1 Introduction
Geometric and functional inequalities have a wide range of applications and play a crucial role in geometric analysis, partial differential equations and other branches of modern mathematics. In many situations, the validity of the inequality and some explicit bounds for its best constant are enough to run the process. However, there are numerous circumstances where we need to know the exact sharp constant and information on extremal functions.
Among those inequalities, the Caffarelli–Kohn–Nirenberg (CKN) inequality is one of the most important and interesting ones. It is worth noting that many well-known and important inequalities such as Gagliardo–Nirenberg inequalities, Sobolev inequalities, Hardy–Sobolev inequalities, Nash inequalities, etc. are just special cases of CKN inequalities.
The CKN inequalities were first introduced in 1984 by Caffarelli, Kohn and Nirenberg in their celebrated work [6].
Theorem A.
There exists a positive constant C=C(N,r,p,q,γ,α,β) such that for all u∈C0∞(RN),
(1.1)
∥
|
x
|
γ
u
∥
r
≤
C
∥
|
x
|
α
|
∇
u
|
∥
p
a
∥
|
x
|
β
u
∥
q
1
-
a
,
where
p
,
q
≥
1
,
r
>
0
,
0
≤
a
≤
1
,
1
p
+
α
N
,
1
q
+
β
N
,
1
r
+
γ
N
>
0
,
γ
=
a
σ
+
(
1
-
a
)
β
,
1
r
+
γ
N
=
a
(
1
p
+
α
-
1
N
)
+
(
1
-
a
)
(
1
q
+
β
N
)
and
0
≤
α
-
σ
if
a
>
0
,
α
-
σ
≤
1
if
a
>
0
and
1
p
+
α
-
1
N
=
1
r
+
γ
N
.
If we perform, as in [37], the change of exponents
α
=
-
μ
p
,
β
=
-
θ
q
,
γ
=
-
s
r
,
then (1.1) can be written in the following equivalent form:
(1.2)
(
∫
ℝ
N
|
u
|
r
d
x
|
x
|
s
)
1
r
≤
C
(
∫
ℝ
N
|
∇
u
|
p
d
x
|
x
|
μ
)
a
p
(
∫
ℝ
N
|
u
|
q
d
x
|
x
|
θ
)
1
-
a
q
,
where
a
=
[
(
N
-
θ
)
r
-
(
N
-
s
)
q
]
p
[
(
N
-
θ
)
p
-
(
N
-
μ
-
p
)
q
]
r
.
In this paper, we will restrict our consideration to the case 1<p<N.
When s=μ=θ=0 and a=1, we recover the well-known Sobolev inequality
(
∫
ℝ
N
|
u
|
p
∗
d
x
)
1
p
∗
≤
S
(
N
,
p
)
(
∫
ℝ
N
|
∇
u
|
p
d
x
)
1
p
,
where p∗=NpN-p. When p>1 the best constant S(N,p) was found in the works of Aubin [3] and Talenti [34], using rather classical tools such as Schwartz rearrangement. The case p=2 was explored more by Beckner in [4], due to its conformal invariance. For p=1, it has been known that the Sobolev inequality is equivalent to the classical Euclidean isoperimetric inequality.
When a=1, μ=0, 0≤s≤p<N and r=p∗(s)=N-sN-pp, the CKN inequality becomes the Hardy–Sobolev (HS) inequality:
(1.3)
(
∫
ℝ
N
|
u
|
p
∗
(
s
)
d
x
|
x
|
s
)
1
p
∗
(
s
)
≤
HS
(
N
,
p
,
s
)
(
∫
ℝ
N
|
∇
u
|
p
d
x
)
1
p
.
In this situation, Lieb in [26] applied the symmetrization argument to study (1.3) in the case p=2 and gave the best constants and explicit maximizers. The study of the best constant HS(N,p,s) and extremal functions for inequalities (1.3) in the general range goes back to Ghoussoub and Yuan [25] and maybe even earlier (see the references in [25]). The maximizers for the HS inequality when 0≤s<p<N are the functions
u
c
,
λ
(
x
)
=
c
(
λ
+
|
x
|
p
-
s
p
-
1
)
-
N
-
p
p
-
s
for some
c
≠
0
,
λ
>
0
.
Actually, uc,λ (after rescaling) is the only positive radial solution of
-
div
(
|
∇
u
|
p
-
2
∇
u
)
=
u
p
∗
(
s
)
-
1
|
x
|
s
on
ℝ
N
.
When a=1 and 0<μ,s<N, the CKN inequality does not contain the interpolation term. There are great efforts to investigate the sharp constants, existence/nonexistence and symmetry/symmetry breaking of maximizers in this situation, especially when p=2. See [5, 9, 12, 21, 35], among others. For instance, Chou and Chu [12] considered the case p=2 and μ2≤sr≤μ2+1, and provided the best constants and explicit optimizers. In [35], Wang and Willem studied the compactness of all maximizing sequences up to dilations in the spirit of Lions [28, 29, 30, 31]. In [9], Catrina and Wang investigated the class of p=2 and μ<0, and established the attainability/unattainability and symmetry breaking of extremal functions. In [7], Caldiroli and Musina studied the symmetry breaking of extremals for CKN inequalities in a non-Hilbertian setting. In a recent paper, [16], Dolbeault, Esteban and Loss studied the characterization of the optimal symmetry breaking region in HS inequalities with p=2. As a consequence, maximizers and best constants are calculated in the symmetry region. Their result solves a longstanding conjecture on the optimal symmetry range.
In the case 0<a<1, the CKN inequality includes the interpolation term. This situation is much harder to study. When there is no singular term, i.e., s=θ=μ=0, the nonweighted CKN inequality, namely, the Gagliardo–Nirenberg inequality, has been studied at length by many authors. Especially, for very particular classes, the best constant and the maximizers for the Gagliardo–Nirenberg inequality are provided explicitly by Del Pino and Dolbeault in [14, 15]. Indeed, in the special class r=pq-1p-1, Del Pino and Dolbeault proved that the maximizers for the Gagliardo–Nirenberg inequality have the form A(1+B|x-x¯|pp-1)-p-1q-p, while in the case q=pr-1p-1, the maximizers are A(1-B|x-x¯|pp-1)+-p-1r-p for some A∈ℝ, B>0 and x¯∈ℝN. See also [1, 2], where Agueh gives a proof by studying a p-Laplacian type equation by transforming the unknown of the equation via some change of functions. We also cite [13] where Cordero-Erausquin, Nazaret, and Villani set up a beautiful link between optimal transportation and certain Sobolev inequalities and Gagliardo–Nirenberg inequalities.
However, as far as we know, there are only a few papers concerning the full weighted CKN inequalities (i.e., 0<a<1 and at least one of s,μ,θ is nonzero). Compared with the special cases of Gagliardo–Nirenberg inequalities without the interpolation term (i.e., a=1), dealing with such CKN inequalities is considerably more difficult. For instance, the Fourier analysis techniques cannot be applied in this setting. Moreover, the classical Schwarz rearrangement, which is based on an isoperimetric inequality, is unavailable due to the presence of singular terms (i.e., the weights 1|x|s, 1|x|θ and 1|x|μ). It is worth noting that symmetrization has been a very useful and efficient (and almost inevitable) method when dealing with the sharp geometric inequalities. Hence, in general, we are not able to reduce our problem on CKN inequalities to a radial setting. Actually, the problem of symmetry and symmetry breaking of optimizers for CKN inequalities has been investigated by many researchers, see, for instance, [7, 10, 11, 16].
Concerning inequality (1.2), for the special class
q
=
p
(
r
-
1
)
p
-
1
,
1
<
p
<
r
,
N
-
θ
<
(
1
+
μ
p
-
θ
p
)
(
r
-
1
)
p
r
-
p
,
s
=
μ
p
+
1
+
p
-
1
p
θ
,
Xia could guess and then verify in [36] that (λ+|x|1+μp-θp)-p-1r-p, λ>0, are extremal functions. But he could not prove that these are all possible optimizers. Moreover, this case does not cover the interesting situations in [14, 15].
In [8], Catrina and Costa studied best constants and explicit optimizers for the CKN inequality when p=q=r=2, μ=2a, θ=2b and s=a+b+1. They were able to show that when a<b+1, b≤N-22, or a>b+1, b≥N-22, the optimizers are of the form
D
exp
(
t
|
x
|
b
+
1
-
a
b
+
1
-
a
)
,
while if a>b+1, b≤N-22, or a<b+1, b≥N-22, the extremal functions are
D
|
x
|
2
(
b
+
1
)
-
N
exp
(
t
|
x
|
b
+
1
-
a
b
+
1
-
a
)
.
We note that the case a=b+1 was treated in [10]. More precisely, in this case, the best constant is 2|N-s(b+1)|, and is not achieved.
In [37], Zhong and Zou studied the existence of extremal functions for the CKN inequality under a wider region, and used it to set up the continuity and compactness of embeddings on weighted Sobolev spaces. However, there is no information about the maximizers provided there.
In a very recent paper, [17], Dolbeault, Muratori and Nazaret studied the CKN inequality in the regime s=θ>0, p=2 and r=2(q-1)>2. In this case, they were able to show that for s=θ>0 small enough, the CKN inequality can be achieved by optimizers of the form (1+|x|2-s)-1q-2, up to multiplications by a constant and scalings.
In [18, 19], when dealing with the sharp singular Trudinger–Moser inequalities, which can be considered as limiting Sobolev embeddings, where again the classical Schwarz rearrangement could not be used, the authors in collaboration with Dong proposed a new approach. Namely, we used suitable quasi-conformal mappings to convert those sharp singular weighted Trudinger–Moser inequalities to the nonweighted ones. (We will not discuss in detail weighted Trudinger–Moser inequalities here, but we refer the interested reader to, e.g., [32] for more references on weighted Trudinger–Moser inequalities for singular weights.) Moreover, in [19], we established the existence of the optimizers for weighted Trudinger–Moser inequalities for all functions which are not necessarily spherically symmetric by using this type of quasi-conformal mapping to reduce to the case of inequalities for spherically symmetric ones, and in [18], we treated successfully CKN inequalities in the special case p=N, μ=0, 0≤s=θ<N, 1≤q<r and a=1-qr, using this new transform. Especially, for a 1-parameter family of inequalities, the best constants and the maximizers for the CKN inequality are calculated explicitly there.
Motivated by the results in [18, 19] and [1, 2, 13, 15, 17], in this paper, we will use convenient vector fields to investigate the CKN inequality in some special regions. Our main idea is that under our suitable transforms, CKN inequalities can be converted to simpler versions, namely, the Hardy–Sobolev inequalities and the Gagliardo–Nirenberg inequalities. Since the sharp constants and optimizers of those inequalities are easier to study, and are known in some particular classes, we can get the best constants and maximizers for CKN inequalities in the corresponding regions.
More precisely, we study the extremal functions for the CKN inequality involving the interpolation term (i.e., 0<a<1). We will consider the following class:
(C1)
{
1
<
p
<
p
+
μ
<
N
,
θ
≤
N
μ
N
-
p
≤
s
<
N
,
1
≤
q
<
r
<
N
p
N
-
p
,
a
=
[
(
N
-
θ
)
r
-
(
N
-
s
)
q
]
p
[
(
N
-
θ
)
p
-
(
N
-
μ
-
p
)
q
]
r
.
We denote by Dμ,θp,q(ℝN) the completion of the space of smooth compactly supported functions with the norm
(
∫
ℝ
N
|
∇
u
|
p
d
x
|
x
|
μ
)
1
p
+
(
∫
ℝ
N
|
u
|
q
d
x
|
x
|
θ
)
1
q
,
and set
(1.4)
CKN
(
N
,
μ
,
θ
,
s
,
p
,
q
,
r
)
=
sup
u
∈
D
μ
,
θ
p
,
q
(
ℝ
N
)
(
∫
ℝ
N
|
u
|
r
d
x
|
x
|
s
)
1
r
(
∫
ℝ
N
|
∇
u
|
p
d
x
|
x
|
μ
)
a
p
(
∫
ℝ
N
|
u
|
q
d
x
|
x
|
θ
)
1
-
a
q
.
Then we have the following result.
Theorem 1.1.
Assume that (C1) holds. Then CKN(N,μ,θ,s,p,q,r) can be achieved. Moreover, all the extremal functions of CKN(N,μ,θ,s,p,q,r) are radially symmetric.
We will also give the explicit forms for all maximizers and the exact best constant for CKN(N,μ,θ,s,p,q,r) in the following special cases.
Theorem 1.2.
Assume that (C1) holds with θ=s=NμN-p. If p<r=pq-1p-1<NpN-p, then, for δ=Np-q(N-p), we have
CKN
(
N
,
μ
,
θ
,
s
,
p
,
q
,
r
)
=
(
N
-
p
N
-
p
-
μ
)
1
r
+
p
-
1
p
-
1
-
a
q
-
p
-
1
p
(
1
-
a
)
(
q
-
p
p
π
)
a
(
p
q
N
(
q
-
p
)
)
a
p
(
δ
p
q
)
1
r
(
Γ
(
q
p
-
1
q
-
p
)
Γ
(
N
2
+
1
)
Γ
(
p
-
1
p
δ
q
-
p
)
Γ
(
N
p
-
1
p
+
1
)
)
a
N
,
and all the maximizers have the form
V
0
(
x
)
=
A
(
1
+
B
|
x
|
N
-
p
-
μ
N
-
p
p
p
-
1
)
-
p
-
1
q
-
p
for some
A
∈
ℝ
,
B
>
0
.
Theorem 1.3.
Assume that (C1) holds with θ=s=NμN-p. If 1<q=pr-1p-1<p, then, for δ=Np-r(N-p), we have
CKN
(
N
,
μ
,
θ
,
s
,
p
,
q
,
r
)
=
(
N
-
p
N
-
p
-
μ
)
1
r
+
p
-
1
p
-
1
-
a
q
-
p
-
1
p
(
1
-
a
)
(
p
-
r
p
π
)
a
(
p
r
N
(
p
-
r
)
)
a
p
(
p
r
δ
)
1
-
a
q
(
Γ
(
p
-
1
p
δ
p
-
r
+
1
)
Γ
(
N
2
+
1
)
Γ
(
r
p
-
1
p
-
r
+
1
)
Γ
(
N
p
-
1
p
+
1
)
)
a
N
.
If r>2-1p, then all the maximizers have the form
V
0
(
x
)
=
A
(
1
-
B
|
x
|
N
-
p
-
μ
N
-
p
p
p
-
1
)
+
-
p
-
1
r
-
p
for some
A
∈
ℝ
,
B
>
0
.
We also provide the explicit optimizers for CKN inequalities in the following regime:
(C2)
{
p
=
2
<
2
+
μ
<
N
,
2
<
r
=
2
(
q
-
1
)
<
2
N
N
-
2
,
μ
+
2
>
s
=
θ
>
N
μ
N
-
2
,
a
=
(
N
-
s
)
[
q
-
2
]
[
(
N
-
s
)
2
-
(
N
-
μ
-
2
)
q
]
(
q
-
1
)
.
Again, we define
(1.5)
CKN
(
N
,
μ
,
s
,
q
)
=
sup
u
∈
D
μ
,
s
2
,
q
(
ℝ
N
)
(
∫
ℝ
N
|
u
|
2
(
q
-
1
)
d
x
|
x
|
s
)
1
2
(
q
-
1
)
(
∫
ℝ
N
|
∇
u
|
2
d
x
|
x
|
μ
)
a
2
(
∫
ℝ
N
|
u
|
q
d
x
|
x
|
s
)
1
-
a
q
.
We will prove the following theorem.
Theorem 1.4.
There exists s∗=s∗(N,q,μ)∈(0,N-(q-1)(N-2-μ)) such that CKN(N,μ,s,q) is attained for all NμN-2<s<s∗ by optimizers of the form
V
0
(
x
)
=
A
(
1
+
B
|
x
|
μ
+
2
-
s
)
-
1
q
-
2
for some
A
∈
ℝ
,
B
>
0
.
2 Preliminaries and Some Important Lemmata
To carry through our argument, it is necessary to show that our quasi-conformal changes of variable can indeed be used to reduce CKN inequalities with more complicated weights to simpler ones and vice versa. This interchange is nicely done through the following lemmata which are of independent interests and can be found useful in other settings as well.
Lemma 2.1.
We have that |x⋅∇u(x)|=|x||∇u(x)| for a.e. x∈RN if and only if u is radially symmetric, that is, u(x)=u(y) when |x|=|y|.
Proof.
If u is radial, then we have
∂
u
∂
x
j
(
x
)
=
u
′
(
|
x
|
)
x
j
|
x
|
.
Hence, |∇u(x)|=|u′(|x|)|. Also,
|
∑
j
=
1
N
x
j
∂
u
∂
x
j
(
x
)
|
=
|
u
′
(
|
x
|
)
|
|
∑
j
=
1
N
x
j
x
j
|
x
|
|
=
|
u
′
(
|
x
|
)
|
|
x
|
=
|
x
|
|
∇
u
(
x
)
|
.
Now, assume that |x⋅∇u(x)|=|x||∇u(x)| for all x. This means that ∇u(x) has the same direction with x. That is, we can find a scalar function g(x) such that ∇u(x)=g(x)x. Now, let a and b be two points on the sphere with radius r>0 (that is, |a|=|b|=r). We connect x and y by a piecewise smooth curve r(t) on this sphere, i.e., |r(t)|=r, r(0)=a and r(1)=b. Then we have ∇u(r(t))=g(r(t))r(t). Note that since |r(t)|=r for all t, we can get that ∇r(t)⋅r(t)=0. Thus,
∇
u
(
r
(
t
)
)
⋅
∇
r
(
t
)
=
g
(
r
(
t
)
)
r
(
t
)
⋅
∇
r
(
t
)
=
0
.
So,
u
(
b
)
-
u
(
a
)
=
u
(
r
(
1
)
)
-
u
(
r
(
0
)
)
=
∫
0
1
∇
u
(
r
(
t
)
)
⋅
∇
r
(
t
)
𝑑
t
=
0
.
This completes the proof of the lemma. ∎
Let d>0. We define the vector-valued function LN,d:ℝN→ℝN by
L
N
,
d
(
x
)
=
|
x
|
d
-
1
x
.
This is a quasi-conformal mapping type of transform, which was used earlier in [20] to establish weighted Poincaré and Sobolev type inequalities for powers of the Jacobian of a quasi-conformal mapping (these are not necessarily of Muckenhoupt Ap type weights), in particular, for appropriate power weights. The results of [20] have been subsequently extended in a greater generality to weighted Sobolev inequalities with a product of power weights by Gatto and Wheeden [22], and then further by Chanillo and Wheeden for weighted Poincaré inequalities [10]. We note that the best constants and maximizers of the inequalities were not of concern in the aforementioned works. The determinant of the Jacobian of this type of map was already calculated in the literature (see, e.g., [23, 20]). Since the calculation is quite elementary, we include below another elementary and simple way of calculation by using the characteristic polynomials of a matrix for the reader’s convenience.
The Jacobian matrix of the function LN,d is
𝐉
L
N
,
d
=
|
x
|
d
-
1
𝐈
N
+
𝐀
,
where
𝐀
=
(
(
d
-
1
)
|
x
|
d
-
3
x
1
2
(
d
-
1
)
|
x
|
d
-
3
x
1
x
2
…
(
d
-
1
)
|
x
|
d
-
3
x
1
x
N
(
d
-
1
)
|
x
|
d
-
3
x
2
x
1
(
d
-
1
)
|
x
|
d
-
3
x
2
2
…
(
d
-
1
)
|
x
|
d
-
3
x
2
x
N
⋮
⋮
⋱
⋮
(
d
-
1
)
|
x
|
d
-
3
x
N
x
1
(
d
-
1
)
|
x
|
d
-
3
x
N
x
2
…
(
d
-
1
)
|
x
|
d
-
3
x
N
2
)
.
It is easy to check that
rank
(
𝐀
)
=
1
and
tr
(
𝐀
)
=
(
d
-
1
)
|
x
|
d
-
1
.
Hence, its characteristic polynomial is
det
(
λ
𝐈
N
-
𝐀
)
=
λ
N
-
(
d
-
1
)
|
x
|
d
-
1
λ
N
-
1
.
For λ=-|x|d-1, we get det(𝐉LN,d)=d|x|N(d-1).
We now define the mapping DN,d,p, with p>1, by
D
N
,
d
,
p
u
(
x
)
:=
(
1
d
)
p
-
1
p
u
(
L
N
,
d
(
x
)
)
=
(
1
d
)
p
-
1
p
u
(
|
x
|
d
-
1
x
)
.
We also define
D
N
,
d
,
p
-
1
u
=
v
if
u
=
D
N
,
d
,
p
v
.
Under the transform DN,d,p, we have the following result that will play an important role in our paper. This is a powerful replacement of the classical Polyá–Szegö inequality in the weighted case with power weights.
Lemma 2.2.
For a continuous function
f
, we have
∫
ℝ
N
f
(
(
1
d
)
p
-
1
p
u
(
x
)
)
|
x
|
t
𝑑
x
=
d
∫
ℝ
N
f
(
D
N
,
d
,
p
u
(
x
)
)
|
x
|
N
+
t
d
-
N
d
𝑑
x
.
In particular, we obtain that
u
∈
L
s
(
d
x
/
|
x
|
t
)
if and only if
D
N
,
d
,
p
u
∈
L
s
(
d
x
/
|
x
|
N
+
t
d
-
N
d
)
.
For
d
>
1
, if
∇
u
∈
L
p
(
d
x
/
|
x
|
μ
)
, then
∇
D
N
,
d
,
p
u
∈
L
p
(
d
x
/
|
x
|
d
(
p
+
μ
-
N
)
+
N
-
p
)
. Moreover,
∫
ℝ
N
|
∇
D
N
,
d
,
p
u
(
x
)
|
p
|
x
|
d
(
p
+
μ
-
N
)
+
N
-
p
𝑑
x
≤
∫
ℝ
N
|
∇
u
(
x
)
|
p
|
x
|
μ
𝑑
x
.
The equality occurs if and only if
u
is radially symmetric.
Proof.
(1) We have
∫
ℝ
N
f
(
D
N
,
d
,
p
u
(
x
)
)
|
x
|
N
+
t
d
-
N
d
𝑑
x
=
∫
ℝ
N
f
(
(
1
d
)
p
-
1
p
u
(
|
x
|
d
-
1
x
)
)
|
x
|
N
+
t
d
-
N
d
𝑑
x
.
Using the change of variables yi=|x|d-1xi, i=1,2,…,N, we have
d
y
=
det
(
𝐉
L
N
,
d
)
d
x
=
d
|
x
|
N
(
d
-
1
)
d
x
and
d
x
=
1
d
d
y
|
y
|
N
d
-
1
d
.
Hence,
∫
ℝ
N
f
(
D
N
,
d
,
p
u
(
x
)
)
|
x
|
N
+
t
d
-
N
d
𝑑
x
=
1
d
∫
ℝ
N
f
(
(
1
d
)
p
-
1
p
u
(
y
)
)
|
y
|
N
d
-
1
d
|
y
|
N
+
t
d
-
N
d
d
𝑑
y
=
1
d
∫
ℝ
N
f
(
(
1
d
)
p
-
1
p
u
(
y
)
)
|
y
|
t
𝑑
y
.
(2) Now we begin to consider the gradient of DN,d,pu. After calculations, we have
(
∂
D
N
,
d
,
p
u
∂
x
1
(
x
)
∂
D
N
,
d
,
p
u
∂
x
2
(
x
)
⋮
∂
D
N
,
d
,
p
u
∂
x
N
(
x
)
)
=
∇
D
N
,
d
,
p
u
(
x
)
=
(
1
d
)
p
-
1
p
∇
(
u
(
|
x
|
d
-
1
x
)
)
=
(
1
d
)
p
-
1
p
𝐉
L
N
,
d
T
(
∂
u
∂
x
1
(
|
x
|
d
-
1
x
)
∂
u
∂
x
2
(
|
x
|
d
-
1
x
)
⋮
∂
u
∂
x
N
(
|
x
|
d
-
1
x
)
)
.
Hence, we have
∂
D
N
,
d
,
p
u
∂
x
i
(
x
)
=
(
1
d
)
p
-
1
p
(
|
x
|
d
-
1
∂
u
∂
x
i
(
|
x
|
d
-
1
x
)
+
A
i
)
for i=1,2,…,N, where
A
i
:=
∑
j
=
1
N
(
d
-
1
)
|
x
|
d
-
3
x
i
x
j
∂
u
∂
x
j
(
|
x
|
d
-
1
x
)
.
Hence, we obtain
|
∇
D
N
,
d
,
p
u
(
x
)
|
2
=
∑
i
=
1
N
(
∂
D
N
,
d
,
p
u
∂
x
i
(
x
)
)
2
=
d
-
2
p
-
1
p
∑
i
=
1
N
(
|
x
|
d
-
1
∂
u
∂
x
i
(
|
x
|
d
-
1
x
)
+
A
i
)
2
=
d
-
2
p
-
1
p
[
∑
i
=
1
N
|
x
|
2
(
d
-
1
)
(
∂
u
∂
x
i
(
|
x
|
d
-
1
x
)
)
2
+
∑
i
=
1
N
2
A
i
|
x
|
d
-
1
∂
u
∂
x
i
(
|
x
|
d
-
1
x
)
+
∑
i
=
1
N
A
i
2
]
=:
d
-
2
p
-
1
p
(
I
1
+
I
2
+
I
3
)
.
Direct computations show
I
1
=
∑
i
=
1
N
|
x
|
2
(
d
-
1
)
(
∂
u
∂
x
i
(
|
x
|
d
-
1
x
)
)
2
=
|
x
|
2
(
d
-
1
)
|
∇
u
(
|
x
|
t
N
-
t
x
)
|
2
.
By applying the Cauchy–Schwarz inequality to estimate the second term, we get
I
2
=
∑
i
=
1
N
2
A
i
|
x
|
d
-
1
∂
u
∂
x
i
(
|
x
|
d
-
1
x
)
=
∑
i
=
1
N
2
|
x
|
d
-
1
∂
u
∂
x
i
(
|
x
|
d
-
1
x
)
∑
j
=
1
N
(
d
-
1
)
|
x
|
d
-
3
x
i
x
j
∂
u
∂
x
j
(
|
x
|
d
-
1
x
)
=
2
(
d
-
1
)
|
x
|
2
d
-
2
∑
i
=
1
N
∑
j
=
1
N
x
i
x
j
|
x
|
2
∂
u
∂
x
j
(
|
x
|
d
-
1
x
)
∂
u
∂
x
i
(
|
x
|
d
-
1
x
)
=
2
(
d
-
1
)
|
x
|
2
d
-
2
(
∑
i
=
1
N
x
i
|
x
|
∂
u
∂
x
i
(
|
x
|
d
-
1
x
)
)
2
≤
2
(
d
-
1
)
|
x
|
2
d
-
2
[
∑
i
=
1
N
(
x
i
|
x
|
)
2
]
[
∑
i
=
1
N
(
∂
u
∂
x
i
(
|
x
|
d
-
1
x
)
)
2
]
=
2
(
d
-
1
)
|
x
|
2
d
-
2
|
∇
u
(
|
x
|
d
-
1
x
)
|
2
.
Similarly, for the last term, we have
I
3
=
∑
i
=
1
N
A
i
2
=
∑
i
=
1
N
(
∑
j
=
1
N
(
d
-
1
)
|
x
|
d
-
3
x
i
x
j
∂
u
∂
x
j
(
|
x
|
d
-
1
x
)
)
2
≤
(
d
-
1
)
2
|
x
|
2
d
-
6
∑
i
=
1
N
[
∑
j
=
1
N
(
x
i
x
j
)
2
]
[
∑
j
=
1
N
(
∂
u
∂
x
j
(
|
x
|
t
N
-
t
x
)
)
2
]
=
(
d
-
1
)
2
|
x
|
2
d
-
6
∑
i
=
1
N
|
x
|
2
x
i
2
|
∇
u
(
|
x
|
d
-
1
x
)
|
2
=
(
d
-
1
)
2
|
x
|
2
d
-
2
|
∇
u
(
|
x
|
d
-
1
x
)
|
2
.
Combining them together, we have
|
∇
D
N
,
d
,
p
u
(
x
)
|
2
≤
d
-
2
p
-
1
p
d
2
|
x
|
2
d
-
2
|
∇
u
(
|
x
|
d
-
1
x
)
|
2
.
This leads to
|
∇
D
N
,
d
,
p
u
(
x
)
|
≤
d
1
p
|
x
|
d
-
1
|
∇
u
(
|
x
|
d
-
1
x
)
|
.
Using the change of variables again, we get
∫
ℝ
N
|
∇
u
(
y
)
|
p
|
y
|
μ
𝑑
y
=
∫
ℝ
N
|
∇
u
(
|
x
|
d
-
1
x
)
|
p
|
|
x
|
d
-
1
x
|
μ
d
|
x
|
N
(
d
-
1
)
𝑑
x
≥
1
d
∫
ℝ
N
|
∇
D
N
,
d
,
p
u
(
x
)
|
p
|
x
|
p
(
d
-
1
)
|
|
x
|
d
-
1
x
|
μ
d
|
x
|
N
(
d
-
1
)
𝑑
x
=
∫
ℝ
N
|
∇
D
N
,
d
,
p
u
(
x
)
|
p
|
x
|
d
(
p
+
μ
-
N
)
+
N
-
p
𝑑
x
.
Finally, by Lemma 2.1, it is easy to check that the equalities hold if and only if u is radial. ∎
3 The CKN Inequality when 0<a<1 Under Condition (C1)
Theorems 1.1–1.3 will be proved via the following series of lemmata. Recall that the conditions on the parameters are given in (C1), and CKN(N,μ,θ,s,p,q,r) is defined in (1.4). We now set
GN
(
N
,
p
,
q
,
r
,
μ
,
θ
,
s
)
=
sup
u
∈
D
0
,
N
+
θ
d
-
N
d
p
,
q
(
ℝ
N
)
(
∫
ℝ
N
|
u
|
r
|
x
|
N
+
s
d
-
N
d
𝑑
x
)
1
r
(
∫
ℝ
N
|
∇
u
|
p
d
x
)
a
p
(
∫
ℝ
N
|
u
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
)
1
-
a
q
,
where
d
=
N
-
p
N
-
p
-
μ
.
It is important to note here that since θ≤NμN-p≤s<N, we have
N
+
θ
d
-
N
d
≤
0
≤
N
+
s
d
-
N
d
<
N
.
Lemma 3.1.
The variational problem
A
(
N
,
p
,
q
,
r
,
μ
,
θ
,
s
)
=
inf
{
I
(
u
)
=
1
p
∫
ℝ
N
|
∇
u
|
p
d
x
+
1
q
∫
ℝ
N
|
u
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
:
u
∈
D
0
,
N
+
θ
d
-
N
d
p
,
q
(
ℝ
N
)
and
∫
ℝ
N
|
u
|
r
|
x
|
N
+
s
d
-
N
d
d
x
=
1
}
has a minimizer. Moreover, A(N,p,q,r,μ,θ,s)>0.
Proof.
By the classical Schwarz rearrangement, we can assume that there exists a sequence of radial functions (un) such that
I
(
u
n
)
↓
A
(
N
,
p
,
q
,
r
,
μ
,
θ
,
s
)
and
∫
ℝ
N
|
u
n
|
r
|
x
|
N
+
s
d
-
N
d
𝑑
x
=
1
.
We can assume, without loss of generality, that un⇀u in u∈D0,N+θd-Ndp,q(ℝN). Since it is evident that I(u)≤A(N,p,q,r,μ,θ,s), it is enough to show
∫
ℝ
N
|
u
|
r
|
x
|
N
+
s
d
-
N
d
𝑑
x
=
1
.
But this is easy to observe since for R>0 sufficiently large, we can write
∫
ℝ
N
|
u
n
-
u
|
r
|
x
|
N
+
s
d
-
N
d
𝑑
x
=
∫
B
R
+
∫
B
R
c
|
u
n
-
u
|
r
|
x
|
N
+
s
d
-
N
d
𝑑
x
.
Then, by the radial lemma, we get
∫
B
R
c
|
u
n
-
u
|
r
|
x
|
N
+
s
d
-
N
d
𝑑
x
→
0
.
Also, by the compactness of Sobolev embeddings, we can deduce
∫
B
R
|
u
n
-
u
|
r
|
x
|
N
+
s
d
-
N
d
𝑑
x
→
0
.
As a consequence, u≠0 and A(N,p,q,r,μ,θ,s)>0. Moreover, noting that uλ(x)=λNd-sdru(λx) for λ>0, we have
∥
∇
u
λ
∥
p
=
λ
N
d
-
s
d
r
+
p
-
N
p
∥
∇
u
∥
p
,
∥
u
λ
∥
k
=
λ
N
d
-
s
d
r
-
N
k
∥
u
∥
k
and
∫
ℝ
N
|
u
λ
|
r
|
x
|
N
+
s
d
-
N
d
𝑑
x
=
1
.
Also,
I
(
u
λ
)
=
1
p
∫
ℝ
N
|
∇
u
λ
|
p
d
x
+
1
q
∫
ℝ
N
|
u
λ
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
=
1
p
λ
N
d
-
s
d
r
p
+
p
-
N
∥
∇
u
∥
p
p
+
1
q
λ
q
N
d
-
s
d
r
-
N
+
N
+
θ
d
-
N
d
∫
ℝ
N
|
u
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
=
λ
m
A
+
λ
-
n
B
,
with
m
=
N
d
-
s
d
r
p
+
p
-
N
,
n
=
N
d
-
θ
d
-
q
N
d
-
s
d
r
,
A
=
1
p
∥
∇
u
∥
p
p
,
B
=
1
q
∫
ℝ
N
|
u
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
.
Hence,
A
(
N
,
p
,
q
,
r
,
μ
,
θ
,
s
)
=
inf
λ
>
0
I
(
u
λ
)
=
I
(
u
λ
0
)
,
where
λ
0
=
(
n
B
m
A
)
1
m
+
n
.
This means that
A
(
N
,
p
,
q
,
r
,
μ
,
θ
,
s
)
=
m
+
n
m
(
n
m
)
-
n
m
+
n
A
n
m
+
n
B
m
m
+
n
.
∎
Lemma 3.2.
GN
(
N
,
p
,
q
,
r
)
can be achieved and
GN
(
N
,
p
,
q
,
r
,
μ
,
θ
,
s
)
=
[
m
+
n
m
(
n
m
)
-
n
m
+
n
(
1
p
)
n
m
+
n
(
1
q
)
m
m
+
n
A
(
N
,
p
,
q
,
r
,
μ
,
θ
,
s
)
]
a
/
p
n
/
(
m
+
n
)
.
Proof.
For any v with
∫
ℝ
N
|
v
|
r
|
x
|
N
+
s
d
-
N
d
𝑑
x
=
1
,
we use the above process and get
A
(
N
,
p
,
q
,
r
,
μ
,
θ
,
s
)
≤
inf
λ
>
0
I
(
v
λ
)
=
m
+
n
m
(
n
m
)
-
n
m
+
n
(
1
p
∥
∇
v
∥
p
p
)
n
m
+
n
(
1
q
∫
ℝ
N
|
u
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
)
m
m
+
n
.
Noting that n/(m+n)m/(m+n)=a/p(1-a)/q, we obtain
(
∫
ℝ
N
|
v
|
r
|
x
|
N
+
s
d
-
N
d
𝑑
x
)
1
r
(
∫
ℝ
N
|
∇
v
|
p
d
x
)
a
p
(
∫
ℝ
N
|
v
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
)
1
-
a
q
≤
[
m
+
n
m
(
n
m
)
-
n
m
+
n
(
1
p
)
n
m
+
n
(
1
q
)
m
m
+
n
A
(
N
,
p
,
q
,
r
,
μ
,
θ
,
s
)
]
a
/
p
n
/
(
m
+
n
)
.
Combining this with the previous lemma, we conclude that GN(N,p,q,r,μ,θ,s) can be achieved and
GN
(
N
,
p
,
q
,
r
,
μ
,
θ
,
s
)
=
[
m
+
n
m
(
n
m
)
-
n
m
+
n
(
1
p
)
n
m
+
n
(
1
q
)
m
m
+
n
A
(
N
,
p
,
q
,
r
,
μ
,
θ
,
s
)
]
a
/
p
n
/
(
m
+
n
)
.
This completes the proof. ∎
Using Lemma 3.1 and Lemma 3.2, we will now show that CKN(N,μ,θ,s,p,q,r) can be achieved.
Lemma 3.3.
Under condition (C1), CKN(N,μ,θ,s,p,q,r) can be attained and
CKN
(
N
,
μ
,
θ
,
s
,
p
,
q
,
r
)
=
(
N
-
p
N
-
p
-
μ
)
1
r
+
p
-
1
p
-
1
-
a
q
-
p
-
1
p
(
1
-
a
)
GN
(
N
,
p
,
q
,
r
,
μ
,
θ
,
s
)
.
Proof.
We begin by the observation that if u≥0 is a maximizer for GN(N,p,q,r,μ,θ,s), then we can assume that u is radial. Indeed, this fact is just a consequence of the Schwarz rearrangement; see, for instance, [27]. Now, let us assume that U0≥0 is a radial maximizer of GN(N,p,q,r,μ,θ,s). We set V0=DN,d,p-1U0 with d=N-pN-p-μ. This means that U0=DN,d,pV0. We will show that V0 is a maximizer of CKN(N,μ,θ,s,p,q,r). Indeed, for any v, we need to show
(
∫
ℝ
N
|
v
|
r
d
x
|
x
|
s
)
1
r
(
∫
ℝ
N
|
∇
v
|
p
d
x
|
x
|
μ
)
a
p
(
∫
ℝ
N
|
v
|
q
d
x
|
x
|
θ
)
1
-
a
q
≤
(
∫
ℝ
N
|
V
0
|
r
d
x
|
x
|
s
)
1
r
(
∫
ℝ
N
|
∇
V
0
|
p
d
x
|
x
|
μ
)
a
p
(
∫
ℝ
N
|
V
0
|
q
d
x
|
x
|
θ
)
1
-
a
q
.
By Lemma 2.2, by noting that when d=N-pN-p-μ, i.e., d(p+μ-N)+N-p=0, we get
∫
ℝ
N
|
v
|
r
d
x
|
x
|
s
=
d
1
+
p
-
1
p
r
∫
ℝ
N
|
D
N
,
d
,
p
v
|
r
|
x
|
N
+
s
d
-
N
d
d
x
,
∫
ℝ
N
|
v
|
q
d
x
|
x
|
θ
=
d
1
+
p
-
1
p
q
∫
ℝ
N
|
D
N
,
d
,
p
v
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
,
∫
ℝ
N
|
∇
v
|
p
|
x
|
μ
d
x
≥
∫
ℝ
N
|
∇
D
N
,
d
,
p
v
|
p
d
x
,
and
∫
ℝ
N
|
V
0
|
r
d
x
|
x
|
s
=
d
1
+
p
-
1
p
r
∫
ℝ
N
|
U
0
|
r
|
x
|
N
+
s
d
-
N
d
d
x
,
∫
ℝ
N
|
V
0
|
q
d
x
|
x
|
θ
=
d
1
+
p
-
1
p
q
∫
ℝ
N
|
U
0
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
,
∫
ℝ
N
|
∇
V
0
|
p
|
x
|
μ
d
x
=
∫
ℝ
N
|
∇
U
0
|
p
d
x
.
Hence,
(
∫
ℝ
N
|
v
|
r
d
x
|
x
|
s
)
1
r
(
∫
ℝ
N
|
∇
v
|
p
d
x
|
x
|
μ
)
a
p
(
∫
ℝ
N
|
v
|
q
d
x
|
x
|
θ
)
1
-
a
q
≤
(
d
1
+
p
-
1
p
r
∫
ℝ
N
|
D
N
,
d
,
p
v
|
r
|
x
|
N
+
s
d
-
N
d
𝑑
x
)
1
r
(
∫
ℝ
N
|
∇
D
N
,
d
,
p
v
|
p
d
x
)
a
p
(
d
1
+
p
-
1
p
q
∫
ℝ
N
|
D
N
,
d
,
p
v
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
)
1
-
a
q
≤
d
1
r
+
p
-
1
p
-
1
-
a
q
-
p
-
1
p
(
1
-
a
)
(
∫
ℝ
N
|
U
0
|
r
|
x
|
N
+
s
d
-
N
d
𝑑
x
)
1
r
(
∫
ℝ
N
|
∇
U
0
|
p
d
x
)
a
p
(
∫
ℝ
N
|
U
0
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
)
1
-
a
q
=
(
∫
|
V
0
|
r
d
x
|
x
|
s
)
1
r
(
∫
|
∇
V
0
|
p
d
x
|
x
|
μ
)
a
p
(
∫
|
V
0
|
q
d
x
|
x
|
θ
)
1
-
a
q
.
We note that the last equality holds because U0 and V0 are radial. Hence, CKN(N,μ,θ,s,p,q,r) is attained. Moreover, it is easy to see that
CKN
(
N
,
μ
,
θ
,
s
,
p
,
q
,
r
)
=
d
1
r
+
p
-
1
p
-
1
-
a
q
-
p
-
1
p
(
1
-
a
)
GN
(
N
,
p
,
q
,
r
,
μ
,
θ
,
s
)
.
∎
Lemma 3.4.
Assume that (C1) holds. If V0 is a maximizer of CKN(N,μ,θ,s,p,q,r), then V0 is radially symmetric.
Proof.
let V0 be a maximizer of CKN(N,μ,θ,s,p,q,r). We set U0=DN,d,pV0 where d=N-pN-p-μ. We will show that U0 is a maximizer of GN(N,p,q,r,μ,θ,s). Indeed, for any radial function u (we can just choose radial functions because of symmetrization arguments), we define
v
=
D
N
,
d
,
p
-
1
u
,
i.e.,
u
=
D
N
,
d
,
p
v
.
By Lemma 2.2, we get
∫
ℝ
N
|
v
|
r
d
x
|
x
|
s
=
d
1
+
p
-
1
p
r
∫
ℝ
N
|
u
|
r
|
x
|
N
+
s
d
-
N
d
d
x
,
∫
ℝ
N
|
v
|
q
d
x
|
x
|
θ
=
d
1
+
p
-
1
p
q
∫
ℝ
N
|
u
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
,
∫
ℝ
N
|
∇
v
|
p
|
x
|
μ
d
x
=
∫
ℝ
N
|
∇
u
|
p
d
x
,
and
∫
ℝ
N
|
V
0
|
r
d
x
|
x
|
s
=
d
1
+
p
-
1
p
r
∫
ℝ
N
|
U
0
|
r
|
x
|
N
+
s
d
-
N
d
d
x
,
∫
ℝ
N
|
V
0
|
q
d
x
|
x
|
θ
=
d
1
+
p
-
1
p
q
∫
ℝ
N
|
U
0
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
,
∫
ℝ
N
|
∇
V
0
|
p
|
x
|
μ
d
x
≥
∫
ℝ
N
|
∇
U
0
|
p
d
x
.
Hence,
d
1
r
+
p
-
1
p
-
1
-
a
q
-
p
-
1
p
(
1
-
a
)
(
∫
ℝ
N
|
U
0
|
r
|
x
|
N
+
s
d
-
N
d
𝑑
x
)
1
r
(
∫
ℝ
N
|
∇
U
0
|
p
d
x
)
a
p
(
∫
ℝ
N
|
U
0
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
)
1
-
a
q
≥
(
∫
ℝ
N
|
V
0
|
r
d
x
|
x
|
s
)
1
r
(
∫
ℝ
N
|
∇
V
0
|
p
|
x
|
μ
d
x
)
a
p
(
∫
ℝ
N
|
V
0
|
q
d
x
|
x
|
θ
)
1
-
a
q
≥
(
∫
ℝ
N
|
v
|
r
d
x
|
x
|
s
)
1
r
(
∫
ℝ
N
|
∇
v
|
p
|
x
|
μ
d
x
)
a
p
(
∫
ℝ
N
|
v
|
q
d
x
|
x
|
θ
)
1
-
a
q
=
d
1
r
+
p
-
1
p
-
1
-
a
q
-
p
-
1
p
(
1
-
a
)
(
∫
ℝ
N
|
u
|
r
|
x
|
N
+
s
d
-
N
d
𝑑
x
)
1
r
(
∫
ℝ
N
|
∇
u
|
p
d
x
)
a
p
(
∫
ℝ
N
|
u
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
)
1
-
a
q
.
Moreover, it is easy to see that
d
1
r
+
p
-
1
p
-
1
-
a
q
-
p
-
1
p
(
1
-
a
)
(
∫
ℝ
N
|
U
0
|
r
|
x
|
N
+
s
d
-
N
d
𝑑
x
)
1
r
(
∫
ℝ
N
|
∇
U
0
|
p
d
x
)
a
p
(
∫
ℝ
N
|
U
0
|
q
|
x
|
N
+
θ
d
-
N
d
d
x
)
1
-
a
q
=
(
∫
ℝ
N
|
V
0
|
r
d
x
|
x
|
s
)
1
r
(
∫
ℝ
N
|
∇
V
0
|
p
|
x
|
μ
d
x
)
a
p
(
∫
ℝ
N
|
V
0
|
q
d
x
|
x
|
θ
)
1
-
a
q
.
Hence,
∫
ℝ
N
|
∇
V
0
|
p
|
x
|
μ
d
x
=
∫
ℝ
N
|
∇
U
0
|
p
d
x
.
So, V0 is radial. ∎
Lemma 3.5.
Assume that (C1) holds with s=θ=NμN-p. If p<r=pq-1p-1<NpN-p, then, with δ=Np-q(N-p), we have
CKN
(
N
,
s
,
μ
,
p
,
q
,
r
)
=
(
N
-
p
N
-
p
-
μ
)
1
r
+
p
-
1
p
-
1
-
a
q
-
p
-
1
p
(
1
-
a
)
(
q
-
p
p
π
)
a
(
p
q
N
(
q
-
p
)
)
a
p
(
δ
p
q
)
1
r
(
Γ
(
q
p
-
1
q
-
p
)
Γ
(
N
2
+
1
)
Γ
(
p
-
1
p
δ
q
-
p
)
Γ
(
N
p
-
1
p
+
1
)
)
a
N
,
and all the maximizers have the form
V
0
(
x
)
=
A
(
1
+
B
|
x
|
1
d
p
p
-
1
)
-
p
-
1
q
-
p
for some
A
∈
ℝ
,
B
>
0
,
where d=N-pN-p-μ.
Proof.
When r=pq-1p-1 and s=θ=NμN-p, from [1, 2, 14, 15], we have that
GN
(
N
,
p
,
q
,
r
)
=
(
q
-
p
p
π
)
a
(
p
q
N
(
q
-
p
)
)
a
p
(
δ
p
q
)
1
r
(
Γ
(
q
p
-
1
q
-
p
)
Γ
(
N
2
+
1
)
Γ
(
p
-
1
p
δ
q
-
p
)
Γ
(
N
p
-
1
p
+
1
)
)
a
N
,
and all the maximizers have the form
U
0
(
x
)
=
A
(
1
+
B
|
x
-
x
¯
|
p
p
-
1
)
-
p
-
1
q
-
p
for some
A
∈
ℝ
,
B
>
0
,
x
¯
∈
ℝ
N
.
Now, let V0 be a maximizer of CKN(N,s,μ,p,q,r). By Lemma 3.4, DN,d,pV0 is a maximizer of GN(N,p,q,r). Hence,
D
N
,
d
,
p
V
0
(
x
)
=
A
(
1
+
B
|
x
-
x
¯
|
p
p
-
1
)
-
p
-
1
q
-
p
.
This means that
V
0
(
x
)
=
A
′
(
1
+
B
|
|
x
|
1
d
-
1
x
-
x
¯
|
p
p
-
1
)
-
p
-
1
q
-
p
.
Noting that V0 is radial, we conclude that x¯=0. ∎
Lemma 3.6.
Assume that (C1) holds with s=θ=NμN-p. If 1<q=pr-1p-1<p, then, with δ=Np-r(N-p), we have
CKN
(
N
,
s
,
μ
,
p
,
q
,
r
)
=
(
N
-
p
N
-
p
-
μ
)
1
r
+
p
-
1
p
-
1
-
a
q
-
p
-
1
p
(
1
-
a
)
(
p
-
r
p
π
)
a
(
p
r
N
(
p
-
r
)
)
a
p
(
p
r
δ
)
1
-
a
q
(
Γ
(
p
-
1
p
δ
p
-
r
+
1
)
Γ
(
N
2
+
1
)
Γ
(
r
p
-
1
p
-
r
+
1
)
Γ
(
N
p
-
1
p
+
1
)
)
a
N
.
If r>2-1p, then all the maximizers of GN(N,p,q,r) have the form
V
0
(
x
)
=
A
(
1
-
B
|
x
|
N
-
p
-
μ
N
-
p
p
p
-
1
)
+
-
p
-
1
r
-
p
for some
A
∈
ℝ
,
B
>
0
.
Proof.
When q=pr-1p-1 and s=θ=NμN-p, from [1, 2, 14, 15], we have that
GN
(
N
,
p
,
q
,
r
)
=
(
p
-
r
p
π
)
a
(
p
r
N
(
p
-
r
)
)
a
p
(
p
r
δ
)
1
-
a
q
(
Γ
(
p
-
1
p
δ
p
-
r
+
1
)
Γ
(
N
2
+
1
)
Γ
(
r
p
-
1
p
-
r
+
1
)
Γ
(
N
p
-
1
p
+
1
)
)
a
N
.
Also, when r>2-1p, all the maximizers of GN(N,p,q,r) have the form
U
0
(
x
)
=
A
(
1
-
B
|
x
-
x
¯
|
p
p
-
1
)
+
-
p
-
1
r
-
p
for some
A
∈
ℝ
,
B
>
0
,
x
¯
∈
ℝ
N
.
Now, let V0 be a maximizer of CKN(N,s,μ,p,q,r). By Lemma 3.4, DN,d,pV0 is a maximizer of GN(N,p,q,r). Hence,
D
N
,
d
,
p
V
0
(
x
)
=
A
(
1
-
B
|
x
-
x
¯
|
p
p
-
1
)
+
-
p
-
1
r
-
p
.
This means that
V
0
(
x
)
=
A
′
(
1
-
B
|
|
x
|
1
d
-
1
x
-
x
¯
|
p
p
-
1
)
+
-
p
-
1
r
-
p
.
Noting that V0 is radially symmetric, we conclude that x¯=0. ∎
5 The CKN Inequality Without the Interpolation Term: The Case a=1
In this section, we will also consider CKN inequalities without the interpolation term for all 1<p<N, and we will be concerned with the following range:
(C3)
{
1
<
p
<
p
+
μ
<
N
,
μ
p
≤
s
r
<
μ
p
+
1
,
r
=
(
N
-
s
)
p
N
-
μ
-
p
,
a
=
1
.
Note that the condition μp≤sr≤μp+1 comes from the constraints of the CKN inequality. In this case, we define
D
μ
1
,
p
(
ℝ
N
;
d
x
/
|
x
|
s
)
=
{
u
∈
L
r
(
d
x
/
|
x
|
s
)
:
∫
ℝ
N
|
∇
u
|
p
d
x
|
x
|
μ
<
∞
}
and
CKN
(
N
,
p
,
μ
,
s
)
=
sup
u
∈
D
μ
1
,
p
(
ℝ
N
;
d
x
/
|
x
|
s
)
(
∫
ℝ
N
|
u
|
r
d
x
|
x
|
s
)
1
r
(
∫
ℝ
N
|
∇
u
|
p
d
x
|
x
|
μ
)
1
p
.
We will prove in this section the following result.
Theorem 5.1.
Assume that (C3) holds. Then CKN(N,p,μ,s) is achieved with the extremals being of the following form:
V
c
,
λ
(
x
)
=
c
(
λ
+
|
x
|
p
+
μ
-
s
p
-
1
)
-
N
-
p
-
μ
p
+
μ
-
s
for some
c
≠
0
,
λ
>
0
.
Theorem 5.1 was studied in [33] by solving the corresponding ODE. In this section, we will provide another proof using the transform DN,d,p.
We note that μp≤sr<μp+1 means NμN-p≤s<p+μ.
We also define
HS
(
N
,
p
,
μ
,
s
)
=
sup
D
0
1
,
p
(
ℝ
N
;
d
x
/
|
x
|
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
)
(
∫
ℝ
N
|
u
|
p
∗
(
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
)
d
x
|
x
|
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
)
1
p
∗
(
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
)
(
∫
ℝ
N
|
∇
u
|
p
d
x
)
1
p
.
Note that
p
∗
(
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
)
=
N
-
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
N
-
p
p
=
(
N
-
s
)
p
N
-
μ
-
p
and
0
≤
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
<
p
.
Lemma 5.2.
CKN
(
N
,
p
,
μ
,
s
)
can be attained.
Proof.
We will use the fact that HS(N,p,μ,s) is attained by some radial functions U0. Set V0=DN,d,p-1U0 with d=N-pN-p-μ. This means that U0=DN,d,pV0. We will show that V0 is a maximizer of CKN(N,p,μ,s). Indeed, for any v∈Dμ1,p(ℝN), we set u=DN,d,pv. Then, by Lemma 2.2, we get
∫
ℝ
N
|
∇
v
|
p
d
x
|
x
|
μ
≥
∫
ℝ
N
|
∇
D
N
,
d
,
p
v
|
p
|
x
|
d
(
p
+
μ
-
N
)
+
N
-
p
d
x
=
∫
ℝ
N
|
∇
u
|
p
d
x
,
∫
ℝ
N
|
v
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
=
d
d
p
-
1
p
(
N
-
s
)
p
N
-
μ
-
p
∫
ℝ
N
|
u
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
N
+
s
d
-
N
d
=
d
1
+
(
N
-
s
)
(
p
-
1
)
N
-
μ
-
p
∫
ℝ
N
|
u
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
,
∫
ℝ
N
|
∇
V
0
|
p
d
x
|
x
|
μ
=
∫
ℝ
N
|
∇
U
0
|
p
d
x
and
∫
ℝ
N
|
V
0
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
=
d
1
+
(
N
-
s
)
(
p
-
1
)
N
-
μ
-
p
∫
ℝ
N
|
U
0
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
.
Hence,
(
∫
ℝ
N
|
v
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
)
N
-
μ
-
p
(
N
-
s
)
p
(
∫
ℝ
N
|
∇
v
|
p
d
x
|
x
|
μ
)
1
p
≤
d
(
1
+
(
N
-
s
)
(
p
-
1
)
N
-
μ
-
p
)
N
-
μ
-
p
(
N
-
s
)
p
(
∫
ℝ
N
|
u
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
)
N
-
μ
-
p
(
N
-
s
)
p
(
∫
ℝ
N
|
∇
u
|
p
𝑑
x
)
1
p
≤
d
(
1
+
(
N
-
s
)
(
p
-
1
)
N
-
μ
-
p
)
N
-
μ
-
p
(
N
-
s
)
p
(
∫
ℝ
N
|
U
0
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
)
N
-
μ
-
p
(
N
-
s
)
p
(
∫
ℝ
N
|
∇
U
0
|
p
𝑑
x
)
1
p
=
(
∫
ℝ
N
|
V
0
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
)
N
-
μ
-
p
(
N
-
s
)
p
(
∫
ℝ
N
|
∇
V
0
|
p
d
x
|
x
|
μ
)
1
p
.
In other words, V0 is the maximizer for CKN(N,p,μ,s). Moreover, we also deduce that
CKN
(
N
,
p
,
μ
,
s
)
=
d
(
1
+
(
N
-
s
)
(
p
-
1
)
N
-
μ
-
p
)
N
-
μ
-
p
(
N
-
s
)
p
HS
(
N
,
p
,
μ
,
s
)
.
∎
Lemma 5.3.
All the optimizers for CKN(N,p,μ,s) are radially symmetric.
Proof.
Assume that V0 is a maximizer for CKN(N,p,μ,s) and U0=DN,d,pV0 where d=N-pN-p-μ. Again, by Lemma 2.2, we get
∫
ℝ
N
|
∇
V
0
|
p
d
x
|
x
|
μ
≥
∫
ℝ
N
|
∇
U
0
|
p
d
x
and
∫
ℝ
N
|
V
0
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
=
d
1
+
(
N
-
s
)
(
p
-
1
)
N
-
μ
-
p
∫
ℝ
N
|
U
0
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
.
We will now prove that U0 is a maximizer for HS(N,p,μ,s). Indeed, for any radial function u (we can assume that u is radial by the Schwarz rearrangement argument), if we set v=DN,d,p-1u, that is, u=DN,d,pv, then, by Lemma 2.2, we obtain
∫
ℝ
N
|
∇
v
|
p
d
x
|
x
|
μ
=
∫
ℝ
N
|
∇
u
|
p
d
x
and
∫
ℝ
N
|
v
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
=
d
1
+
(
N
-
s
)
(
p
-
1
)
N
-
μ
-
p
∫
ℝ
N
|
u
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
.
Hence,
(
∫
ℝ
N
|
U
0
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
)
N
-
μ
-
p
(
N
-
s
)
p
(
∫
ℝ
N
|
∇
U
0
|
p
𝑑
x
)
1
p
≥
(
1
d
)
(
1
+
(
N
-
s
)
(
p
-
1
)
N
-
μ
-
p
)
N
-
μ
-
p
(
N
-
s
)
p
(
∫
ℝ
N
|
V
0
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
)
N
-
μ
-
p
(
N
-
s
)
p
(
∫
ℝ
N
|
∇
V
0
|
p
d
x
|
x
|
μ
)
1
p
≥
(
1
d
)
(
1
+
(
N
-
s
)
(
p
-
1
)
N
-
μ
-
p
)
N
-
μ
-
p
(
N
-
s
)
p
(
∫
ℝ
N
|
v
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
)
N
-
μ
-
p
(
N
-
s
)
p
(
∫
ℝ
N
|
∇
v
|
p
d
x
|
x
|
μ
)
1
p
=
(
∫
ℝ
N
|
u
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
)
N
-
μ
-
p
(
N
-
s
)
p
(
∫
ℝ
N
|
∇
u
|
p
d
x
)
1
p
.
Thus, U0 is a maximizer for HS(N,p,μ,s). Moreover, it is easy to see that the equality must occur in the first line, that is,
(
∫
ℝ
N
|
U
0
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
)
N
-
μ
-
p
(
N
-
s
)
p
(
∫
ℝ
N
|
∇
U
0
|
p
d
x
)
1
p
=
(
1
d
)
(
1
+
(
N
-
s
)
(
p
-
1
)
N
-
μ
-
p
)
N
-
μ
-
p
(
N
-
s
)
p
(
∫
ℝ
N
|
V
0
|
(
N
-
s
)
p
N
-
μ
-
p
d
x
|
x
|
s
)
N
-
μ
-
p
(
N
-
s
)
p
(
∫
ℝ
N
|
∇
V
0
|
p
d
x
|
x
|
μ
)
1
p
.
This means that
∫
ℝ
N
|
∇
V
0
|
p
d
x
|
x
|
μ
=
∫
ℝ
N
|
∇
U
0
|
p
d
x
,
and thus V0 is radial. ∎
Proof of Theorem 5.1.
From Lemmata 5.2 and 5.3, we see that CKN(N,p,μ,s) is attained,
CKN
(
N
,
p
,
μ
,
s
)
=
d
(
1
+
(
N
-
s
)
(
p
-
1
)
N
-
μ
-
p
)
N
-
μ
-
p
(
N
-
s
)
p
HS
(
N
,
p
,
μ
,
s
)
,
and all maximizers for CKN(N,p,μ,s) are radially symmetric. Furthermore, we can conclude that V0 is a maximizer for CKN(N,p,mu,s) only if U0=DN,d,pV0 is a maximizer for HS(N,p,μ,s), where d=N-pN-p-μ. It is known (see, for instance, [24]) that HS(N,p,μ,s) is attained with the maximizers being the functions
U
c
,
λ
(
x
)
=
c
(
λ
+
|
x
|
p
-
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
p
-
1
)
-
N
-
p
p
-
s
(
N
-
p
)
-
N
μ
N
-
p
-
μ
=
c
(
λ
+
|
x
|
(
N
-
p
)
(
p
+
μ
-
s
)
(
p
-
1
)
(
N
-
p
-
μ
)
)
-
N
-
p
-
μ
p
+
μ
-
s
for some
c
≠
0
,
λ
>
0
.
Hence, CKN(N,p,μ,s) could be achieved with the optimizers being the functions
V
c
,
λ
(
x
)
=
D
N
,
d
,
p
-
1
U
c
,
λ
(
x
)
=
c
(
λ
+
|
x
|
p
+
μ
-
s
p
-
1
)
-
N
-
p
-
μ
p
+
μ
-
s
for some
c
≠
0
,
λ
>
0
.
∎
Remark 5.4.
If we have sr=μp+1 in condition (C2), then s=p+μ and s(N-p)-NμN-p-μ=p. So in this case, after applying the transform DN,d,p, where d=N-pN-p-μ, the CKN inequality corresponds to the Hardy inequality. Hence, the best constant in this case is
CKN
(
N
,
p
,
μ
,
s
)
=
p
N
-
p
-
μ
,
and it is never achieved.
6 CKN Inequalities with Arbitrary Norm
In this section, we will investigate CKN inequalities under arbitrary norms in ℝN in the spirit of Cordero-Erausquin, Nazaret and Villani [13]. More precisely, let E=(ℝN,∥⋅∥), where ∥⋅∥ is an arbitrary norm on ℝN. Then its dual space is E∗=(ℝN,∥⋅∥∗), where for X∈E∗,
∥
X
∥
∗
=
sup
Y
∈
E
:
∥
Y
∥
≤
1
X
⋅
Y
.
For simplicity, we will assume that |{∥x∥∗≤1}|=ωN and set κN=|{∥x∥≤1}|. We will assume that for any X∈ℝN, there exists a unique X∗∈ℝN such that ∥X∗∥∗=1 and
X
⋅
X
∗
=
∥
X
∥
=
sup
Y
∈
ℝ
N
:
∥
Y
∥
∗
≤
1
X
⋅
Y
.
It is clear that ∥⋅∥ is Lipschitz with Lipschitz constant 1, and thus, differentiable a.e. From the fact that ∥λx∥=λ∥x∥ for all λ>0, we can see that the gradient of ∥⋅∥ at x∈ℝN is the unique vector ∇(∥⋅∥)(x)=x∗. Recall that
∥
x
∗
∥
∗
=
1
,
x
⋅
x
∗
=
∥
x
∥
=
sup
∥
y
∥
∗
≤
1
x
⋅
y
.
Actually, first we will consider a more general situation. More precisely, we suppose that C is q-homogeneous, that is, there exists q>1 such that
(6.1)
C
(
λ
x
)
=
λ
q
C
(
x
)
for all
λ
≥
0
and all
x
∈
ℝ
N
.
Then C∗, the Legendre transform of C defined by
C
∗
(
x
)
=
sup
y
{
〈
x
,
y
〉
-
C
(
y
)
}
,
is even, strictly convex function and is p-homogeneous with p=qq-1.
We have that 〈X,Y〉≤C∗(X)+C(Y) for all X,Y. Hence, 〈X,Y〉≤λpC∗(X)+λ-qC(Y) for all λ>0, X,Y. Minimizing the right-hand side with respect to λ gives the Cauchy–Schwarz inequality
X
⋅
Y
≤
[
q
C
(
Y
)
]
1
q
[
p
C
∗
(
X
)
]
1
p
.
By Young’s inequality, we have
X
⋅
Y
≤
[
q
C
(
Y
)
]
1
q
[
p
C
∗
(
X
)
]
1
p
≤
C
∗
(
x
)
+
C
(
y
)
.
Hence, we also have that
[
p
C
∗
(
X
)
]
1
p
=
sup
Y
X
⋅
Y
[
q
C
(
Y
)
]
1
q
.
In other words,
C
∗
(
X
)
=
sup
Y
|
X
⋅
Y
|
p
p
[
q
C
(
Y
)
]
p
q
.
We will assume that for all x∈ℝN, there exists a unique vector x∗ such that
x
⋅
x
∗
=
q
C
(
x
)
and
C
∗
(
x
∗
)
=
(
q
-
1
)
C
(
x
)
=
q
p
C
(
x
)
.
In other words, for all x∈ℝN, there exists a unique vector x∗ such that the equality in the Cauchy–Schwarz inequality occurs.
Note that, from (6.1), we get that C(⋅) is differentiable a.e. We will assume that the gradient of C(⋅) at x∈ℝN is the unique vector x∗. The example that we have in mind is C(x)=1q|x|q and C∗(x)=1p|x|p, with |⋅| being the regular Euclidean norm on ℝN. Another example is the pair C(x)=1q∥x∥q and C∗(x)=1p∥x∥∗p.
6.1 A Change of Variables
As in Lemma 2.1, we have the following result.
Lemma 6.1.
We have
|
x
⋅
∇
u
(
x
)
|
=
[
q
C
(
x
)
]
1
q
[
p
C
∗
(
∇
u
)
]
1
p
for a.e
.
x
∈
ℝ
N
if and only if u is C-radial, i.e., u(x)=u(y) when C(x)=C(y).
Proof.
If u is C-radial, then recalling that ∇(C(⋅))(x)=x∗, we have
∂
u
∂
x
j
(
x
)
=
u
′
(
C
(
x
)
)
x
j
∗
.
Hence,
C
∗
(
∇
u
)
=
C
∗
(
u
′
(
C
(
x
)
)
x
∗
)
=
|
u
′
(
C
(
x
)
)
|
p
C
∗
(
x
∗
)
=
|
u
′
(
C
(
x
)
)
|
p
q
p
C
(
x
)
and
[
q
C
(
x
)
]
1
q
[
p
C
∗
(
∇
u
)
]
1
p
=
[
q
C
(
x
)
]
1
q
[
|
u
′
(
C
(
x
)
)
|
p
q
C
(
x
)
]
1
p
=
|
u
′
(
C
(
x
)
)
|
q
C
(
x
)
.
Also,
|
x
⋅
∇
u
(
x
)
|
=
|
∑
j
=
1
N
x
j
∂
u
∂
x
j
(
x
)
|
=
|
u
′
(
∥
x
∥
)
|
|
∑
j
=
1
N
x
j
x
j
∗
|
=
|
u
′
(
∥
x
∥
)
|
q
C
(
x
)
.
Now, if for all x∈ℝN,
|
x
⋅
∇
u
(
x
)
|
=
[
q
C
(
x
)
]
1
q
[
p
C
∗
(
∇
u
)
]
1
p
,
then ∇u(x) has the same direction with x∗. That is, we can find a function f(x) such that ∇u(x)=f(x)x∗. Now let a and b be two points on the C-sphere with radius r>0, that is, C(a)=C(b)=r. We connect a and b by a piecewise smooth curve r(t) on the sphere, i.e., C(r(t))=r and C(r(0))=a, C(r(1))=b. Then we have
∇
u
(
r
(
t
)
)
=
f
(
r
(
t
)
)
(
r
(
t
)
)
∗
.
Using that fact that C(r(t))=r for all t, we get (r(t))∗⋅∇r(t)=0. Hence,
∫
0
1
∇
u
(
r
(
t
)
)
⋅
∇
r
(
t
)
𝑑
t
=
∫
0
1
f
(
r
(
t
)
)
(
r
(
t
)
)
∗
⋅
∇
r
(
t
)
𝑑
t
=
0
.
In other words,
u
(
b
)
-
u
(
a
)
=
u
(
C
(
r
(
1
)
)
)
-
u
(
C
(
r
(
0
)
)
)
=
0
.
The proof is completed. ∎
Let d>0. We define a vector-valued function LN,d:ℝN→ℝN by
L
N
,
d
(
x
)
=
C
(
x
)
d
x
.
The Jacobian matrix of this function LN,d is
𝐉
L
N
,
d
=
C
(
x
)
d
𝐈
N
+
𝐀
,
where
𝐀
=
(
d
C
(
x
)
d
-
1
x
1
x
1
∗
d
C
(
x
)
d
-
1
x
1
x
2
∗
⋯
d
C
(
x
)
d
-
1
x
1
x
N
∗
d
C
(
x
)
d
-
1
x
2
x
1
∗
d
C
(
x
)
d
-
1
x
2
x
2
∗
⋯
d
C
(
x
)
d
-
1
x
2
x
N
∗
⋮
⋮
⋱
⋮
d
C
(
x
)
d
-
1
x
N
x
1
∗
d
C
(
x
)
d
-
1
x
N
x
2
⋯
d
C
(
x
)
d
-
1
x
N
x
N
∗
)
.
Then we get
det
(
𝐉
L
N
,
d
)
=
(
-
1
)
N
det
(
-
C
(
x
)
d
𝐈
N
-
𝐀
)
=
(
1
+
d
q
)
C
(
x
)
N
d
.
We now define the mapping DN,d,p, with p>1, by
D
N
,
d
,
p
u
(
x
)
:=
(
1
1
+
d
q
)
p
-
1
p
u
(
L
N
,
d
(
x
)
)
=
(
1
1
+
d
q
)
p
-
1
p
u
(
C
(
x
)
d
x
)
.
We also define
D
N
,
d
,
p
-
1
u
=
v
if
u
=
D
N
,
d
,
p
v
.
Under the transform DN,d,p, we also have the following result.
Lemma 6.2.
For continuous function
f
, we have
∫
ℝ
N
f
(
(
1
1
+
d
q
)
p
-
1
p
u
(
x
)
)
C
(
x
)
t
𝑑
x
=
(
1
+
d
q
)
∫
ℝ
N
f
(
D
N
,
d
,
p
u
(
x
)
)
C
(
x
)
t
(
d
q
+
1
)
-
N
d
𝑑
x
.
In particular, we obtain that
u
∈
L
s
(
d
x
/
C
(
x
)
t
)
if and only if
D
N
,
d
,
p
u
∈
L
s
(
d
x
/
C
(
x
)
t
(
d
q
+
1
)
-
N
d
)
.
For smooth functions
u
, we have
∫
ℝ
N
C
∗
(
∇
D
N
,
d
,
p
u
(
x
)
)
C
(
x
)
(
q
d
+
1
)
μ
+
p
d
-
N
d
𝑑
x
≤
∫
ℝ
N
C
∗
(
∇
u
(
y
)
)
C
(
y
)
μ
𝑑
y
.
The equality occurs if and only if
u
is
C
-radially symmetric.
Proof.
(1) We have
∫
ℝ
N
f
(
D
N
,
d
,
p
u
(
x
)
)
C
(
x
)
t
(
d
q
+
1
)
-
N
d
𝑑
x
=
1
1
+
d
q
∫
ℝ
N
f
(
(
1
1
+
d
q
)
p
-
1
p
u
(
y
)
)
C
(
y
)
t
𝑑
y
.
Using the change of variables yi=C(x)dxi, i=1,2,…,N, we have
d
y
=
det
(
𝐉
L
N
,
d
)
d
x
=
(
1
+
d
q
)
C
(
x
)
N
d
d
x
and
d
x
=
1
(
1
+
d
q
)
d
y
C
(
y
)
N
d
d
q
+
1
.
Hence,
∫
ℝ
N
f
(
D
N
,
d
,
p
u
(
x
)
)
C
(
x
)
t
(
d
q
+
1
)
-
N
d
𝑑
x
=
∫
ℝ
N
f
(
(
1
1
+
d
q
)
p
-
1
p
u
(
C
(
x
)
d
x
)
)
C
(
x
)
t
(
d
q
+
1
)
-
N
d
𝑑
x
=
1
1
+
d
q
∫
ℝ
N
f
(
(
1
1
+
d
q
)
p
-
1
p
u
(
y
)
)
C
(
y
)
t
(
d
q
+
1
)
-
N
d
d
q
+
1
d
y
C
(
y
)
N
d
d
q
+
1
=
1
1
+
d
q
∫
ℝ
N
f
(
(
1
1
+
d
q
)
p
-
1
p
u
(
y
)
)
C
(
y
)
t
𝑑
y
.
(2) Now we begin to consider the gradient of DN,d,pu. After calculations, we have
(
∂
D
N
,
d
,
p
u
∂
x
1
(
x
)
∂
D
N
,
d
,
p
u
∂
x
2
(
x
)
⋮
∂
D
N
,
d
,
p
u
∂
x
N
(
x
)
)
=
∇
D
N
,
d
,
p
u
(
x
)
=
(
1
1
+
d
q
)
p
-
1
p
∇
(
u
(
C
(
x
)
d
x
)
)
=
(
1
1
+
d
q
)
p
-
1
p
𝐉
L
N
,
d
T
(
∂
u
∂
x
1
(
C
(
x
)
d
x
)
∂
u
∂
x
2
(
C
(
x
)
d
x
)
⋮
∂
u
∂
x
N
(
C
(
x
)
d
x
)
)
.
Hence, we have
∂
u
(
C
(
x
)
d
x
)
∂
x
i
=
(
C
(
x
)
d
∂
u
∂
x
i
(
C
(
x
)
d
x
)
+
A
i
)
for i=1,2,…,N, where
A
i
:=
∑
j
=
1
N
d
C
(
x
)
d
-
1
x
i
∗
x
j
∂
u
∂
x
j
(
C
(
x
)
d
x
)
and
C
∗
(
X
)
=
sup
|
X
⋅
Y
|
p
p
[
q
C
(
Y
)
]
p
q
.
Hence, we obtain
C
∗
(
∇
D
N
,
d
,
p
u
(
x
)
)
=
C
∗
(
(
1
1
+
d
q
)
p
-
1
p
∇
(
u
(
C
(
x
)
d
x
)
)
)
=
(
1
1
+
d
q
)
p
-
1
C
∗
(
∇
(
u
(
C
(
x
)
d
x
)
)
)
=
(
1
1
+
d
q
)
p
-
1
sup
y
{
(
∇
(
u
(
C
(
x
)
d
x
)
)
⋅
y
)
p
p
[
q
C
(
y
)
]
p
q
}
=
(
1
1
+
d
q
)
p
-
1
sup
y
{
[
∑
i
=
1
N
[
C
(
x
)
d
∂
u
∂
x
i
(
C
(
x
)
d
x
)
y
i
+
A
i
y
i
]
]
p
p
[
q
C
(
y
)
]
p
q
}
.
The first term is easy to compute. We have
I
1
=
∑
i
=
1
N
C
(
x
)
d
∂
u
∂
x
i
(
C
(
x
)
d
x
)
y
i
=
C
(
x
)
d
∇
u
(
C
(
x
)
d
x
)
⋅
y
≤
C
(
x
)
d
[
q
C
(
y
)
]
1
q
[
p
C
∗
(
∇
u
(
C
(
x
)
d
x
)
)
]
1
p
.
Applying the Cauchy–Schwarz inequality
X
⋅
Y
≤
[
q
C
(
Y
)
]
1
q
[
p
C
∗
(
X
)
]
1
p
,
we can estimate the second term as follows:
I
2
=
∑
i
=
1
N
A
i
y
i
=
∑
i
=
1
N
∑
j
=
1
N
d
C
(
x
)
d
-
1
x
i
∗
x
j
∂
u
∂
x
j
(
C
(
x
)
d
x
)
y
i
=
d
C
(
x
)
d
-
1
∑
i
=
1
N
x
i
∗
y
i
∑
j
=
1
N
x
j
∂
u
∂
x
j
(
C
(
x
)
d
x
)
≤
d
C
(
x
)
d
-
1
|
x
∗
⋅
y
|
|
x
⋅
∇
u
(
C
(
x
)
d
x
)
|
≤
d
C
(
x
)
d
-
1
[
q
C
(
y
)
]
1
q
[
p
C
∗
(
x
∗
)
]
1
p
[
q
C
(
x
)
]
1
q
[
p
C
∗
(
∇
u
(
C
(
x
)
d
x
)
)
]
1
p
≤
d
C
(
x
)
d
-
1
[
q
C
(
y
)
]
1
q
[
q
C
(
x
)
]
1
p
[
q
C
(
x
)
]
1
q
[
p
C
∗
(
∇
u
(
C
(
x
)
d
x
)
)
]
1
p
≤
q
d
C
(
x
)
d
[
q
C
(
y
)
]
1
q
[
p
C
∗
(
∇
u
(
C
(
x
)
d
x
)
)
]
1
p
.
Therefore,
sup
y
{
[
∑
i
=
1
N
[
C
(
x
)
d
∂
u
∂
x
i
(
C
(
x
)
d
x
)
y
i
+
A
i
y
i
]
]
p
p
[
q
C
(
y
)
]
p
q
}
≤
sup
y
{
[
(
1
+
q
d
)
]
p
C
(
x
)
p
d
[
q
C
(
y
)
]
p
q
p
C
∗
(
∇
u
(
C
(
x
)
d
x
)
)
p
[
q
C
(
y
)
]
p
q
}
=
[
(
1
+
q
d
)
]
p
C
(
x
)
p
d
C
∗
(
∇
u
(
C
(
x
)
d
x
)
)
.
In conclusion, we get
C
∗
(
∇
D
N
,
d
,
p
u
(
x
)
)
≤
(
1
+
q
d
)
C
(
x
)
p
d
C
∗
(
∇
u
(
C
(
x
)
d
x
)
)
.
Using the change of variables again, we get
∫
ℝ
N
C
∗
(
∇
u
(
y
)
)
C
(
y
)
μ
𝑑
y
=
∫
ℝ
N
C
∗
(
∇
u
(
C
(
x
)
d
x
)
)
C
(
C
(
x
)
d
x
)
μ
(
1
+
d
q
)
C
(
x
)
N
d
𝑑
x
≥
∫
ℝ
N
C
∗
(
∇
D
N
,
d
,
p
u
(
x
)
)
C
(
x
)
(
q
d
+
1
)
μ
C
(
x
)
p
d
C
(
x
)
N
d
𝑑
x
=
∫
ℝ
N
C
∗
(
∇
D
N
,
d
,
p
u
(
x
)
)
C
(
x
)
(
q
d
+
1
)
μ
+
p
d
-
N
d
𝑑
x
.
Finally, it is easy to check that the equalities hold if and only if the equality in the Cauchy–Schwarz inequality occurs. This means that u is C-radially symmetric. ∎
We note here that we will mainly apply the above change of variables with C(x)=1q∥x∥q and C∗(x)=1p∥x∥∗p. In this case, for ease of reference, we will use the transform
T
N
,
d
,
p
u
(
x
)
:=
(
1
d
)
p
-
1
p
u
(
∥
x
∥
d
-
1
x
)
.
We also define
T
N
,
d
,
p
-
1
u
=
v
if
u
=
T
N
,
d
,
p
v
.
The following lemma is a restatement of Lemma 6.2.
Lemma 6.3.
For continuous function
f
, we have
∫
ℝ
N
f
(
(
1
d
)
p
-
1
p
u
(
x
)
)
∥
x
∥
t
𝑑
x
=
d
∫
ℝ
N
f
(
T
N
,
d
,
p
u
(
x
)
)
∥
x
∥
N
+
t
d
-
N
d
𝑑
x
.
In particular, we obtain that
u
∈
L
s
(
d
x
/
∥
x
∥
t
)
if and only if
T
N
,
d
,
p
u
∈
L
s
(
d
x
/
∥
x
∥
N
+
t
d
-
N
d
)
.
If
∇
u
∈
L
p
(
d
x
/
∥
x
∥
μ
)
, then
∇
T
N
,
d
,
p
∈
L
p
(
d
x
/
∥
x
∥
d
(
p
+
μ
-
N
)
+
N
-
p
)
. Moreover,
∫
ℝ
N
∥
∇
T
N
,
d
,
p
u
(
x
)
∥
∗
p
∥
x
∥
d
(
p
+
μ
-
N
)
+
N
-
p
𝑑
x
≤
∫
ℝ
N
∥
∇
u
(
x
)
∥
∗
p
∥
x
∥
μ
𝑑
x
.
The equality occurs if and only if
u
is
∥
⋅
∥
-radial.
6.2 Maximizers for CKN Inequalities with Arbitrary Norms
Consider the following class:
{
1
<
p
<
p
+
μ
<
N
,
θ
≤
N
μ
N
-
p
≤
s
<
N
,
1
≤
q
<
r
<
N
p
N
-
p
,
a
=
[
(
N
-
θ
)
r
-
(
N
-
s
)
q
]
p
[
(
N
-
θ
)
p
-
(
N
-
μ
-
p
)
q
]
r
.
We denote by Dμ,θp,q(ℝN) the completion of the space of smooth compactly supported functions with the norm
(
∫
ℝ
N
∥
∇
u
∥
∗
p
d
x
∥
x
∥
μ
)
1
p
+
(
∫
ℝ
N
|
u
|
q
d
x
∥
x
∥
θ
)
1
q
,
and we set
CKN
(
N
,
μ
,
θ
,
s
,
p
,
q
,
r
)
=
sup
u
∈
D
μ
,
θ
p
,
q
(
ℝ
N
)
(
∫
ℝ
N
|
u
|
r
d
x
∥
x
∥
s
)
1
r
(
∫
ℝ
N
∥
∇
u
∥
∗
p
d
x
∥
x
∥
μ
)
a
p
(
∫
ℝ
N
|
u
|
q
d
x
∥
x
∥
θ
)
1
-
a
q
,
GN
(
N
,
p
,
q
,
r
,
μ
,
θ
,
s
)
=
sup
u
∈
D
0
,
N
+
θ
d
-
N
d
p
,
q
(
ℝ
N
)
(
∫
ℝ
N
|
u
|
r
∥
x
∥
N
+
s
d
-
N
d
𝑑
x
)
1
r
(
∫
ℝ
N
∥
∇
u
∥
∗
p
d
x
)
a
p
(
∫
ℝ
N
|
u
|
q
∥
x
∥
N
+
θ
d
-
N
d
d
x
)
1
-
a
q
.
Then, similarly as in Section 3, we can prove the following theorem.
Theorem 6.4.
Assume that (C3) holds. Then CKN(N,μ,θ,s,p,q,r) can be achieved. Moreover, all the extremal functions of CKN(N,μ,θ,s,p,q,r) are ∥⋅∥-radially symmetric.
The proof of Theorem 6.4 is similar to that of Theorem 1.1 and will be omitted.
Furthermore, we can provide the maximizers for CKN(N,μ,θ,s,p,q,r) in the following two classes.
Theorem 6.5.
Assume that (C2) holds with θ=s=NμN-p. If p<r=pq-1p-1<NpN-p, then CKN(N,μ,θ,s,p,q,r) is achieved by maximizers of the form
V
0
(
x
)
=
A
(
1
+
B
∥
x
∥
N
-
p
-
μ
N
-
p
p
p
-
1
)
-
p
-
1
q
-
p
for some
A
∈
ℝ
,
B
>
0
.
Theorem 6.6.
Assume that (C2) holds with θ=s=NμN-p. If 1<q=pr-1p-1<p, then CKN(N,μ,θ,s,p,q,r) is achieved if r>2-1p, by maximizers of the form
V
0
(
x
)
=
A
(
1
-
B
∥
x
∥
N
-
p
-
μ
N
-
p
p
p
-
1
)
+
-
p
-
1
r
-
p
for some
A
∈
ℝ
,
B
>
0
.
Proofs of Theorems 6.5–6.6.
When r=pq-1p-1 and s=θ=NμN-p, from [13], we have that GN(N,p,q,r) is achieved by maximizers of the form
U
0
(
x
)
=
A
(
1
+
B
∥
x
-
x
¯
∥
p
p
-
1
)
-
p
-
1
q
-
p
for some
A
∈
ℝ
,
B
>
0
,
x
¯
∈
ℝ
N
.
Now, let V0 be a maximizer of CKN(N,s,μ,p,q,r). Then TN,d,pV0 is a maximizer of GN(N,p,q,r) with d=N-pN-p-μ. Hence, CKN(N,s,μ,p,q,r) can be attained by
V
0
(
x
)
=
T
N
,
d
,
p
-
1
A
(
1
+
B
∥
x
-
x
¯
∥
p
p
-
1
)
-
p
-
1
q
-
p
.
This means that
V
0
(
x
)
=
A
′
(
1
+
B
∥
|
x
|
1
d
-
1
x
-
x
¯
∥
p
p
-
1
)
-
p
-
1
q
-
p
.
Noting that V0 is ∥⋅∥-radial, we conclude that x¯=0, that is,
V
0
(
x
)
=
A
′
(
1
+
B
∥
x
∥
N
-
p
-
μ
N
-
p
p
p
-
1
)
-
p
-
1
q
-
p
.
Similarly, when θ=s=NμN-p, if q=pr-1p-1 and r>2-1p, then CKN(N,μ,θ,s,p,q,r) is achieved by maximizers of the form
V
0
(
x
)
=
A
(
1
-
B
∥
x
∥
N
-
p
-
μ
N
-
p
p
p
-
1
)
+
-
p
-
1
r
-
p
for some
A
∈
ℝ
,
B
>
0
.
∎
