Boundedness of multilinear Calderón-Zygmund singular operators on weighted Lebesgue spaces and Morrey-Herz spaces with variable exponents

1.   Introduction

In last decades there have been many works on the boundedness for the multilinear singular integral operators in the product of Lebesgue spaces, Morrey type spaces and Herz spaces etc. Let $T$ be a multilinear singular integral operator which is initially defined on the $m$-fold product of the Schwartz space $\mathcal S(\mathbb R^n)$ with values in the space of tempered distributions $\mathcal S'(\mathbb R^n)$ such that for $x\notin \cap_{i = 1}^m\text{supp}f_i,$

where $f_1, \cdots, f_m$ are in $L_c^{\infty}(\mathbb R^n),$ the space of compactly supported bounded functions. The kernel $K$ is a function in $(\mathbb R^n)^{m+1}$ away from the diagonal $y_0 = y_1 = \cdots = y_m$ and satisfies the standard estimates

and for some $\varepsilon > 0,$

provided that $|x-x'|\le \frac 12 \max\{|x-y_1|, \cdots, |x-y_m|\}$ and

provided that $|y_i-y'_i|\le \frac 12 \max\{|x-y_1|, \cdots, |x-y_m|\}$ for all $1\le i\le m.$

Such kernels are called Calderón-Zygmund kernel and the collection of such functions is denoted by $m-CZK(C, \varepsilon)$ in ^[1]. Let $T$ be as in (1.1) with an $m-CZK(C, \varepsilon)$ kernel. If $T$ is bounded from $L~~^{p_1}\times \cdots \times L~~^{p_m}$ to $L^p$ for some $1 < p_1, \cdots, p_m < \infty$ and $\frac 1p = \frac 1{p_1}+\cdots +\frac 1{p_m},$ then we say that $T$ is an $m-$linear Calderón-Zygmund operator. If $T$ is an $m-$linear Calderón-Zygmund operator, Grafakos and Torres in ^[1] proved its boundedness from $L^{q_1}\times \cdots \times L^{q_m}$ to $L^q$ for all $1 < q_1, \cdots, q_m < \infty$ and $\frac 1q = \frac 1{q_1}+\cdots +\frac 1{q_m}$ and from $L^1\times \cdots \times L^1$ to $L^{1/m, \infty}$. Curbera et al. in ^[2] obtained its weighted inequalities. P$\acute{e}$rez and Torres in ^[3] presented a new proof of a weighted norm inequality for multilinear singular integral of Calderón-Zygmund type and studied an application of certain multilinear commutators. Recently, Tao and Zhang in ^[4] obtained the boundedness of the multilinear singular integral operators on weighted Morrey-Herz spaces and gave their weak estimates on endpoints.

Function spaces with variable exponents have been intensively studied in the recent years by a significant number of authors, e.g. ^{[5,6,7,8,9,10,11]}. The motivation for the increasing interest in such spaces comes not only from theoretical purpose, but also from applications to fluid dynamics, image restoration and PDE with non-standard growth conditions. The results above had been extended to the variable exponent Lebesgue spaces $L~~^{p(\cdot)}(\Omega)$ and the variable exponent Morrey spaces $M_{q(\cdot)}^{p(\cdot)}(\Omega)$ over an open set $\Omega \subseteq \mathbb{R}^{n}$ in ^[12,13,14]. Lu and Zhu in ^[15] discussed the boundedness of multilinear Calderón-Zygmund singular operators on Morrey-Herz spaces $M\dot K^{\alpha(\cdot), \lambda}_{q, p(\cdot)}(\mathbb R^n)$ and Herz spaces $\dot{K}^{\alpha{(\cdot)}}_{q, p(\cdot)}(\mathbb{R}^{n})$ with two variable exponents $\alpha(\cdot)$ and $p(\cdot)$.

