr\Vert _{\ell ^{q_{0}}({L^{p(\cdot )}(w )})}+2^{-L\lambda } \bigl\Vert \bigl(2^{k\alpha _{\infty }}f\chi _{k}\bigr)_{k= 0}^{L} \bigr\Vert _{\ell ^{q_{ \infty }}({L^{p(\cdot )}(w )})} \bigr] \Bigr\} , \end{aligned}$$where throughout \(q_{0}:=q(0)\).
-
(ii)
If \(\alpha (\cdot )\), \(q(\cdot )\in \mathcal{P}^{\log } _{\infty }(\mathbb{R}^{n})\), then
$$ MK^{\alpha (\cdot ),q(\cdot )}_{p(\cdot ),\lambda }(w )=MK^{ \alpha _{\infty },q_{\infty }}_{p(\cdot ),\lambda }( w ). $$
Proof
Obviously,
When \(L \leq 0\), from Lemma 2 we know that
When \(L>0\), from Lemma 2 again we also obtain
Thus we obtain (i). Similarly, we obtain (ii). □
Lemmas 3 and 4 below have been proved by Izuki and Noi in [30, 31].
Lemma 3
If \(p(\cdot )\in \mathcal{P}^{\log }(\mathbb{R}^{n}) \cap \mathcal{P}( \mathbb{R}^{n})\) and \(w \in A_{p(\cdot )}\), then there exists a constant \(C>0\) such that, for all balls B in \(\mathbb{R}^{n}\) and all measurable subsets \(S\subset B\),
Lemma 4
If \(p(\cdot )\in \mathcal{P}^{\log }(\mathbb{R}^{n})\cap \mathcal{P}( \mathbb{R}^{n})\) and \(w \in A_{p(\cdot )}\), then there exist constants \(\delta _{1}\), \(\delta _{2}\in (0,1)\) and \(C>0\) such that, for all balls B in \(\mathbb{R}^{n}\) and all measurable subsets \(S\subset B\),
Lemma 5
(see [30, Lemma 4])
If \(p(\cdot )\in \mathcal{P}^{\log }(\mathbb{R}^{n})\cap \mathcal{P}( \mathbb{R}^{n})\) and \(w \in A_{p(\cdot )}\), then there exists a positive constant C such that, for all balls B in \(\mathbb{R}^{n}\),
Our main result is as follows.
Theorem 1
Assume that T is a bilinear Calderón–Zygmund operator, \(p_{1}(\cdot )\), \(p_{2}(\cdot )\in \mathcal{P}^{\log }(\mathbb{R}^{n}) \cap \mathcal{P}(\mathbb{R}^{n})\) satisfying \(1/p(x)=1/{p_{1}(x)}+1/ {p_{2}(x)}\) for \(x \in \mathbb{R}^{n}\). Let \(w_{1}\), \(w_{2}\) be weights, \(w=w_{1} w_{2}\), \(w_{i} \in A_{p_{i}(\cdot )}\), \(i=1, 2\). Suppose that \(\alpha (\cdot ) \in L^{\infty }(\mathbb{R}^{n})\cap \mathcal{P}^{ \log }_{0}(\mathbb{R}^{n})\cap \mathcal{P}^{\log }_{\infty }( \mathbb{R}^{n})\), \(\alpha (0)=\alpha _{1}(0)+\alpha _{2}(0)\), \(\alpha _{\infty }=\alpha _{1\infty }+\alpha _{2\infty }\), \(q(\cdot ) \in \mathcal{P}^{\log }_{0}(\mathbb{R}^{n})\cap \mathcal{P}^{\log } _{\infty }(\mathbb{R}^{n})\), \(1/q(0)=1/{q_{1}(0)}+1/{q_{2}(0)}\), \(1/q_{\infty }=1/{q_{1\infty }}+1/{q_{2\infty }}\), \(\lambda =\lambda _{1}+\lambda _{2}\), \(0 \leq \lambda _{i}< \infty \), \(\delta _{i1}\), \(\delta _{i2}\in (0,1)\) are the constants in Lemma 4 for exponents \(p_{i}(\cdot )\) and weights \(w_{i}\), \(i=1, 2\). If \(\lambda _{i}+n \delta _{i2}>\alpha _{i\infty }\geq \alpha _{i}(0)\) for \(i=1, 2\), then
From Theorem 1, we obtain the following corollary.
