Compactness for iterated commutators of general bilinear fractional integral operators on Morrey spaces with non-doubling measures

1.   Introduction

For $0 < \alpha < n$, the fractional integral operator,

has many applications in harmonic analysis, PDEs and the theory of Sobolev embeddings (see ^[2,15]). For the boundedness of the fractional integral operator on function spaces, Adams ^[1] first obtained the boundedness of $I_\alpha$ on Morrey space. Recently it has been shown that many results of the classical singular integral theory also hold without assuming the doubling property, considerable attention has been paid to the study of when the underlying measures only satisfy the polynomial growth condition, namely, there exists a positive constant $C_0 > 0$ and $n \in(0, d ]$ such that

for every $x\in \mathbb{R}^d$ and $l > 0$, $B(x, l)$ being the Euclidean ball with center $x$ and radius $l$. In 2005, Sawano and Tanaka ^[19] introduced the definition of Morrey spaces under non-doubling measure conditions, and considered the boundedness of some classic operators including fractional integral $I_\alpha$ on Morrey spaces. Let us recall the definition of Morrey spaces $\mathcal{M}_{q}^{p}(\mu)$, where $\mu$ satisfies the growth condition (1.1). Let $\kappa > 1$ and $1 \leq q \leq p < \infty$. We define the Morrey spaces $\mathcal{M}_{q}^{p}(\kappa, \mu)$ as

where the norm $\left\|f \right\|_{\mathcal{M}_{q}^{p}(\kappa, \mu)}$ is given by

Applying Hölder's inequality, it is easy to see that

for $1 \leq s \leq t \leq p < \infty$. Moreover, the definition of Morrey spaces is independent of the constant $\kappa > 1$, and the norms for different choice of $\kappa > 1$ are equivalent, see ^[17,19,20] for details. So we will denote $\mathcal{M}_{q}^{p}(2, \mu)$ by $\mathcal{M}_{q}^{p}(\mu)$. One can see from the definition of Morrey spaces that, if $f_i\in \mathcal{M}_{s_i}^{p_i}(\mu)$, $1\leq s_i\leq p_i < \infty, i = 1, 2$, and $\frac 1p = \frac 1{p_1} +\frac 1{p_2}, \frac 1s = \frac 1{s_1} +\frac 1{s_2}$, we have $\|f_1f_2\|_{\mathcal{M}_{s}^{p}(\mu)}\leq \|f_1\|_{\mathcal{M}_{s_1}^{p_1}(\mu)} \|f_2\|_{\mathcal{M}_{s_2}^{p_2}(\mu)}$.

In recent decades, there have been many works ^[3,12,14,24] about the properties of commutators. Chanillo ^[5] obtained the boundedness of the commutator $[b, I_\alpha]$ on the classical Lebesgue space, the operator $[b, I_\alpha]$ given by

Subsequently, Di Fazio and Ragusa ^[9] studied the boundedness of the commutator $[b, I_\alpha]$ on the classical Morrey space. These conclusions can be extended to the case of non-doubling measures. Tolsa ^[23] introduced the concept of BMO space with non-double measures and called it as RBMO. In order to describe RBMO, we use the follow notations. Here and below we mean by a "cube" a compact cube with the sides parallel to the coordinate axis. $Q(x, l(Q))$ will be the cube centered at $x$ with side length $l(Q)$. Next, we define the set of doubling cubes:

where $c_{1}$ is a constant independent of $Q \in \mathcal{Q}(\mu)$. $\mathcal{Q}(\mu, 2)$ denotes the set of doubling cubes. For $Q \in \mathcal{Q}(\mu)$ be a cube which is not always doubling, one can find $N$ large enough such that $NQ$ is a doubling cube. Let $j \geq 0$ be the smallest integer such that $2^{j}Q$ is doubling, we denote this cube by $Q^{*}$.

Let $0 \leq \gamma < n$. Denote by $c_{Q}$ the center of $Q \in \mathcal{Q}(\mu)$. Then we define

and $K_{Q, R}: = K_{Q, R}^{(0)}$ for $Q, R \in \mathcal{Q}(\mu, 2)$ with $Q \subset R$. The coefficient $K_{Q, R}$ was introduced by Tolsa in ^[23] and the modified coefficient $K_{Q, R}^{(\gamma)}$ was defined later by Chen and Sawyer ^[7]. For $Q \in \mathcal{Q}(\mu, 2)$ we denote by $m_{Q}b$ the average of the $L_{l o c}^{1}(\mu){\rm{-function}}\; b$, namely, $m_{Q}b = \frac{1}{\mu(Q)}\int_{Q}b(x)d\mu$. Let $\eta > 1$ be a fixed constant, we say that $b \in L_{loc}^{1}(\mu)$ is in RBMO($\mu$) if there exists a constant $\mathcal{A}$ such that

for any cube $Q$, and

for any two doubling cubes $Q \subset R$. The minimal constant $\mathcal{A}$ is the RBMO($\mu$) norm of $b$, and it will be denoted by $\left\| b \right\|_{*}$.

By using an uniform boundedness property for $I_{\alpha}$ associated with non-doubling measures in $\mathbb{R}^d$, Betancor and Fari$\tilde{\rm n}$a ^[4] obtained the compactness of the commutator $[b, I_{\alpha}]$ when $b \in \mathrm{RVMO}(\mu)$, where $\mathrm{RVMO}(\mu)$ denotes the closure of the space $C_{c}^{\infty}(\mathbb{R}^d)$ of the smooth and compact supported functions on $\mathbb{R}^d$, in $\mathrm{RBMO}(\mu)$.

In 2004, Garcia-Cuerva and Gatto ^[13] generalized the classic Hardy-Littlewood-Sobolev theorem, for $1 < p < \frac{n}{\alpha}$, $\|I_{\alpha}f\|_{L^{q}(\mu)} \leq C\|f\|_{L^{p}(\mu)}$, where $\frac{1}{q} = \frac{1}{p}-\frac{\alpha}{n}$. They also introduced a general class of fractional integrals operators $K_\alpha$ for the purpose of obtaining boundedness of the operators from $L^{p}(\mu)$ to $Lip(\alpha-\frac{n}{p})$, $\frac{n}{\alpha} < p\leq \infty$, which defined as follows

the kernel function $\mathfrak{K}_{\alpha}: \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{C}$ is said to be a general fractional kernel of order $\alpha$ and regularity $\delta (0 < \delta \leq 1)$ if it satisfies the following two conditions:

and

for $|x-y| \geq 2 |x-x'|$. We can see that $I_\alpha$ is a fractional integral operator with a kernel having regularity 1. In 2008, Sawano and Shirai ^[18] proved that if operator $K_{\alpha}$ with the kernel $\mathfrak{K}_{\alpha}$ which satisfying the conditions (1.4) and (1.5), then the commutator $[b, K_\alpha]$ with $b\in {{\rm{RVMO}}}(\mu)$ is a compact operator on the Morrey spaces with non-doubling measures. For the multilinear fractional integral $\mathcal{I}_{\alpha}$, we can also see ^[6,8,11,16]. Tao and Zheng in ^[22] discussed the Morrey spaces boundedness of the iterated commutators $\left[b_1, b_2, \mathcal{I}_{\alpha}\right]$ under the setting with non-doubling measures for some indexes, which is defined by

where $b_1, b_2 \in {\rm{RBMO}}(\mu)$, and

For $b \in {\rm{VMO}}$, the BMO-closure of $C_c^\infty$, Ding and Mei ^[10] established the compactness of the bilinear fractional integrals commutators $[\mathcal{I}_\alpha, b]_{i}(i = 1, 2)$ on Morrey spaces, the commutators of $\mathcal{I}_{\alpha}$ are defined by

In this paper, we consider a general bilinear factional integral operator $\mathcal{K}_{\alpha}$ based on (1.3), which is given by

where $0 < \alpha < 2n$ and $0 < \delta \leq 1$. A function $k_{\alpha}: \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{C}$ is said to be a general bilinear fractional kernel of order $\alpha$ and regularity $\delta$ if it satisfies the following two conditions:

for $\max\limits_{i = 1, 2} {|x-y_i|} \geq 2 |x-x'| > 0$.

In this paper, under non-doubling measure conditions we will study the $L^p$-boundedness and Morrey-boundedness of the general bilinear factional integral operator $\mathcal{K}_{\alpha}$, and the Morrey-boundedness and compactness of $\mathcal{K}_{\alpha}$ associated commutators. Firstly, we give the boundedness of $\mathcal{K}_{\alpha}$ on Lebesgue spaces:

Lemma 1.1. Let $0 < \alpha < 2n$, $f_i\in L^{p_i}(\mu)$ with $1 < p_{i} < \frac{\alpha}{n}$, $i = 1, 2$. If $\frac{1}{p} = \frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{\alpha}{n} > 0$, then there exists a constant $C > 0$ independent of $f_1, f_2$ such that

Proof. By a simple inequality $(a+b)^{c} \geq a^{c/2}b^{c/2}, a, b, c > 0$, we have

Applying Hölder's inequality with $\frac{1}{t_1} = \frac{1}{p_{1}}-\frac{\alpha}{2n}$, $\frac{1}{t_2} = \frac{1}{p_{2}}-\frac{\alpha}{2n}$ and $\left\| I_{\frac{\alpha}{2}} f \right\|_{L^{t_i}(\mu)} \leq C\left\| f \right\|_{L^{p_i}(\mu)}$, $i = 1, 2$ (see^[13]HY__HY, Corollary3]), we get

Next we will illustrate the boundedness properties of the general bilinear fractional integrals on Morrey spaces with non-doubling measures. For the convenience of the following, the modified fractional maximal operator is defined by

and $M f(x): = M_{0, \frac{9}{8}} f(x)$.

Lemma 1.2. ^[19,20] Suppose that the parameters $p, q, s, t$ and $\gamma$ satisfy $1 < q \leq p < \infty$, $1 < t \leq s < \infty$, $0 \leq \gamma < n$, $\frac{1}{s} = \frac{1}{p}-\frac{\gamma}{n}$ and $\frac{q}{p} = \frac{t}{s}$. Then there exists a constant $C > 0$ such that, for all $f \in \mathcal{M}_{q}^{p}(\mu)$,

Lemma 1.3. ^[19] Let $1 < q \leq p < \infty$, $1 < t \leq s < \infty$, $0 < \alpha < n$ and $\frac{q}{p} = \frac{t}{s}$, $\frac{1}{s} = \frac{1}{p}-\frac{\alpha}{n}$. Then there exists a constant $C > 0$ such that, for every $f \in \mathcal{M}_{q}^{p}(\mu)$,

Lemma 1.4. Support $0 < \alpha < 2n$, $1 < q_{i} \leq p_{i} < \infty$, $\frac{1}{p} = \frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{\alpha}{n} > 0$ and $\frac{1}{q} = \frac{1}{q_{1}}+\frac{1}{q_{2}}-\frac{\alpha}{n} > 0$, $i = 1, 2$. Then, the operators $\mathcal{K}_{\alpha}$ are bounded from $\mathcal{M}_{q_{1}}^{p_{1}}(\mu) \times \mathcal{M}_{q_{2}}^{p_{2}}(\mu)$ to $\mathcal{M}_{q}^{p}(\mu)$, that is to say,

where $C > 0$ is the constant independent of $f_1, f_2$.

Proof. For all $y\in \mathbb{R}^{d}, y\neq x$, an elementary calculation yields

Thus, by (1.10) and Fubini's theorem, we get

Applying the Hölder inequality and the growth condition (1.1), and letting cube $Q = Q(x, 2l)$, we get

and

Then, by (1.11), we have

And by (1.12), we get

By choosing $\epsilon = \left(\frac{\left\|f_{1}\right\|_{\mathcal{M}_{q_{1}}^{p_{1}}(\mu)}\left\|f_{2}\right\|_{\mathcal{M}_{q_{2}}^{p_{2}}(\mu)}}{Mf_1(x)Mf_2(x)}\right)^{\frac{n}{h}}$, where $\frac{1}{h} = \frac{1}{p_1}+\frac{1}{p_2}$, we can get

