© Wuhan University 2025

0 Introduction

Let ${\mathbb{R}}^n,n\ge \mathrm{2},$ be the $n$-dimensional Euclidean spaces. This paper aims to investigate the compactness of the commutator of the Adams type bilinear fractional integral operator (or Riesz potential) $I_{\alpha },\mathrm{0}<\alpha <n.$ The bilinear fractional integral operator $I_{\alpha }$ is defined by

$I_{\alpha }(f,g)(x)={\int }_{{\mathbb{R}}^n}^{}{\int }_{{\mathbb{R}}^n}^{}\frac{f(y)g(z)}{{\left(\left|x-y\right|+\left|x-z\right|\right)}^{\mathrm{2}n-\alpha }}\mathrm{d}y\mathrm{d}z,$

where $x\in {\mathbb{R}}^n,$and $f,g$ are locally integrable functions.

When considering a bilinear operator $I_{\alpha }(f,g),$ we define the commutators $\{[b,I_{\alpha }{]}_i{\}}_{i=\mathrm{1}}^{\mathrm{2}}$ to be

$[b,I_{\alpha }{]}_{\mathrm{1}}(f,g)=b{I}_{\alpha }(f,g)-I_{\alpha }(bf,g),$

and

$[b,I_{\alpha }{]}_{\mathrm{2}}(f,g)=b{I}_{\alpha }(f,g)-I_{\alpha }(f,bg),$

where $b\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{1}}({\mathbb{R}}^n)$.

Riesz^[1] developed a theory of multidimensional fractional integration and derived a powerful method for solving linear differential equations of the hyperbolic type from it. In 1978, Uchiyama^[2] showed that linear commutators of Calderón-Zygmund operators and pointwise multiplication with a symbol belonging to an appropriate subspace of the John-Nirenberg space $(\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n))$ are compact. For a proof of the boundedness of multilinear fractional integrals, see the papers by Chen and Xue^[3] and Chen and Wu^[4]. Bényi et al^[5] established the compactness of commutators of bilinear fractional integrals from $L_{}^p({\mathbb{R}}^n)\times L_{}^q({\mathbb{R}}^n)$ to $L_{}^r({\mathbb{R}}^n)$ with $b\in \mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)$. In 2015, Ding and Mei^[6] obtained the compactness of commutators of bilinear Riesz potential on Morrey spaces, from $L_{}^{(p_{\mathrm{1}},{\lambda }_{\mathrm{1}})}({\mathbb{R}}^n)\times L_{}^{(p_{\mathrm{2}},{\lambda }_{\mathrm{2}})}({\mathbb{R}}^n)$ to $L_{}^{(q,\lambda )}({\mathbb{R}}^n)$ with $b\in \mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)$. Subsequently, the compactness of commutators of multilinear operators was studied extensively, see Refs. [7-10].

On the other hand, Arai and Nakai^[11-12] studied the commutators $[b,I_{\rho }]$ of the fractional integral operator $I_{\rho }$ on the generalized Morrey spaces and showed that if $b$ is a function of generalized Campanato spaces ${\epsilon }^{(\mathrm{1},\psi )}({\mathbb{R}}^n)$, which contain the $\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)$ and Lipschitz spaces as special examples, then $[b,I_{\rho }]$ is bounded and compact on the generalized Morrey spaces. The corresponding result for the commutator of a general fractional integral was also obtained.

Based on the results above, it is natural to ask the following question:

Question: What are the mapping properties of $[b,I_{\alpha }{]}_i,i=\mathrm{1,2},$ on the generalized Morrey spaces when $b$ is a function in the generalized Campanato spaces? Moreover, whether it is compact or not in generalized Morrey spaces?

The main purpose of this paper is to address this question. To state our main results, we first recall some relevant definitions and notations. Let $B(x,r)$ be the open ball centered at $x\in {\mathbb{R}}^n$ and of radius $r$, that is,

$B(x,r)=\{y\in {\mathbb{R}}^n:\left|y-x\right|<r\}.$

For a measurable set $E\subset {\mathbb{R}}^n,$ we denote by $\left|E\right|$ and ${\chi }_E$ the Lebesgue measure of $E$ and the characteristic function of $E$, respectively. For a function $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{1}}({\mathbb{R}}^n)$ and a ball $B,$ let $f_B=\frac{\mathrm{1}}{\left|B\right|}{\int }_B^{}f(y)\mathrm{d}y.$

Moreover, for a measurable function $\varphi :{\mathbb{R}}^n\times (\mathrm{0},\mathrm{\infty })\to (\mathrm{0},\mathrm{\infty })$, while a ball $B=B(x,r)$, we denote by $\varphi (B)=\varphi (x,r).$

Definition 1^[11] Let $\varphi (x,r)$ be a positive measurable function on ${\mathbb{R}}^n\times (\mathrm{0},\mathrm{\infty })$ and $p\in [\mathrm{1},\mathrm{\infty })$, the generalized Morrey space $L_{}^{(p,\varphi )}({\mathbb{R}}^n)$ is denoted by:

$L_{}^{(p,\varphi )}({\mathbb{R}}^n):=\{f:{\Vert f\Vert }_{L_{}^{(p,\varphi )}({\mathbb{R}}^n)}=\underset{B}{\mathrm{s}\mathrm{u}\mathrm{p}}{\left(\frac{\mathrm{1}}{\varphi (B)\left|B\right|}{\int }_B^{}{\left|f(y)\right|}^{p}dy\right)}^{\mathrm{1}/p}<\mathrm{\infty }\},$

where the supremum is taken over all balls $B$ in ${\mathbb{R}}^n.$

We know that ${\Vert f\Vert }_{L_{}^{(p,\varphi )}({\mathbb{R}}^n)}$ is a norm and $L_{}^{(p,\varphi )}({\mathbb{R}}^n)$ is a Banach space. More generally, if ${\varphi }_{\lambda }(x,r)=r^{\lambda }$ for $\lambda \in [-n,\mathrm{0}]$, then $L_{}^{(p,\varphi )}({\mathbb{R}}^n)$ is the classical Morrey space, that is,

$\begin{array}{l}{\Vert f\Vert }_{L_{}^{(p,{\varphi }_{\lambda })}({\mathbb{R}}^n)}=\underset{B}{\mathrm{s}\mathrm{u}\mathrm{p}}{\left(\frac{\mathrm{1}}{{\varphi }_{\lambda }(B)\left|B\right|}{\int }_B^{}{\left|f(y)\right|}^{p}dy\right)}^{\mathrm{1}/p}\\ =\underset{B=B(x,r)}{\mathrm{s}\mathrm{u}\mathrm{p}}{\left(\frac{\mathrm{1}}{r^{\lambda }\left|B\right|}{\int }_B^{}{\left|f(y)\right|}^{p}dy\right)}^{\mathrm{1}/p},\end{array}$

In particular, $L_{}^{(p,{\varphi }_{-n})}({\mathbb{R}}^n)=L_{}^p({\mathbb{R}}^n)$ and $L_{}^{(p,{\varphi }_{\mathrm{0}})}({\mathbb{R}}^n)=L_{}^{\mathrm{\infty }}({\mathbb{R}}^n).$

We recall the definition of $\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)$, denoted by

$\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n):=\{b\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{1}}({\mathbb{R}}^n):{\Vert b\Vert }_{\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)}$

$=\underset{B}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{\mathrm{1}}{\left|B\right|}{\int }_B^{}\left|b(x)-b_B\right|\mathrm{d}x<\mathrm{\infty }\},$

where the supremum is taken over all balls $B\subset {\mathbb{R}}^n.$

We also consider the generalized Campanato spaces with variable growth condition, which are defined as follows.

Definition 2^[11] Let $\psi (x,r)$ be a positive measurable function on ${\mathbb{R}}^n\times (\mathrm{0},\mathrm{\infty })$ and $p\in [\mathrm{1},\mathrm{\infty })$, the generalized Campanato spaces ${\epsilon }^{(p,\psi )}({\mathbb{R}}^n)$ are denoted by

${\epsilon }^{(p,\psi )}({\mathbb{R}}^n):=\{f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{1}}({\mathbb{R}}^n):{\Vert f\Vert }_{{\epsilon }^{(p,\psi )}({\mathbb{R}}^n)}<\mathrm{\infty }\},$

where ${\Vert f\Vert }_{{\epsilon }^{(p,\psi )}({\mathbb{R}}^n)}=\underset{B}{\mathrm{s}\mathrm{u}\mathrm{p}}{\left(\frac{\mathrm{1}}{\psi (B)\left|B\right|}{\int }_B{\left|f(y)-f_B\right|}^{p}dy\right)}^{\mathrm{1}/p},$ the supremum is taken over all balls $B\subset {\mathbb{R}}^n.$

It is easy to check that ${\Vert f\Vert }_{{\epsilon }^{(p,\psi )}({\mathbb{R}}^n)}$ is a normed modulo constant functions and thereby ${\epsilon }^{(p,\psi )}({\mathbb{R}}^n)$ is a Banach space. If $p=\mathrm{1}$ and $\psi \equiv \mathrm{1},$ then ${\epsilon }^{(p,\psi )}({\mathbb{R}}^n)=\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)$. If $p=\mathrm{1}$ and $\psi (x,r)=r^{\alpha }$$(\mathrm{0}<\alpha \le \mathrm{1}),$ then coincide with $\mathrm{L}\mathrm{i}{\mathrm{p}}_{\alpha }({\mathbb{R}}^n).$

We say that a function $\theta :{\mathbb{R}}^n\times (\mathrm{0},\mathrm{\infty })\to (\mathrm{0},\mathrm{\infty })$ satisfies the doubling condition if there exists a positive constant $C,$ for all $x\in {\mathbb{R}}^n$ and $r,s\in (\mathrm{0},\mathrm{\infty }),$ such that

$\frac{\mathrm{1}}{C}\le \frac{\theta (x,r)}{\theta (x,s)}\le C{,}_{}\mathrm{i}{\mathrm{f}}_{}\frac{\mathrm{1}}{\mathrm{2}}\le \frac{r}{s}\le \mathrm{2}.$(1)

We say that $\theta$ is almost increasing (resp. almost decreasing) if there exists a positive constant $C,$ for all $x\in {\mathbb{R}}^n$ and $r,s\in (\mathrm{0},\mathrm{\infty }),$ such that $\theta (x,r)\le C\theta (x,s),(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}.$

$\theta (x,s)\le C\theta (x,r)),\mathrm{i}\mathrm{f}r\le s.$

We also consider the following condition: there exists a positive constant $C,$ for all $x,y\in {\mathbb{R}}^n$ and $r\in (\mathrm{0},\mathrm{\infty }),$ such that

$\frac{\mathrm{1}}{C}\le \frac{\theta (x,r)}{\theta (y,r)}\le C{,}_{}\mathrm{i}{\mathrm{f}}_{}\left|x-y\right|\le r.$(2)

For two functions $\theta ,\kappa :{\mathbb{R}}^n\times (\mathrm{0},\mathrm{\infty })\to (\mathrm{0},\mathrm{\infty })$, we write $\theta ~\kappa$ if there exists a positive constant $C,$ for all $x\in {\mathbb{R}}^n$ and $r\in (\mathrm{0},\mathrm{\infty }),$ such that

$\frac{\mathrm{1}}{C}\le \frac{\theta (x,r)}{\kappa (x,r)}\le C.$(3)

Definition 3 (i)   Let ${\varsigma }^{\mathrm{d}\mathrm{e}\mathrm{c}}$ be the set of all functions $\varphi :{\mathbb{R}}^n\times (\mathrm{0},\mathrm{\infty })\to (\mathrm{0},\mathrm{\infty })$ such that $\varphi$ is almost decreasing and that $r\mapsto \varphi (x,r)r^n$ is almost increasing. That is, there exists a positive constant $C,$for all $x\in {\mathbb{R}}^n$ and $r,s\in (\mathrm{0},\mathrm{\infty }),$ such that

$C\varphi (x,r)\ge \varphi (x,s){,}_{}\varphi (x,r)r^n\le C\varphi (x,s)s^n,\mathrm{i}{\mathrm{f}}_{}r<s.$

(ii) Let ${\varsigma }^{\mathrm{i}\mathrm{n}\mathrm{c}}$ be the set of all functions $\varphi :{\mathbb{R}}^n\times (\mathrm{0},\mathrm{\infty })\to (\mathrm{0},\mathrm{\infty })$ such that $\varphi$ is almost increasing and that $r\mapsto \varphi (x,r)/r$ is almost decreasing. That is, there exists a positive constant $C,$ for all $x\in {\mathbb{R}}^n$ and $r,\mathrm{s}\in (\mathrm{0},\mathrm{\infty }),$ such that $\varphi (x,r)\le C\varphi (x,s){,}_{}C\varphi (x,r)/r\ge \varphi (x,s)/s{,}_{}\mathrm{i}{\mathrm{f}}_{}r<s.$

Remark 1 (i)   If $\varphi \in {\varsigma }^{\mathrm{d}\mathrm{e}\mathrm{c}}$ or $\varphi \in {\varsigma }^{\mathrm{i}\mathrm{n}\mathrm{c}},$ then $\varphi$ satisfies the doubling condition (1).

(ii) It follows from Ref. [11] that for $\varphi \in {\varsigma }^{\mathrm{d}\mathrm{e}\mathrm{c}}$, if $\varphi$ satisfies

$\underset{r\to +\mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\varphi (x,r)=\mathrm{\infty }{,}_{}\underset{r\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\varphi (x,r)=\mathrm{0},$(4)

then there exists $\tilde{\varphi }\in {\varsigma }^{\mathrm{d}\mathrm{e}\mathrm{c}}$ such that $\varphi ~\tilde{\varphi }$ and that $\tilde{\varphi }(x,\cdot )$ is continuous, strictly decreasing and bijective from $(\mathrm{0},\mathrm{\infty })$ to itself for each $x.$

(iii) Assume that $\varphi \in {\varsigma }^{\mathrm{d}\mathrm{e}\mathrm{c}}$ and there exists a positive constant $C,$ for all $x\in {\mathbb{R}}^n$ and $r\in (\mathrm{0},\mathrm{\infty }),$we get

${\int }_r^{\mathrm{\infty }}\frac{\varphi (x,\mathrm{t})}{t}\mathrm{d}t\le C\varphi (x,r).$(5)

For $f\in L^{(p_{\mathrm{1}},{\varphi }_{\mathrm{1}})}({\mathbb{R}}^n),g\in L^{(p_{\mathrm{2}},{\varphi }_{\mathrm{2}})}({\mathbb{R}}^n),\mathrm{1}<p_{\mathrm{1}},p_{\mathrm{2}}<\mathrm{\infty },$ we denote $f{\chi }_{\mathrm{2}B}(y):=f_{\mathrm{1}}(y),f{\chi }_{{(\mathrm{2}B)}^C}(y):=f_{\mathrm{2}}(y),$ $g{\chi }_{\mathrm{2}B}(\mathrm{z}):=g_{\mathrm{1}}(z),$

$g{\chi }_{{(\mathrm{2}B)}^C}(z):=g_{\mathrm{2}}(z),$ We define ${{\rm I}}_{\alpha }(f,g)(x)$ on each ball $B$ by

${{\rm I}}_{\alpha }(f,g)(x)={{\rm I}}_{\alpha }(f_{\mathrm{1}},g_{\mathrm{1}})(x)+{{\rm I}}_{\alpha }(f_{\mathrm{1}},g_{\mathrm{2}})(x)+{{\rm I}}_{\alpha }(f_{\mathrm{2}},g_{\mathrm{1}})(x)+{{\rm I}}_{\alpha }(f_{\mathrm{2}},g_{\mathrm{2}})(x).$(6)

Note that ${{\rm I}}_{\alpha }(f_{\mathrm{1}},g_{\mathrm{1}})(x)$ is well defined since $f{\chi }_{\mathrm{2}B}\in L^{p_{\mathrm{1}}}({\mathbb{R}}^n)$, and $g{\chi }_{\mathrm{2}B}\in L^{p_{\mathrm{2}}}({\mathbb{R}}^n),$ it is easy to check that $I_{\alpha }(f_{\mathrm{1}},g_{\mathrm{2}})(x),I_{\alpha }(f_{\mathrm{2}},g_{\mathrm{1}})(x),\mathrm{a}\mathrm{n}\mathrm{d}{I}_{\alpha }(f_{\mathrm{2}},g_{\mathrm{2}})(x)$ converge absolutely. $I_{\alpha }(f,g)(x)$ defined in (6) is independent of the choice of the ball containing $x.$ Moreover, we can show that $I_{\alpha }(f,g)(x)$ is bounded from $L^{(p_{\mathrm{1}},{\varphi }_{\mathrm{1}})}({\mathbb{R}}^n)\times L^{(p_{\mathrm{2}},{\varphi }_{\mathrm{2}})}({\mathbb{R}}^n)$ to $L^{(\tilde{p},\varphi )}({\mathbb{R}}^n)$, see Lemma 4.1 in Ref. [13].

For $f\in L^{(p_{\mathrm{1}},{\varphi }_{\mathrm{1}})}({\mathbb{R}}^n),g\in L^{(p_{\mathrm{2}},{\varphi }_{\mathrm{2}})}({\mathbb{R}}^n),\mathrm{1}<p_{\mathrm{1}},p_{\mathrm{2}}<\mathrm{\infty }$, employing the notation as in (6), we will define $[b,I_{\alpha }{]}_i(f,g)(x),i=\mathrm{1,2},$on each ball $B$ by

$[b,I_{\alpha }{]}_i(f,g)(x)=[b,I_{\alpha }{]}_i(f_{\mathrm{1}},g_{\mathrm{1}})(x)+[b,I_{\alpha }{]}_i(f_{\mathrm{1}},g_{\mathrm{2}})(x)+[b,I_{\alpha }{]}_i(f_{\mathrm{2}},g_{\mathrm{1}})(x)+[b,I_{\alpha }{]}_i(f_{\mathrm{2}},g_{\mathrm{2}})(x).$(7)

Let $b\in {\epsilon }^{(\mathrm{1},\psi )}({\mathbb{R}}^n),$ note that $[b,I_{\alpha }{]}_i(f_{\mathrm{1}},g_{\mathrm{1}})(x)$ is well defined since $f{\chi }_{\mathrm{2}B}\in L^{p_{\mathrm{1}}}({\mathbb{R}}^n)$, and $g{\chi }_{\mathrm{2}B}\in L^{p_{\mathrm{2}}}({\mathbb{R}}^n),$ it is easy to check that $[b,I_{\alpha }{]}_i(f_{\mathrm{1}},g_{\mathrm{2}})(x),[b,I_{\alpha }{]}_i(f_{\mathrm{2}},g_{\mathrm{1}})(x),\mathrm{a}\mathrm{n}\mathrm{d}$

$[b,I_{\alpha }{]}_i(f_{\mathrm{2}},g_{\mathrm{2}})(x)$ converge absolutely, then (7) is well defined. Moreover, we can show that $[b,I_{\alpha }{]}_i(f,g)(x),$ defined in (7), is independent of the choice of the ball containing $x,$see Lemma 4.2 in Ref. [13]. Furthermore, we can show that $[b,I_{\alpha }{]}_i(f,g)(x),i=\mathrm{1,2},$ is bounded from $L^{(p_{\mathrm{1}},{\varphi }_{\mathrm{1}})}({\mathbb{R}}^n)\times L^{(p_{\mathrm{2}},{\varphi }_{\mathrm{2}})}({\mathbb{R}}^n)$ to $L^{(q,\varphi )}({\mathbb{R}}^n)$. See Lemma 1 for the details.

For the commutators $[b,I_{\alpha }{]}_i(f,g)(x),i=\mathrm{1,2},$ we have Lemma 1.

Lemma 1^[13] Let $p,p_{\mathrm{1}},p_{\mathrm{2}},q\in (\mathrm{1},\mathrm{\infty }),p<q,\frac{\mathrm{1}}{p}+\frac{\mathrm{1}}{q}<\mathrm{1},\alpha \in (\mathrm{0,2}n)$ satisfies $\alpha /n<\mathrm{1}/p,$$\mathrm{1}/q=\mathrm{1}/p-\alpha /n,$ and $\varphi ,{\varphi }_{\mathrm{1}},{\varphi }_{\mathrm{2}},\psi :{\mathbb{R}}^n\times (\mathrm{0},\mathrm{\infty })\to (\mathrm{0},\mathrm{\infty })$.

(i) Assume that $\varphi ,{\varphi }_{\mathrm{1}},{\varphi }_{\mathrm{2}}\in {\varsigma }^{\mathrm{d}\mathrm{e}\mathrm{c}}$ and $\psi \in {\varsigma }^{\mathrm{i}\mathrm{n}\mathrm{c}}$, $\psi$ satisfies (2), $\varphi ,{\varphi }_{\mathrm{1}},{\varphi }_{\mathrm{2}}$ satisfy (5) and there exist positive constants $C_{\mathrm{0}},C_{\mathrm{1}}$ and an exponent $\tilde{p}\in (p,q],$ for all $x\in {\mathbb{R}}^n$ and $r\in (\mathrm{0},\mathrm{\infty }),$ we get

$\frac{\mathrm{1}}{p_{\mathrm{1}}}+\frac{\mathrm{1}}{p_{\mathrm{2}}}=\frac{\mathrm{1}}{p_{}},$(8)

${\varphi }_{\mathrm{1}}^{\mathrm{1}/p_{\mathrm{1}}}{\varphi }_{\mathrm{2}}^{\mathrm{1}/p_{\mathrm{2}}}={\varphi }_{}^{\mathrm{1}/p},$(9)

${\int }_{\mathrm{0}}^r{t}^{\alpha -\mathrm{1}}\mathrm{d}t\varphi {(x,r)}^{\mathrm{1}/p}+{\int }_r^{\mathrm{\infty }}{t}^{\mathrm{\alpha }-\mathrm{1}}\varphi {(x,t)}^{\mathrm{1}/p}\mathrm{d}t\le C_{\mathrm{0}}\varphi {(x,r)}^{\mathrm{1}/\tilde{p}},$(10)

$\psi (x,r)\varphi {(x,r)}^{\mathrm{1}/\tilde{p}}\le C_{\mathrm{1}}\varphi {(x,r)}^{\mathrm{1}/q}$(11)

If $b\in {\epsilon }^{(\mathrm{1},\psi )}({\mathbb{R}}^n),$ then $[b,I_{\alpha }{]}_i(f,g)(x),i=\mathrm{1,2}$ is well defined for all $f\in L^{(p_{\mathrm{1}},{\varphi }_{\mathrm{1}})}({\mathbb{R}}^n),$$g\in$$L^{(p_{\mathrm{2}},{\varphi }_{\mathrm{2}})}({\mathbb{R}}^n).$ That is, for all $f,g,$ there exists a positive constant $C,$ independent of $b,f,g,$ such that

${\Vert [b,I_{\alpha }{]}_i(f,g)\Vert }_{L^{(q,\varphi )}}\le C{\Vert b\Vert }_{{\epsilon }^{(\mathrm{1},\psi )}}^{}{\Vert f\Vert }_{L^{(p_{\mathrm{1}},{\varphi }_{\mathrm{1}})}}{\Vert g\Vert }_{L^{(p_{\mathrm{2}},{\varphi }_{\mathrm{2}})}}.$

(ii) Conversely, assume that $\varphi$ satisfies (2), and there exist positive constants $C_{\mathrm{2}},$ for all $x\in {\mathbb{R}}^n$ and $r\in (\mathrm{0},\mathrm{\infty }),$we get

$C_{\mathrm{2}}\psi (x,r)r^{\alpha }\varphi {(x,r)}^{\mathrm{1}/p}\ge \varphi {(x,r)}^{\mathrm{1}/q}.$(12)

If $[b,I_{\alpha }{]}_i(f,g)(x),i=\mathrm{1,2},$ is bounded from $L^{(p_{\mathrm{1}},{\varphi }_{\mathrm{1}})}({\mathbb{R}}^n)\times L^{(p_{\mathrm{2}},{\varphi }_{\mathrm{2}})}({\mathbb{R}}^n)$ to $L^{(q,\varphi )}({\mathbb{R}}^n),$ then $b\in {\epsilon }^{(\mathrm{1},\psi )}({\mathbb{R}}^n),$ such that ${\Vert b\Vert }_{{\epsilon }^{(\mathrm{1},\psi )}}^{}\prec {\Vert [b,I_{\alpha }{]}_i(f,g)\Vert }_{L^{(p_{\mathrm{1}},{\varphi }_{\mathrm{1}})}\times L^{(p_{\mathrm{2}},{\varphi }_{\mathrm{2}})}\to L^{(q,\varphi )}}.$

In this paper, we use the method of Arai and Nakai^[12], and give sufficient conditions for the compactness of commutators of bilinear fractional integral operator.

We denote by ${\overline{C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{\infty }}({\mathbb{R}}^n)}}^{{\epsilon }^{(\mathrm{1},\psi )}({\mathbb{R}}^n)},$ the closure of $C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{\infty }}({\mathbb{R}}^n)$ with respect to ${\epsilon }^{(\mathrm{1},\psi )}({\mathbb{R}}^n)$. If $\psi \equiv \mathrm{1}$, then ${\epsilon }^{(\mathrm{1},\psi )}({\mathbb{R}}^n)=\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)$ and ${\overline{C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{\infty }}({\mathbb{R}}^n)}}^{\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)}=\mathrm{C}\mathrm{M}\mathrm{O}({\mathbb{R}}^n).$

In order to obtain the compactness of the commutators $[b,I_{\alpha }{]}_i(f,g)(x),i=\mathrm{1,2},$ we consider the following condition on $\psi$: there exists a positive constant $C,$ for all $x\in {\mathbb{R}}^n$ and $r\in (\mathrm{0},\mathrm{\infty }),$ such that

${\int }_r^{\mathrm{\infty }}\frac{\psi (x,t)}{t^{\mathrm{2}}}\mathrm{d}t\le C\frac{\psi (x,r)}{r}.$(13)

In the end, the main results of this work are stated as follows.

Theorem 1   Let $p,p_{\mathrm{1}},p_{\mathrm{2}}\in (\mathrm{1},\mathrm{\infty }),p<q$ and $\varphi ,{\varphi }_{\mathrm{1}},{\varphi }_{\mathrm{2}}:{\mathbb{R}}^n\times (\mathrm{0},\mathrm{\infty })\to (\mathrm{0},\mathrm{\infty })$, assume the same condition as Lemma 1, $\varphi$ and $\psi$ satisfy (2) and (13), respectively. If $b\in {\overline{C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{\infty }}({\mathbb{R}}^n)}}^{{\epsilon }^{(\mathrm{1},\psi )}({\mathbb{R}}^n)},$ then $[b,I_{\alpha }{]}_i(f,g)(x),i=\mathrm{1,2},$ are compact from $L^{(p_{\mathrm{1}},{\varphi }_{\mathrm{1}})}({\mathbb{R}}^n)\times L^{(p_{\mathrm{2}},{\varphi }_{\mathrm{2}})}({\mathbb{R}}^n)$ to $L^{(q,\varphi )}({\mathbb{R}}^n).$

Remark 2   From the theorem above, we have the following several corollaries.