Corollary 1
Assume that T is a bilinear Calderón–Zygmund operator, \(p_{1}(\cdot )\), \(p_{2}(\cdot )\in \mathcal{P}^{\log }(\mathbb{R}^{n}) \cap \mathcal{P}(\mathbb{R}^{n})\) satisfying \(1/p(x)=1/{p_{1}(x)}+1/ {p_{2}(x)}\) for \(x \in \mathbb{R}^{n}\). Let \(w_{1}\), \(w_{2}\) be weights, \(w=w_{1} w_{2}\), \(w_{i} \in A_{p_{i}(\cdot )}\), \(i=1, 2\). Suppose that \(\alpha (\cdot ) \in L^{\infty }(\mathbb{R}^{n})\cap \mathcal{P}^{ \log }_{0}(\mathbb{R}^{n})\cap \mathcal{P}^{\log }_{\infty }( \mathbb{R}^{n})\), \(\alpha (0)=\alpha _{1}(0)+\alpha _{2}(0)\), \(\alpha _{\infty }=\alpha _{1\infty }+\alpha _{2\infty }\), \(q(\cdot ) \in \mathcal{P}^{\log }_{0}(\mathbb{R}^{n})\cap \mathcal{P}^{\log } _{\infty }(\mathbb{R}^{n})\), \(1/q(0)=1/{q_{1}(0)}+1/{q_{2}(0)}\), \(1/q_{\infty }=1/{q_{1\infty }}+1/{q_{2\infty }}\), \(\delta _{i1}\), \(\delta _{i2}\in (0,1)\) are the constants in Lemma 4 for exponents \(p_{i}(\cdot )\) and weights \(w_{i}\), \(i=1, 2\). If \(\lambda _{i}+n \delta _{i2}>\alpha _{i\infty }\geq \alpha _{i}(0)\) for \(i=1, 2\), then
3 Proof of Theorem 1
To prove Theorem 1, we need a series of lemmas.
Lemma 6
(see [16, Theorem 2.3])
Let \(p(\cdot )\), \(p_{1}(\cdot )\), \(p_{2}(\cdot )\in \mathcal{P}_{0}( \mathbb{R}^{n})\) such that \(1/p(x)=1/p_{1}(x)+1/p_{2}(x)\) for \(x \in \mathbb{R}^{n}\). Then there exists a constant \(C_{p,p_{1}}\) independent of functions f and g such that
holds for every \(f\in L^{p_{1}(\cdot )}(\mathbb{R}^{n})\) and \(g\in L^{p_{2}(\cdot )}(\mathbb{R}^{n})\). If \(p \in \mathcal{P}( \mathbb{R}^{n})\), w be weight with \(w =w _{1} \times w _{2}\), then
Lemma 7
(see [32, Proposition 1.2])
Let \(0< p\leq \infty \), \(\delta >0\). Then there is a positive constant C such that
for non-negative sequences \(\{a_{j}\}^{\infty }_{j= -\infty }\). Here, when \(p=\infty \), it is understood that (4) stands for
The following lemma is a corollary of [1, Theorem 2.8].
Lemma 8
Let \(p_{1}(\cdot )\), \(p_{1}(\cdot ) \in \mathcal{P}(\mathbb{R}^{n})\), \(1<(p_{i})_{-}\leq (p_{i})_{+}<\infty \) and \(p_{i}(\cdot )\in \mathcal{P}^{\log }(\mathbb{R}^{n})\cap \mathcal{P}(\mathbb{R}^{n})\) satisfying \(1/p(x)=1/{p_{1}(x)}+1/{p_{2}(x)}\) for \(x \in \mathbb{R} ^{n}\), \(i=1, 2\). Let \(w_{1} \in A_{p_{1}(\cdot )}, w_{2} \in A_{p _{2}(\cdot )}\) and \(w=w_{1}w_{2}\). If T is a bilinear Calderón–Zygmund operator, then
Proof of Theorem 1
Let \(f_{1}\) and \(f_{2}\) be bounded functions with compact support and write
By Proposition 1, we have
where
Since to estimate F is essentially similar to estimate E, so it suffices for us to show that E and G are bounded in weighted Herz–Morrey space with variable exponents. It is easy to see that
where
We shall use the following estimates. If \(l\leq k-1\), then, by Hölder’s inequality and Lemmas 4 and 5, we have
If \(l=k\), then
If \(l\geq k+1\), then
By the symmetry of \(f_{1}\) and \(f_{2}\), it is only necessary to estimate \(E_{1}\), \(E_{2}\), \(E_{3}\), \(E_{5}\), \(E_{6}\), and \(E_{9}\).
To estimate \(E_{1}\), since l, \(j \leq k-2\), we deduce that, for \(i=1,2\),
Therefore, for \(x\in D_{k}\), we have
Thus, \(\forall x \in D_{k}\) and l, \(j \leq k-2\), we have
Therefore, by Hölder’s inequality, we obtain
Since \(1/q(0)=1/{q_{1}(0)}+1/{q_{2}(0)}\), \(\lambda =\lambda _{1}+ \lambda _{2}\), by Hölder’s inequality, we have
where
Since \(n\delta _{2i}-\alpha _{i}(0)>0\), by (5) and Lemma 7 we obtain
where we wrote \(2^{-|k-l|(n\delta _{2i}-\alpha _{i}(0))}\lesssim 2^{-|k-l| \varepsilon _{i}}\) for some \(\varepsilon _{i} \in (0,n\delta _{2i}-\alpha _{i}(0))\). Thus, we obtain
To estimate \(E_{2}\), since \(l \leq k-2\), \(k-1 \leq j \leq k+1\) for \(i=1, 2\), we have
Therefore, by Hölder’s inequality, we obtain
Since \(1/q(0)=1/{q_{1}(0)}+1/{q_{2}(0)}\), \(\lambda =\lambda _{1}+ \lambda _{2}\), by Hölder’s inequality, we have
It is obvious that
Now we estimate \(E_{2,2}\). Taking (5), (6) and (7) together, we have
where we used \(2^{-n\delta _{22}} <1\) and \(2^{(j-k)n(1-\delta _{12})} < 2^{(j-k)n} \), \(j\in \{ k-1, k, k+1\}\) for (5) and (7), respectively. Thus, we obtain
To estimate \(E_{3}\), since \(l \leq k-2\), \(j \geq k+2\), then we have
Therefore, \(\forall x \in D_{k}\), \(l \leq k-2\), \(j \geq k+2\), we get
Thus, by Hölder’s inequality, we have
Since \(1/q(0)=1/{q_{1}(0)}+1/{q_{2}(0)}\), \(\lambda =\lambda _{1}+ \lambda _{2}\), by Hölder’s inequality, we have
It is obvious that
Since \(n\delta _{12}+\alpha _{2}(0)>0\), by (7) and Lemma 7 we obtain
where we wrote \(2^{-|k-j|(n\delta _{12}+\alpha _{2}(0))}\lesssim 2^{-|k-j| \eta _{2}}\) for some \(\eta _{2} \in (0,n\delta _{12}+\alpha _{2}(0))\). Thus, we have
To estimate \(E_{5}\), using Hölder’s inequality and Lemma 8, we have
To estimate \(E_{6}\), since \(k-1 \leq l \leq k+1\) and \(j \geq k+2\), we obtain
Thus, \(\forall x \in D_{k}\), \(k-1 \leq l \leq k+1\) and \(j \geq k+2\), we obtain
Therefore, by Hölder’s inequality, we obtain
Since \(1/q(0)=1/{q_{1}(0)}+1/{q_{2}(0)}\), \(\lambda =\lambda _{1}+ \lambda _{2}\), by Hölder’s inequality, we have
By the symmetry of \(f_{1}\) and \(f_{2}\), we can know that the estimate \(E_{6,1}\) is similar to the estimated \(E_{2,2}\) and \(E_{6,2} =E_{3,2}\).
To estimate \(E_{9}\), since l, \(j \geq k+2\), for \(i=1,2\), we get
Therefore, \(\forall x \in D_{k}\), l, \(j \geq k+2\), we have
Thus, by Hölder’s inequality, we have
Since \(1/q(0)=1/{q_{1}(0)}+1/{q_{2}(0)}\), \(\lambda =\lambda _{1}+ \lambda _{2}\), by Hölder’s inequality, we have
Obviously, the estimate \(E_{9,i}\) is similar to the estimated \(E_{3,2}\) for \(i=1,2\).
