Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Necessary conditions for the boundedness of linear and bilinear
  commutators on Banach function spaces   In this article we extend recent results by the first author on the necessity
of $BMO$ for the boundedness of commutators on the classical Lebesgue spaces.
We generalize these results to a large class of Banach function spaces. We show
that with modest assumptions on the underlying spaces and on the operator $T$,
if the commutator $[b,T]$ is bounded, then the function $b$ is in $BMO$.

14
The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
6
The categories of this paper are: astro-ph
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
34
The categories of this paper are: chem-ph
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
32
The categories of this paper are: atom-ph
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol