# Multidimensional bilinear Hardy inequalities

Research paper by **Nevin Bilgiçli, Rza Mustafayev, Tuğçe Ünver**

Indexed on: **17 May '18**Published on: **17 May '18**Published in: **arXiv - Mathematics - Functional Analysis**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### Abstract

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
$(0,\infty)$.