Taking all estimates for \(E_{i}\) together, \(i=1,2,\ldots,9\), we obtain
To go on, we need some further preparation.
If \(l<0\), by Proposition 1, we have
If \(l\geq 0\), we have
Finally, we estimate G. By the symmetry of \(f_{1}\) and \(f_{2}\), it is only necessary to estimate \(G_{1}\), \(G_{2}\), \(G_{3}\), \(G_{5}\), \(G_{6}\), and \(G_{9}\).
To estimate \(G_{1}\), since l, \(j \leq k-2\), \(1/q_{\infty }=1/{q_{1 \infty }}+1/{q_{2\infty }}\), \(\lambda =\lambda _{1}+\lambda _{2}\), by (8) and Hölder’s inequality, we have
where
Since \(\lambda _{i}+n\delta _{2i}>\alpha _{i\infty } \geq \alpha _{i}(0)\), by (5), (13), (14) and Lemma 7 we obtain
Thus, we get
To estimate \(G_{2}\), since \(l \leq k-2\), \(k-1 \leq j \leq k+1\), \(1/q_{\infty }=1/{q_{1\infty }}+1/{q_{2\infty }}\), \(\lambda =\lambda _{1}+\lambda _{2}\), by (9) and Hölder’s inequality, we have
It is obvious that
Now we estimate \(G_{2,2}\). Combining (5), (6) and (7), we have
where we used \(2^{-n\delta _{22}} <1\) and \(2^{(j-k)n(1-\delta _{12})} < 2^{(j-k)n}\) for (5) and (7), respectively. Thus, we obtain
To estimate \(G_{3}\), since \(l \leq k-2\), \(j \geq k+2\), \(1/q_{\infty }=1/ {q_{1\infty }}+1/{q_{2\infty }}\), \(\lambda =\lambda _{1}+\lambda _{2}\), by (10) and Hölder’s inequality, we have
It is obvious that
Since \(n\delta _{12}+\alpha _{2\infty }>0\), by (7) and Lemma 7 we obtain
where we wrote \(2^{-|k-j|(n\delta _{12}+\alpha _{2\infty })}\lesssim 2^{-|k-j| \vartheta _{2}}\) for some \(\vartheta _{2} \in (0,n\delta _{12}+ \alpha _{2\infty })\). Thus, we get
To estimate \(G_{5}\), using Hölder’s inequality and Lemma 8
To estimate \(G_{6}\), since \(k-1 \leq l \leq k+1\) and \(j \geq k+2\), \(1/q_{\infty }=1/{q_{1\infty }}+1/{q_{2\infty }}\), \(\lambda =\lambda _{1}+\lambda _{2}\), by (11) and Hölder’s inequality, we have
By the symmetry of \(f_{1}\) and \(f_{2}\), we can know that the estimate \(G_{6,1}\) is similar to the estimated \(G_{2,2}\) and \(G_{6,2} =G_{3,2}\).
To estimate \(G_{9}\), since l, \(j \geq k+2\), \(1/q_{\infty }=1/{q_{1 \infty }}+1/{q_{2\infty }}\), \(\lambda =\lambda _{1}+\lambda _{2}\), by (12) and Hölder’s inequality, we have
Obviously, the estimate \(G_{9,i}\) is similar to the estimated \(G_{3,2}\) for \(i=1,2\).
Taking all estimates for \(G_{i}\) together, \(i=1,2,\ldots,9\), we obtain
This completes the proof. □
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The authors would like to thank the referees for their careful reading and suggestions.
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The work is partially supported by Hainan Province Natural Science Foundation of China (2018CXTD338) and the National Natural Science Foundation of China (Grant No. 11761026 and 11761027).
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Wang, S., Xu, J. Weighted norm inequality for bilinear Calderón–Zygmund operators on Herz–Morrey spaces with variable exponents. J Inequal Appl 2019, 251 (2019). https://doi.org/10.1186/s13660-019-2202-8
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DOI: https://doi.org/10.1186/s13660-019-2202-8