Recently the generalized Muckenhoupt weights with variable exponents have been studied. Cruz-Uribe and Wang in ^[6] obtained the boundedness of fractional integrals on weighted Lebesgue spaces with variable exponents by applying the extrapolation. Izuki and Noi in ^[7] proved the the boundedness of fractional integrals on weighted Herz spaces with variable exponents by the theory of Banach functions spaces and Muckenhoupt theory with variable exponents. We refer readers ^{[16,17,18,19,20]} for more references about weights in variable exponent spaces.

Motivated by the results above, we give the definition of weighted Morrey-Herz spaces with variable exponents and prove the boundedness of multilinear Calderón-Zygmund singular operators on the product of weighted Lebesgue spaces and weighted Morrey-Herz spaces with variable exponents.

2.   Preliminaries

2.1.   Variable Lebesgue spaces

Definition 1. Let $p(\cdot):\, \mathbb{R}^{n}\rightarrow [1, \infty)$ be a measurable function. The Lebesgue space with variable exponent $L~~^{p(\cdot)}(\mathbb{R}^{n})$ is defined by

where $\rho_{p}(f) = \int_{\mathbb{R}^{n}}|f(x)|^{p(x)}dx$.

$L~~^{p(\cdot)}(\mathbb{R}^{n})$ is a Banach space with the norm defined by

We denote

The set $\mathcal P(\mathbb{R}^{n})$ consists of all $p(\cdot)$ satisfying $p_{-} > 1$ and $p_{+} < \infty$. $p'(\cdot)$ means the conjugate exponent of $p(\cdot)$, namely $\frac{1}{p(x)}+\frac{1}{p'(x)} = 1$ holds. We also note that generalized Hölder's inequality

is true for all $f\in L~~^{p(\cdot)}(\mathbb{R}^{n})$ and $g\in L~~^{p'(\cdot)}(\mathbb{R}^{n})$, where $r_{p}: = 1+\frac{1}{p_{-}}-\frac{1}{p_{+}}$ (^[5]).

We say that a function $p: \mathbb{R}^{n}\rightarrow \mathbb{R}$ is locally log-Hölder continuous, if it satisfies

If

we say $p$ satisfies the log-Hölder decay condition at infinity. The set of $p(\cdot)$ satisfying (2.1) and (2.2) is denoted by $LH(\mathbb{R}^{n})$. It is also well known that the Hardy-Littlewood maximal operator $M$ defined by

is bounded on $L~~^{p(\cdot)}(\mathbb{R}^{n})$ whenever $p(\cdot)\in \mathcal P(\mathbb{R}^{n})\cap LH(\mathbb{R}^{n})$.

2.2.   The Muckenhoupt weights with variable exponents

Let $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$ and $w$ be a weight. The weighted variable exponent Lebesgue space $L~~^{p(\cdot)}(w)$ is the set of all complex-valued measurable functions $f$ such that $fw^{\frac{1}{p(\cdot)}}\in L~~^{p(\cdot)}(\mathbb{R}^{n})$. The space $L~~^{p(\cdot)}(w)$ is a Banach space equipped with the norm

Now we define the Muckenhoupt classes. We begin with the classical Muckenhoupt $A_{1}$ weights.

Definition 2. A weight is said to be a Muckenhoupt $A_{1}$ weight if $Mw(x)\leq Cw(x)$ holds for almost every $x\in \mathbb{R}^{n}$. The set $A_{1}$ consists of all Muckenhoupt $A_{1}$ weights.

The original Muckenhoupt $A_{p}$ class with constant exponent $p\in (1, \infty)$ established by Muckenhoupt ^[21] can be generalized in terms of a variable exponent as follows.

Definition 3. (^[7,20]) Suppose $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$. A weight $w$ is said to be an $A_{p(\cdot)}$ weight if

Next we state the monotone property of the class $A_{p(\cdot)}$.

Lemma 1. (^[7] Corollary 1) If $p(\cdot), \; q(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\cap LH(\mathbb{R}^{n})$ and $p(\cdot)\leq q(\cdot)$, then we have

Definition 4. (^[7]) Given $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$ and a weight $w$, we say $(p(\cdot), w)$ is an $M$-pair if the maximal operator $M$ is bounded on both $L~~^{p(\cdot)}(w~^{p(\cdot)})$ and $L~~^{p'(\cdot)}(w^{-p'(\cdot)}).$

Lemma 2. If $(p(\cdot), w)$ is an $M$-pair, then we have that for all balls $B$ in $\mathbb{R}^{n}$,

We introduce the sharp maximal function

where the supremum is taken over all cubes $Q$ containing $x$, as usual, $f_{Q}$ denotes the average of over $Q$. For $\delta > 0$, we denote by $M_{\delta}^{\sharp}$ the operator

Similarly as above, for $\delta > 0$, we denote by $M_{\delta}$ the operator $M_{\delta}(f) = M(|f|^{\delta})^{1/\delta}$.

The next lemma gives an estimate for the norm $\|\cdot\|_{L~~^{p(\cdot)}(w~^{p(\cdot)})}$ in terms of $M^{\sharp}$.

Lemma 3. If $(p(\cdot), w)$ is an $M$-pair, then we have that for all $f\in L~~^{p(\cdot)}(w~^{p(\cdot)})$,

The two Lemmas above are the generalization of Lemmas 2 and 4 in ^[22] to the weighted case, which can be found in ^[16] and ^[6] respectively.

Lemma 4. Suppose $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\cap LH(\mathbb{R}^{n})$ and $w\in A_{p(\cdot)}$, then we can take constants $\delta_{1}, \delta_2 \in (0, 1)$ such that

for all balls $B$ and all measurable sets $E\subset B$.

Equations (2.3) and (2.4) are from ^[7]. (2.5) is the generalization of inequality (6) of Lemma 1 in ^[23] to the weighted case. Their proofs are similar, we omit it here.

Lemma 5. (^[3]) Let $T$ be an $m$-linear Calderón-Zygmund operator and let $0 < \delta < 1/m$. Then, there exists a constant $C > 0$ such that for any vector function $\vec{f} = (f_{1}, \ldots, f_{m})$, where each $f_{j}$ is a smooth function and with compact support, the following inequality holds

2.3.   Weighted Morrey-Herz spaces with variable exponents

Let $E\subset\mathbb{R}^{n}$ be a measurable set and $w$ a positive and locally integrable function on $E$. The set $L~~^{p(\cdot)}_{\text{loc}}(E, w)$ consists of all functions f satisfying the following condition: For all compact sets $K\subset E$, there exists a constant $\lambda > 0$ such that

Next we define weighted Morrey-Herz spaces with variable exponents motivated by ^[7,24]. We use the following notations. For each $k\in \mathbb Z$, we denote

Definition 5. Let $0 < q < \infty$, $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})$, $0\leq\lambda < \infty$ and $\alpha\in\mathbb{R}$. The weighted Morrey-Herz space with variable exponents $M\dot{K}^{\alpha, \lambda}_{q, p(\cdot)}(w~^{p(\cdot)})$ is defined by

where

It obviously follows that $M\dot{K}^{\alpha, 0}_{q, p(\cdot)}(w~^{p(\cdot)})$ coincides with the weighted Herz spaces with variable exponents $\dot{K}^{\alpha, q}_{p(\cdot)}(w)$ defined in ^[7].

3.   The main results

Theorem 1. Let $(p_{j}(\cdot), w)$ be $M$-pairs, $j = 1, 2, \ldots, m,$ and $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\cap LH(\mathbb{R}^{n})$ satisfy $\frac 1{p(x)} = \frac 1{p_1(x)}+\frac 1{p_2(x)}+\cdots +\frac 1{p_m(x)}$. Let $w$ be a weight in $A_{p_{0}(\cdot)}$ where $p_{0}(\cdot) = \min\{p_{1}(\cdot), \ldots, p_{m}(\cdot)\}$. Then, there exists a constant $C > 0$ so that for all $\vec{f} = (f_{1}, \ldots, f_{m})$, where each $f_{j}$ is a smooth function and with compact support, the $m$-linear Calderón-Zygmund operator $T$ is bounded on the product of weighted variable exponent Lebesgue spaces. Moreover,

Proof. By using Lemma 3, Lemma 5, Hölder's inequality and the boundedness of $M$ on $L~~^{p_{j}(\cdot)}(w^{p_{j}(\cdot)})$, we get

This yields (3.1).

Theorem 2. Let $(p_{j}(\cdot), w)$ be $M$-pairs, $j = 1, 2, \ldots, m,$ and $p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\cap LH(\mathbb{R}^{n})$ satisfy $\frac 1{p(x)} = \frac 1{p_1(x)}+\frac 1{p_2(x)}+\cdots +\frac 1{p_m(x)}$. Let $w$ be a weight in $A_{p_{0}(\cdot)}$ where $p_{0}(\cdot) = \min\{p_{1}(\cdot), \ldots, p_{m}(\cdot)\}$, $\delta_{1}\in (0, 1)$ be the constant appearing in (2.3). Suppose that $\lambda = \sum_{i = 1}^m \lambda_i, \, \alpha = \sum_{i = 1}^m\alpha_i, \, 0 < \lambda_i < \alpha_i < n\delta_{1}, \, \frac 1q = \sum_{i = 1}^m\frac 1{q_i}.$ Then, there exists a constant $C > 0$ so that for all $\vec{f} = (f_{1}, \ldots, f_{m})$, where each $f_{j}$ is a smooth function and with compact support, the $m-$linear Calderón-Zygmund operator $T$ is bounded on the product of weighted variable exponent Morrey-Herz spaces. Moreover, we have

Let $\lambda_i = 0,$ we immediately get the boundedness of the multilinear Calderón-Zygmund integral operator on the product of weighted Herz spaces with variable exponents.

Proof. Without loss of generality, we only consider the case $m = 2.$ Actually, the similar procedure works for all $m\in N$. When $m = 2$, we have

Write

$L~~^{p(\cdot)}(w~^{p(\cdot)})$ is a Banach space, hence we have

where

Because of the symmetry of $f_{1}$ and $f_{2}$, we see that the estimate of $I_{2}$ is analogous to that of $I_{4}$, the estimate of $I_{3}$ is similar to that of $I_{7}$ and the estimate of $I_{6}$ is similar to that of $I_{8}$. We will estimate $I_{1}-I_{3}, I_{5}, I_{6}$ and $I_{9}$ respectively.

(i) To estimate $I_{1}$, we note $l_{i}\leq k-2$ for $i = 1, 2$, and

Thus, for $x\in C_{k}$, we get

Applying the generalized Hölder's inequality to the last two integrals, we obtain

By $\frac{1}{p(x)} = \frac{1}{p_{1}(x)}+\frac{1}{p_{2}(x)}$, using the generalized Hölder's inequality and Lemmas $2$ and $4$, we get

Therefore, we arrive at the inequality

Since $\frac{1}{q} = \frac{1}{q_{1}}+\frac{1}{q_{2}}$, we have

We consider the two cases $0 < q_{i}\leq 1$ and $1 < q_{i} < \infty$. When $0 < q_{i}\leq 1$, we apply inequality

and obtain

When $1 < q_{i} < \infty$, by Hölder's inequality we obtain

Therefore, for any $0 < q_{i} < \infty$, we could obtain

(ii) To estimate $I_{2}$, we have $l_{1}\leq k-2$, and $k-1\leq l_{2}\leq k+1$, then

From the inequality above, we obtain

By Lemma 4, we have

if $l_{2} = k-1$, $\frac{\|\chi_{B~~_{l_{2}}}~~\|_{(L~~^{p_{2}~(\cdot)~~}(w^{p_{2}(\cdot)~~}))'~~}~~}{\|\chi_{B~~_{k}}\|_{(L~~^{p_{2}~(\cdot)~~}(w^{p_{2}(\cdot)}~~))'}} \leq C\Big(\frac{|B_{l_{2}}|}{|B_{k}|}\Big)^{\delta_{1}} = C2^{(l_{2}-k)n\delta_{1}} = C2^{-n\delta_{1}}$,

if $l_{2} = k$, $\frac{\|\chi_{B~~_{l_{2}}}~~\|_{(L~~^{p_{2}~(\cdot)~~}(w^{p_{2}(\cdot)~~}))'~~}~~}{\|\chi_{B~~_{k}}\|_{(L~~^{p_{2}~(\cdot)~~}(w^{p_{2}(\cdot)}~~))'}} = 1$,

if $l_{2} = k+1$, $\frac{\|\chi_{B~~_{l_{2}}}~~\|_{(L~~^{p_{2}~(\cdot)~~}(w^{p_{2}(\cdot)~~}))'~~}~~}{\|\chi_{B~~_{k}}\|_{(L~~^{p_{2}~(\cdot)~~}(w^{p_{2}(\cdot)}~~))'}}\leq C\frac{|B_{l_{2}}|}{|B_{k}|} = C2^{(l_{2}-k)n} = C2^{n}$.

Combing the estimates above, we arrive the inequality

where $C$ is independent of $l_{2}$ and $k$.

For $I_{2}$, we get

Note that

(iii) To estimate $I_{3}$, for $x\in C_{k}, y_{i}\in C_{l_{i}}$ and $l_{1}\leq k-2$, $l_{2}\geq k+2$, we have

Thus, for $x\in C_{k}$, we get

By Hölder's inequality, Lemmas $2$ and $4$, we obtain

For $I_{3}$, we get

Note that

As for $I_{32}$, we write

Now, we estimate $I^{1}_{32}$ and $I^{2}_{32}$ respectively. For $I^{1}_{32}$, using similar methods as that for $I_{1i}$. Observing that $\alpha_{2} > 0$, we consider the following two cases:

When $0 < q_{2}\leq 1$, by $(3.3)$, it follows that

When $1 < q_{2} < \infty$, we use Hölder's inequality and obtain

For $I^{2}_{32}$, when $0 < q_{2}\leq 1$, by the fact that $0 < \lambda_{2} < \alpha_{2}$, we have

When $1 < q_{2} < \infty$, because $\alpha_{2}-\lambda_{2} > 0$, we have

Therefore we get

(iv) To estimate the term $I_{5}$, by Theorem $1$, the $L~~^{p(\cdot)}(w~^{p(\cdot)})$- boundedness of $T$, we note that

Since $\frac{1}{q} = \frac{1}{q_{1}}+\frac{1}{q_{2}}$, we have

(v) To estimate the term $I_{6}$, for $C_{k}$, $y_{i}\in C_{l_{i}}$, $k-1\leq l_{1}\leq k+1$, $l_{2}\geq k+2$, we have

By the generalized Hölder's inequality, Lemmas $2$ and $4$, we obtain

where

with the constant $C$ independent of $l_{1}$ and $k$.

Hence

Here the estimate of $I_{61}$ is equal to that of $I_{22}$ and $I_{62} = I_{32}.$

(vi) Finally to estimate the term $I_{9}$, we note $l_{2}\geq k+2$ and $|x-y_{i}| > 2^{l_{i}-2}$, \, for $x\in C_{k}$, $y_{i}\in C_{l_{i}}$,

Applying Hölder's inequality to the last integral, we obtain

Thus

where the estimate of $I_{9i} (i = 1, 2)$ is equal to that of $I_{32}$.

Combining all the estimates for $I_{i}$ together $(i = 1, 2, \cdots, 9)$, we get

Consequently, we have proved the Theorem 2.

4.   Conclusions

In this paper we define the Muckenhoupt weights with variable exponent and introduce weighted Morrey-Herz spaces $M\dot K^{\alpha, \lambda}_{q, p(\cdot)}(w~^{p(\cdot)})$ with variable exponent $p(\cdot)$. We investigate the boundedness of multi-linear Calder´on-Zygmund operator on weighted Lebesgue spaces and weighted Morrey-Herz spaces with variable exponents. The results we obtain are the generalization of these in the references, and they could be applied more widely.

Acknowledgments

The authors thank the referees for their valuable comments which helped to improve the paper. The work was supported by NNSF of China grants (11771223).

Conflict of interest

The authors declare no conflict of interest.