Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multidimensional bilinear Hardy inequalities   Our goal in this paper is to find a characterization of $n$-dimensional
bilinear Hardy inequalities \begin{align*} \bigg\| \,\int_{B(0,\cdot)} f \cdot
\int_{B(0,\cdot)} g \,\bigg\|_{q,u,(0,\infty)} & \leq C \,
\|f\|_{p_1,v_1,{\mathbb R}^n} \, \|g\|_{p_2,v_2,{\mathbb R}^n}, \quad f,\,g \in
{\mathfrak M}^+ ({\mathbb R}^n), \end{align*} and \begin{align*} \bigg\|
\,\int_{\,^{^{\mathsf{c}}}\! B(0,\cdot)} f \cdot \int_{\,^{^{\mathsf{c}}}\!
B(0,\cdot)} g \,\bigg\|_{q,u,(0,\infty)} &\leq C \, \|f\|_{p_1,v_1,{\mathbb
R}^n} \, \|g\|_{p_2,v_2,{\mathbb R}^n}, \quad f,\,g \in {\mathfrak M}^+
({\mathbb R}^n), \end{align*} when $0 < q \le \infty$, $1 \le p_1,\,p_2 \le
\infty$ and $u$ and $v_1,\,v_2$ are weight functions on $(0,\infty)$ and
${\mathbb R}^n$, respectively.
  Since the solution of the first inequality can be obtained from the
characterization of the second one by usual change of variables we concentrate
our attention on characterization of the latter. The characterization of this
inequality is easily obtained for the range of parameters when $p_1 \le q$
using the characterizations of multidimensional weighted Hardy-type inequalites
while in the case when $q < p_1$ the problem is reduced to the solution of
multidimensional weighted iterated Hardy-type inequality.
  To achieve the goal, we characterize the validity of multidimensional
weighted iterated Hardy-type inequality $$
\left\|\left\|\int_{\,^{^{\mathsf{c}}}\!
B(0,\cdot)}h(z)dz\right\|_{p,u,(0,t)}\right\|_{q,\mu,(0,\infty)}\leq c
\|h\|_{\theta,v,(0,\infty)},~ h \in \mathfrak{M}^+({\mathbb R}^n) $$ where $0 <
p,\,q < +\infty$, $1 \leq \theta \le \infty$, $u\in {\mathcal W}(0,\infty)$, $v
\in {\mathcal W}({\mathbb R}^n)$ and $\mu$ is a non-negative Borel measure on
$(0,\infty)$.

18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
17
The categories of this paper are: q-fin
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int