Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Revisit on Commutators of linear and bilinear Fractional Integral
  Operator   Let $I_{\alpha}$ be the linear and $\mathcal{I}_{\alpha}$ be the bilinear
fractional integral operators. In the linear setting, it is known that the
two-weight inequality holds for the first order commutators of $I_{\alpha}$.
But the method can't be used to obtain the two weighted norm inequality for the
higher order commutators of $I_{\alpha}$. In this paper, we first give an
alternative proof for the first order commutators of $I_{\alpha}$. This new
approach allows us to consider the higher order commutators. This was done by
showing that the commutator $[b,I_{\alpha}]$ can be represented as a finite
linear combination of some paraproducts. Then, by using the Cauchy integral
theorem, we show that the two-weight inequality holds for the higher order
commutators of $I_{\alpha}$. In the bilinear setting, we present a dyadic proof
for the characterization between $BMO$ and the boundedness of
$[b,\mathcal{I}_{\alpha}]$. Moreover, some bilinear paraproducts are also
treated in order to obtain the boundedness of $[b,\mathcal{I}_{\alpha}]$.

33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
19
The categories of this paper are: eess
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics