Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Compactness Characterizations of Commutators on Ball Banach Function
  Spaces   Let $X$ be a ball Banach function space on ${\mathbb R}^n$. Let $\Omega$ be a
Lipschitz function on the unit sphere of ${\mathbb R}^n$,which is homogeneous
of degree zero and has mean value zero, and let $T_\Omega$ be the convolutional
singular integral operator with kernel $\Omega(\cdot)/|\cdot|^n$. In this
article, under the assumption that the Hardy--Littlewood maximal operator
$\mathcal{M}$ is bounded on both $X$ and its associated space, the authors
prove that the commutator $[b,T_\Omega]$ is compact on $X$ if and only if
$b\in{\rm CMO}({\mathbb R}^n)$. To achieve this, the authors mainly employ
three key tools: some elaborate estimates, given in this article, on the norm
in $X$ of the commutators and the characteristic functions of some measurable
subset,which are implied by the assumed boundedness of ${\mathcal M}$ on $X$
and its associated space as well as the geometry of $\mathbb R^n$; the complete
John--Nirenberg inequality in $X$ obtained by Y. Sawano et al.; the generalized
Fr\'{e}chet--Kolmogorov theorem on $X$ also established in this article. All
these results have a wide range of applications. Particularly, even when
$X:=L^{p(\cdot)}({\mathbb R}^n)$ (the variable Lebesgue space),
$X:=L^{\vec{p}}({\mathbb R}^n)$ (the mixed-norm Lebesgue space),
$X:=L^\Phi({\mathbb R}^n)$ (the Orlicz space), and $X:=(E_\Phi^q)_t({\mathbb
R}^n)$ (the Orlicz-slice space or the generalized amalgam space), all these
results are new.

5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
15
The categories of this paper are: nucl-ex
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics