Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for ^{ }-weighted Hardy inequalities

• Published in: Transactions of the American Mathematical Society. Series B Vol. 5; no. 2; pp. 32 - 62
• Main Authors: Ruzhansky, Michael, Suragan, Durvudkhan, Yessirkegenov, Nurgissa
• Format: Journal Article
• Language: English
• Published: 14-02-2018
• ISSN: 2330-0000; 2330-0000
• DOI: 10.1090/btran/22

In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberg inequalities: we show that for
1
>
p
,
q
>
∞
1>p,q>\infty
,
0
>
r
>
∞
0>r>\infty
with
p
+
q
≥
r
p+q\geq r
,
δ
∈
[
0
,
1
]
∩
[
r
−
q
r
,
p
r
]
\delta \in [0,1]\cap \left [\frac {r-q}{r},\frac {p}{r}\right ]
with
δ
r
p
+
(
1
−
δ
)
r
q
=
1
\frac {\delta r}{p}+\frac {(1-\delta )r}{q}=1
and
a
a
,
b
b
,
c
∈
R
c\in \mathbb {R}
with
c
=
δ
(
a
−
1
)
+
b
(
1
−
δ
)
c=\delta (a-1)+b(1-\delta )
, and for all functions
f
∈
C
0
∞
(
R
n
∖
{
0
}
)
f\in C_{0}^{\infty }(\mathbb {R}^{n}\backslash \{0\})
we have
‖
|
x
|
c
f
‖
L
r
(
R
n
)
≤
|
p
n
−
p
(
1
−
a
)
|
δ
‖
|
x
|
a
∇
f
‖
L
p
(
R
n
)
δ
‖
|
x
|
b
f
‖
L
q
(
R
n
)
1
−
δ
\begin{equation*} \||x|^{c}f\|_{L^{r}(\mathbb {R}^{n})} \leq \left |\frac {p}{n-p(1-a)}\right |^{\delta } \left \||x|^{a}\nabla f\right \|^{\delta }_{L^{p}(\mathbb {R}^{n})} \left \||x|^{b}f\right \|^{1-\delta }_{L^{q}(\mathbb {R}^{n})} \end{equation*}
for
n
≠
p
(
1
−
a
)
n\neq p(1-a)
, where the constant
|
p
n
−
p
(
1
−
a
)
|
δ
\left |\frac {p}{n-p(1-a)}\right |^{\delta }
is sharp for
p
=
q
p=q
with
a
−
b
=
1
a-b=1
or
p
≠
q
p\neq q
with
p
(
1
−
a
)
+
b
q
≠
0
p(1-a)+bq\neq 0
. In the critical case
n
=
p
(
1
−
a
)
n=p(1-a)
we have
‖
|
x
|
c
f
‖
L
r
(
R
n
)
≤
p
δ
‖
|
x
|
a
log
⁡
|
x
|
∇
f
‖
L
p
(
R
n
)
δ
‖
|
x
|
b
f
‖
L
q
(
R
n
)
1
−
δ
.
\begin{equation*} \left \||x|^{c}f\right \|_{L^{r}(\mathbb {R}^{n})} \leq p^{\delta } \left \||x|^{a}\log |x|\nabla f\right \|^{\delta }_{L^{p}(\mathbb {R}^{n})} \left \||x|^{b}f\right \|^{1-\delta }_{L^{q}(\mathbb {R}^{n})}. \end{equation*}
Moreover, we also obtain anisotropic versions of these inequalities which can be conveniently formulated in the language of Folland and Stein’s homogeneous groups. Consequently, we obtain remainder estimates for
L
p
L^{p}
-weighted Hardy inequalities on homogeneous groups, which are also new in the Euclidean setting of
R
n
\mathbb {R}^{n}
. The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are obtained. Moreover, we investigate another improved version of
L
p
L^{p}
-weighted Hardy inequalities involving a distance and stability estimate. The relation between the critical and the subcritical Hardy inequalities on homogeneous groups is also investigated. We also establish sharp Hardy type inequalities in
L
p
L^{p}
,
1
>
p
>
∞
1>p>\infty
, with superweights, i.e., with the weights of the form
(
a
+
b
|
x
|
α
)
β
p
|
x
|
m
\frac {(a+b|x|^{\alpha })^{\frac {\beta }{p}}}{|x|^{m}}
allowing for different choices of
α
\alpha
and
β
\beta
. There are two reasons why we call the appearing weights the superweights: the arbitrariness of the choice of any homogeneous quasi-norm and a wide range of parameters.