For $\frac{1}{h} = \frac{1}{p_1}+\frac{1}{p_2}$, $\frac{1}{r} = \frac{1}{q_1}+\frac{1}{q_2}$, applying the Hölder inequality and Lemma 1.2, we obtain that

Thus, by (1.13) and (1.14), we get

Now let $b_{1}, b_{2} \in {\rm{RBMO}}(\mu)$, then the iterated commutators of general bilinear fractional integral operator $\left[b_1, b_2, \mathcal{K}_{\alpha}\right]$ is formally defined as

Obviously, (1.15) is the form of (1.6) when $k_{\alpha}(x, y_1, y_2) = {1}/{(|x-y_1|+|x-y_2|)^{2n-\alpha}}$. Inspired by ^[18] and ^[10,22], we find that the boundedness and compactness of the iterated commutators of general bilinear fractional integral operators $[b_1, b_2, \mathcal{K}_\alpha]$ on the Morrey space with non-doubling measures have not been established. Our main results can be stated as follows.

Theorem 1.5. Let $0 < \alpha < 2n$, $1 < q_{i} \leq p_{i} < \infty$, $\frac{1}{p} = \frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{\alpha}{n} > 0$ and $\frac{1}{q} = \frac{1}{q_{1}}+\frac{1}{q_{2}}-\frac{\alpha}{n} > 0$, $i = 1, 2$. Suppose $\|\mu\| = \infty$ and $b_{1}, b_{2} \in {\rm{RBMO}}(\mu)$. Then, the commutators $\left[b_1, b_2, \mathcal{K}_{\alpha}\right]$ are bounded from $\mathcal{M}_{q_{1}}^{p_{1}}(\mu) \times \mathcal{M}_{q_{2}}^{p_{2}}(\mu)$ to $\mathcal{M}_{q}^{p}(\mu)$, moreover,

where $C > 0$ is the constant independent of $f_1, f_2, b_1, b_2$.

By using the similar methods in ^[22]HY__HY, Theorem 1.4], we can obtain the proof of Theorem 1.5. Hence, we omit the details.

On the basis of Theorem 1.5, we will pay our attention to the compactness of $[b_1, b_2, \mathcal{K}_\alpha]$ on Morrey space with non-doubling measures. We have the following theorem.

Theorem 1.6. Let $0 < \alpha < 2n$, $1 < q_{i} \leq p_{i} < \infty$, $\frac{1}{p} = \frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{\alpha}{n} > 0$ and $\frac{1}{q} = \frac{1}{q_{1}}+\frac{1}{q_{2}}-\frac{\alpha}{n} > 0$, $i = 1, 2$. Suppose $\|\mu\| = \infty$ and $b_{1}, b_{2} \in {\rm{RVMO}}(\mu)$. Then, the commutator $\left[b_1, b_2, \mathcal{K}_{\alpha}\right]$ is a compact operator from $\mathcal{M}_{q_{1}}^{p_{1}}(\mu) \times \mathcal{M}_{q_{2}}^{p_{2}}(\mu)$ to $\mathcal{M}_{q}^{p}(\mu)$.

2.   Proof of Theorem 1.6

In order to prove Theorem 1.6, we also need a compactness criterion on spaces of $\mathcal{M}_{q_{1}}^{p_{1}}(\mu) \times \mathcal{M}_{q_{2}}^{p_{2}}(\mu)$.

Lemma 2.1. ^[18,21] Let $1 < q \leq p < \infty$, $1 < q_{i} \leq p_{i} < \infty$, $i = 1, 2$. Suppose $\mathcal{T}$ is a bilinear integral operator defined as follows

where the kernel function $k \in L_{c}^{\infty}({\mu \otimes \mu \otimes \mu})$. Then $\mathcal{T}$ is a compact operator from $\mathcal{M}_{q_{1}}^{p_{1}}(\mu) \times \mathcal{M}_{q_{2}}^{p_{2}}(\mu)$ to $\mathcal{M}_{q}^{p}(\mu)$.

Lemma 2.2. Let $x \in {{supp}}(\mu)$, $0\leq\alpha < n$, and let $\varrho > 0$ and $r > 0$. Then there exist constants $C_1, C_2 > 0$ such that

Proof. We will use the same procedure as in the proof of (^[13], Lemmas 2.1 and 2.2). If $n-\alpha \leq \varrho$, (2.2) follows immediately from (1.1). If $\varrho < n-\alpha$, we write

On the other hand, with the similar ways as above we have

Recall the function space ${\rm{RVMO}}(\mu)$, $b \in{\rm{RVMO}}(\mu)$ if and only if there exists a sequence of compactly supported smooth functions $\left\{b_{j}\right\}_{j = 1}^{\infty}$ such that $\lim\limits_{j \to \infty}\left\|b-b_{j}\right\|_{*} = 0$. In order to obtain the compactness in Theorem 1.6, we will establish some function approximations for the iterated commutator. Assume a function $\psi(t)\in C_{0}^{\infty}(\mathbb{R}^{d})$ satisfying $\rm{supp}\; \psi\subset [1, \infty)\; {\rm{and}}\; \psi(t) = 1, \; t\geq2$. For $\varepsilon \leq 1$, we consider the truncation of $\mathcal{K}_{\alpha}$,

Lemma 2.3. Let $b_1, b_2\in C^{\infty}_{c}(\mathbb{R} ^{d})$, the kernel function $k_{\alpha}$ satisfies (1.8) and (1.9). For $f_1 \in \mathcal{M}^{p_1}_{q_1}(\mu)$, $f_2 \in \mathcal{M}^{p_2}_{q_2}(\mu)$ with $1 < q_{i} \leq p_{i} < \infty$, $i = 1, 2$, and let $\frac{1}{p} = \frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{\alpha}{n} > 0,$ $\frac{1}{q} = \frac{1}{q_{1}}+\frac{1}{q_{2}}-\frac{\alpha}{n} > 0$, then the iterated commutators $[b_1, b_2, \mathcal{K}_{\alpha}^{\varepsilon}]$ tends to $[b_1, b_2, \mathcal{K}_{\alpha}]$ uniformly in $\mathcal{M}^{q}_{p}(\mu)$ as $\varepsilon \to 0$.

Proof. One can see that

In view of the size condition of the kernel $k_{\alpha}$, the property of $\psi$, using Lebesgue's convergence theorem and Lemma 2.2, we obtain

where we have chosen $\delta > 0$ and $\alpha_1+\alpha_2 = \alpha$ such that $\frac 1{s_i} = \frac 1{p_i}-\frac{\alpha_i}n > 0$, $i = 1, 2$. We note that $\left\|M_{\alpha_i} f_{i} \right\|_ {\mathcal{M}^{s_i}_{t_i}(\mu)}\leq C\left\| f_{i} \right\|_ {\mathcal{M}^{p_i}_{q_i}(\mu)}$ with $\frac{s_i}{t_i} = \frac{p_i}{q_i}$. Thus, we have that

Therefore, Lemma 2.3 holds.

Lemma 2.4. Under the same conditions and notations, let $\mathcal{K}^{\varepsilon}_{\alpha}$ be the truncation operator defined by (2.4). $\mathcal{K}^{\varepsilon, R}_{\alpha}$ be the truncation of $\mathcal{K}^{\varepsilon}_{\alpha}$,

Then, for $f_1 \in \mathcal{M}^{p_1}_{p_1}(\mu)$, $f_2 \in \mathcal{M}^{p_2}_{p_2}(\mu)$ with $1 < q_{i} \leq p_{i} < \infty$, $i = 1, 2$, the iterated commutator $[b_1, b_2, \mathcal{K}^{\varepsilon, R}_{\alpha}]$ tends to $[b_1, b_2, \mathcal{K}^{\varepsilon}_{\alpha}]$ uniformly in $\mathcal{M}^{p}_{q}(\mu)$ as $R\to \infty$. Moreover, there exist $\delta > 0$ such that

Proof. Note $0 < \alpha < 2n$, $1 < q \leq p < \infty$, $1 < q_{i} \leq p_{i} < \infty$, $i = 1, 2$, $\frac{1}{h} = \frac{1}{p_1}+\frac{1}{p_2}$, $\frac{1}{r} = \frac{1}{q_1}+\frac{1}{q_2}$, $\frac{1}{p} = \frac{1}{h}-\frac{\alpha}{n} > 0$, $\frac{1}{q} = \frac{1}{r}-\frac{\alpha}{n} > 0$ and $b_1, b_2\in C^{\infty}_{c}(\mathbb{R} ^{d})$. Taking $f_1\in \mathcal{M}^{p_1}_{q_1}(\mu)$, $f_2\in \mathcal{M}^{p_2}_{q_2}(\mu)$, we obtain

Next we introduce the following two indices:

Then $\alpha_1$ and $\alpha_2$ satisfy

In fact,

Now, since $p_1, p_2 > 1$, we see that $n > \max(\alpha_1, \alpha_2)$. In particular, this yields

For $b_1, b_2\in C^{\infty}_{c}(\mathbb{R} ^{d})$, there exist $Q_1 \supset {\rm{supp}}\; b_1$ and $Q_2 \supset {\rm{supp}}\; b_2$. We can see that the kernel satisfies

that is

To estimate $D_1$, applying (1.10), the Fubini theorem and (1.12), we get

Furthermore, using the Minkowski inequality, we can deduce that

It is easy to get that

since

For $D_{12}$, using the generalized Minkowski inequality, $\mathcal{M}_{p}^{q}(\mu) \supset L^{p}(\mu)$ and the growth condition (1.1), we have the follow estimate

Similarly, we have

Applying the Hölder inequality, we obtain

where $w_1, w_2, \; {\rm{and}}\; p$ satisfy $\frac{1}{w_1} = \frac{1}{p_1}-\frac{\alpha_1}{n}$, $\frac{1}{w_2} = \frac{1}{p_2}-\frac{\alpha_2}{n}$ and $\frac{1}{p} = \frac{1}{w_1}+\frac{1}{w_2}$.

Therefore, recalling that $\mathcal{M}_{1}^{p_i} \supset \mathcal{M}_{q_i}^{p_i}$, we obtain the norm estimate of $D_1$,

We now estimate $D_2$. Similarly, using (1.10), the Fubini theorem and (1.12), we get

Thus, we get the norm estimate of $D_2$,

where $w_1, w_2, p$ satisfy $\frac{1}{w_1} = \frac{1}{p_1}-\frac{\alpha_1}{n}$, $\frac{1}{w_2} = \frac{1}{p_2}-\frac{\alpha_2}{n}$ and $\frac{1}{p} = \frac{1}{w_1}+\frac{1}{w_2}$.

The same estimate holds for $D_3$ by above argument with the role of $y_1$ and $y_2$ inter changed,

where $w_1, w_2, p$ satisfy $\frac{1}{w_1} = \frac{1}{p_1}-\frac{\alpha_1}{n}$, $\frac{1}{w_2} = \frac{1}{p_2}-\frac{\alpha_2}{n}$ and $\frac{1}{p} = \frac{1}{w_1}+\frac{1}{w_2}$.

Thus, we obtain that

for some constants $\delta > 0$, and the constant $C_\varepsilon$ independent of $R$.

Proof of Theorem 1.6. Lemmas 2.3 and 2.4 reduce the proof of Theorem 1.6 to proving the compactness for $[b_1, b_2, \mathcal{K}_{\alpha}^{\varepsilon, R}]$ if $b_1, b_2 \in C_{c}^{\infty}(\mathbb{R}^{d})$ on product Morrey spaces. We can also get the compactness for the operator $[b_1, b_2, \mathcal{K}_{\alpha}^{\varepsilon, R}]$ by using Lemma 2.1.

3.   Conclusions

In this paper, we mainly obtain the Morrey-boundedness and compactness of the iterated commutators of general bilinear fractional integral operators $\mathcal{K}_{\alpha}$ under the non-doubling measure conditions. Instead of establishing of the compactness of the iterated commutators directly we proved it by means of two truncation.

Acknowledgments

We would like to thank the editors and reviewers for their helpful suggestions. This research was supported by the National Natural Science Foundation of China under grant No.11771399, No.11626213 and No.11961056.

Conflict of interest

All authors declare no conflicts of interest in this paper.