Affiliation:
1. School of Mathematical and Statistics , Beijing Institute of Technology , Beijing 100081 , P. R. China
2. Department of Mathematics , University of Connecticut , Storrs , CT 06269 , USA
3. School of Mathematical Sciences , Beijing Normal University , Beijing 100875 , P. R. China
Abstract
Abstract
The purpose of this paper is four-fold.
First, we employ the reverse weighted Hardy inequality in the form of high dimensions to establish the following reverse Stein–Weiss inequality on the upper half space:
∫
ℝ
+
n
∫
∂
ℝ
+
n
|
x
|
α
|
x
-
y
|
λ
f
(
x
)
g
(
y
)
|
y
|
β
d
y
d
x
≥
C
n
,
α
,
β
,
p
,
q
′
∥
f
∥
L
q
′
(
ℝ
+
n
)
∥
g
∥
L
p
(
∂
ℝ
+
n
)
\int_{\mathbb{R}^{n}_{+}}\int_{\partial\mathbb{R}^{n}_{+}}\lvert x|^{\alpha}|x%
-y|^{\lambda}f(x)g(y)|y|^{\beta}\,dy\,dx\geq C_{n,\alpha,\beta,p,q^{\prime}}\|%
f\/\|_{L^{q^{\prime}}(\mathbb{R}^{n}_{+})}\|g\|_{L^{p}(\partial\mathbb{R}^{n}_%
{+})}
for any nonnegative functions
f
∈
L
q
′
(
ℝ
+
n
)
{f\in L^{q^{\prime}}(\mathbb{R}^{n}_{+})}
,
g
∈
L
p
(
∂
ℝ
+
n
)
{g\in L^{p}(\partial\mathbb{R}^{n}_{+})}
, and
p
,
q
′
∈
(
0
,
1
)
{p,q^{\prime}\in(0,1)}
,
β
<
1
-
n
p
′
{\beta<\frac{1-n}{p^{\prime}}}
or
α
<
-
n
q
{\alpha<-\frac{n}{q}}
,
λ
>
0
{\lambda>0}
satisfying
n
-
1
n
1
p
+
1
q
′
-
α
+
β
+
λ
-
1
n
=
2
.
\frac{n-1}{n}\frac{1}{p}+\frac{1}{q^{\prime}}-\frac{\alpha+\beta+\lambda-1}{n}%
=2.
Second, we show that the best constant of the above inequality can be
attained. Third, for a weighted system analogous to the Euler–Lagrange equations of the reverse Stein–Weiss inequality, we obtain
the necessary conditions of existence for any positive solutions using the Pohozaev identity. Finally, in view of the stereographic projection, we give
a spherical form of the Stein–Weiss inequality and reverse Stein–Weiss inequality on the upper half space
ℝ
+
n
{\mathbb{R}^{n}_{+}}
.
Funder
National Natural Science Foundation of China
Subject
General Mathematics,Statistical and Nonlinear Physics
Cited by
14 articles.
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