Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for 𝐿^{𝑝}-weighted Hardy inequalities
Authors: Ruzhansky Michael , Suragan Durvudkhan , Yessirkegenov Nurgissa 
Published: February 2018

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar vertical-bar x StartAbsoluteValue f EndAbsoluteValue Superscript c Baseline Subscript upper L Sub Superscript r Subscript left-parenthesis double-struck upper R Sub Superscript n Subscript right-parenthesis Baseline less-than-or-equal-to StartAbsoluteValue StartFraction p Over n minus p left-parenthesis 1 minus a right-parenthesis EndFraction EndAbsoluteValue Superscript delta Baseline double-vertical-bar StartAbsoluteValue x EndAbsoluteValue Superscript a Baseline nabla f double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis double-struck upper R Sub Superscript n Subscript right-parenthesis Superscript delta Baseline double-vertical-bar StartAbsoluteValue x EndAbsoluteValue Superscript b Baseline f double-vertical-bar Subscript upper L Sub Superscript q Subscript left-parenthesis double-struck upper R Sub Superscript n Subscript right-parenthesis Superscript 1 minus delta">

‖

|

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar StartAbsoluteValue x EndAbsoluteValue Superscript c Baseline f double-vertical-bar Subscript upper L Sub Superscript r Subscript left-parenthesis double-struck upper R Sub Superscript n Subscript right-parenthesis Baseline less-than-or-equal-to p Superscript delta Baseline double-vertical-bar StartAbsoluteValue x EndAbsoluteValue Superscript a Baseline log StartAbsoluteValue x EndAbsoluteValue nabla f double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis double-struck upper R Sub Superscript n Subscript right-parenthesis Superscript delta Baseline double-vertical-bar StartAbsoluteValue x EndAbsoluteValue Superscript b Baseline f double-vertical-bar Subscript upper L Sub Superscript q Subscript left-parenthesis double-struck upper R Sub Superscript n Subscript right-parenthesis Superscript 1 minus delta Baseline period">

‖

|

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction left-parenthesis a plus b StartAbsoluteValue x EndAbsoluteValue Superscript alpha Baseline right-parenthesis Superscript StartFraction beta Over p EndFraction Baseline Over StartAbsoluteValue x EndAbsoluteValue Superscript m Baseline EndFraction">

(
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.