1) Take $\varphi (x,r)=r^{\lambda },\lambda \in (-n,\mathrm{0}),\psi (x,r)\equiv \mathrm{1}.$ Then we have the result for Morrey spaces $L^{(p_{\mathrm{1}},{\varphi }_{\mathrm{1}})}({\mathbb{R}}^n)=L^{(p_{\mathrm{1}},{\lambda }_{\mathrm{1}})}({\mathbb{R}}^n),L^{(p_{\mathrm{2}},{\varphi }_{\mathrm{2}})}$$({\mathbb{R}}^n)=L^{(p_{\mathrm{2}},{\lambda }_{\mathrm{2}})}({\mathbb{R}}^n),$ ${\epsilon }^{(\mathrm{1},\psi )}({\mathbb{R}}^n)=\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n).$ This case is known by Ding and Mei^[6].

2) Take $\varphi (x,r)=r^{-n},\psi (x,r)\equiv \mathrm{1}.$ Then $L^{(p_{\mathrm{1}},{\varphi }_{\mathrm{1}})}({\mathbb{R}}^n)=L^{p_{\mathrm{1}}}({\mathbb{R}}^n),$ $L^{(p_{\mathrm{2}},{\varphi }_{\mathrm{2}})}({\mathbb{R}}^n)=L^{p_{\mathrm{2}}}({\mathbb{R}}^n),$ ${\epsilon }^{(\mathrm{1},\psi )}({\mathbb{R}}^n)=$$\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n).$ This is the result obtained by Bényi et al^[14].

Remark 3   Under the assumption of Theorem 1, by Ref. [13], for all $f\in L^{(p_{\mathrm{1}},{\varphi }_{\mathrm{1}})}({\mathbb{R}}^n),$ $g\in L^{(p_{\mathrm{2}},{\varphi }_{\mathrm{2}})}({\mathbb{R}}^n),$ and $\mathrm{a}.\mathrm{e}{.}_{}x\in {\mathbb{R}}^n,$ then we get

$[b,I_{\alpha }{]}_i(f,g)(x)=\underset{\epsilon \to +\mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}{\int }_{({\left|x-y\right|}^{\mathrm{2}}+{\left|x-z\right|}^{\mathrm{2}}{)}^{\raisebox{1ex}{$\mathrm{1}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{2}$}\right.}>\epsilon }^{}\frac{(b(x)-b(y))\left|f(x)g(z)\right|}{{(\left|x-y\right|+\left|x-z\right|)}^{\mathrm{2}n-\alpha }}\mathrm{d}y\mathrm{d}z,$(14)

where $i=\mathrm{1,2}.$

The rest of this paper is organized as follows. In Section 1, we will recall Musielak-Orlicz spaces and establish some auxiliary lemmas. The proof of Theorem 1 will be given in Section 2. In Section 3, we discuss the compactness of the commutators of the m-linear fractional integral operator. Finally, we make some conventions on notation. Throughout this paper, we always use $C$ to denote a positive constant that is independent of the main parameters involved but whose value may differ from line to line. Constants with subscripts, such as $C_p$ ,are dependent on the subscripts. We denote $f\prec g$ if $f\le Cg,$ and $f~g$ if $f\prec g\prec f.$ For $\mathrm{1}\le p\le \mathrm{\infty },p\text{'}$ is the conjugate index of $p$, and $\frac{\mathrm{1}}{p}+\frac{\mathrm{1}}{p\text{'}}=\mathrm{1}{.}_{}{E}^C={\mathbb{R}}^n\backslash E$ is the complementary set of any measurable subset $E$ of ${\mathbb{R}}^n.$

1 Preliminaries

In order to obtain our result, we recall Young functions and Musielak-Orlicz spaces. Let ${\Phi }_Y$ be the set of all Young functions, see Ref. [15].

Let ${\Phi }_Y^{\upsilon }$ be the set of all $\Phi :{\mathbb{R}}^n\times [\mathrm{0},\mathrm{\infty }]\to [\mathrm{0},\mathrm{\infty }]$ such that $\Phi (x,\cdot )$ is a Young function for every $x\in {\mathbb{R}}^n,$ and that $\Phi (\cdot ,t)$ is measurable on ${\mathbb{R}}^n$ for every $t\in [\mathrm{0},\mathrm{\infty }].$

For $\Phi \in {\Phi }_Y^{\upsilon }$ and $x\in {\mathbb{R}}^n,$ let

${\Phi }^{-\mathrm{1}}(x,u)=\{\begin{array}{c}\begin{array}{l}\mathrm{i}\mathrm{n}\mathrm{f}\{t\ge \mathrm{0}:\Phi (x,t)>u\}{,}_{}u\in [\mathrm{0},\mathrm{\infty }),\\ \mathrm{\infty },u=\mathrm{\infty }.\end{array}\end{array}$

We also define the complementary function $\tilde{\Phi }:{\mathbb{R}}^n\times [\mathrm{0},\mathrm{\infty }]\to [\mathrm{0},\mathrm{\infty }]$ by

$\tilde{\Phi }(x,t)=\{\begin{array}{c}\begin{array}{l}\mathrm{s}\mathrm{u}\mathrm{p}\{tu-\Phi (x,u):u\in [\mathrm{0},\mathrm{\infty })\},t\in [\mathrm{0},\mathrm{\infty }),\\ \mathrm{\infty },t=\mathrm{\infty }.\end{array}\end{array}$

Definition 4   (Musielak-Orlicz space) For a function $\Phi \in {\Phi }_Y^{\upsilon }$, let

$L^{\Phi }({\mathbb{R}}^n)=\{f:{\int }_{{\mathbb{R}}^n}^{}\Phi (x,\epsilon \left|f(x)\right|\mathrm{d}x<\mathrm{\infty }{,}_{}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}\epsilon >\mathrm{0}\},$

${\Vert f\Vert }_{L^{\Phi }}=\mathrm{i}\mathrm{n}\mathrm{f}\left\{\lambda >\mathrm{0}:{\int }_{{\mathbb{R}}^n}^{}\Phi (x,\frac{\left|f(x)\right|}{\lambda })\mathrm{d}x\le \mathrm{1}\right\}.$

Then ${\Vert \cdot \Vert }_{L^{\Phi }}$ is a norm, which is called the Luxemburg-Nakano norm, and thereby $L^{\Phi }({\mathbb{R}}^n)$ is a Banach space.

Lemma 2^[12] Let $\mathrm{1}\le q<\mathrm{\infty }$ and $\varphi :{\mathbb{R}}^n\times (\mathrm{0},\mathrm{\infty })\to (\mathrm{0},\mathrm{\infty })$, assume that $\varphi$ is in $\varphi \in {\varsigma }^{\mathrm{d}\mathrm{e}\mathrm{c}}$ and satisfies (2) and (4). Then there exists a function ${\Phi }_{q,\varphi }\in {\Phi }_Y^{\upsilon }$ such that

$L^{{\Phi }_{q,\varphi }}({\mathbb{R}}^n)\subset L^{(q,\varphi )}({\mathbb{R}}^n),\mathrm{a}\mathrm{n}\mathrm{d}{\Vert f\Vert }_{L^{(q,\varphi )}}\le C{\Vert f\Vert }_{L^{{\Phi }_{q,\varphi }}},$

where ${\Phi }_{q,\varphi }(x,t)={\Phi }_{\varphi }(x,t^q),$ and $C$ is a positive constant independent of $f\in L^{{\Phi }_{q,\varphi }}({\mathbb{R}}^n).$

Remark 4^[12] By Lemma 2, we see that, for all balls $B,$ then ${\Vert {\chi }_B\Vert }_{L^{{\Phi }_{q,\varphi }}}\prec \frac{\mathrm{1}}{\varphi {(B)}^{\mathrm{1}/q}}.$

For a function $\rho :{\mathbb{R}}^n\times (\mathrm{0},\mathrm{\infty })\to (\mathrm{0},\mathrm{\infty }),$the generalized bilinear Hardy-Littlewood maximal operator is defined by

${\mathbf{{\rm M}}}_{\rho }(f,g)(x)=\underset{x\in B}{\mathrm{s}\mathrm{u}\mathrm{p}}\rho (B)\frac{\mathrm{1}}{{\left|B\right|}^{\mathrm{2}}}{\int }_B^{}\left|f(y)\right|\mathrm{d}y{\int }_B^{}\left|g(z)\right|\mathrm{d}z.$

Clearly, if $\rho \equiv \mathrm{1},$ then ${\mathbf{{\rm M}}}_{\rho }(f,g)(x)$ is the bilinear of the Hardy-Littlewood maximal operator $\mathbf{{\rm M}},$ and if $\rho (B)={\left|B\right|}^{\alpha /n},$ then ${\mathbf{{\rm M}}}_{\rho }(f,g)(x)$ is the fraction maximal operator ${\mathbf{{\rm M}}}_{\alpha }$ defined by

${\mathbf{{\rm M}}}_{\alpha }(f,g)(x)=\underset{x\in B}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{\mathrm{1}}{{\left|B\right|}^{\mathrm{2}-\alpha /n}}{\int }_B^{}\left|f(y)\right|\mathrm{d}y{\int }_B^{}\left|g(z)\right|\mathrm{d}z.$