The Sobolev conjugate of p for
1
≤
p
<
n
{\displaystyle 1\leq p<n}
n is space dimensionality, is
p
∗
=
p
n
n
−
p
>
p
{\displaystyle p^{*}={\frac {pn}{n-p}}>p}
This is an important parameter in the Sobolev inequalities .
A question arises whether u from the Sobolev space
W
1
,
p
(
R
n
)
{\displaystyle W^{1,p}(\mathbb {R} ^{n})}
L
q
(
R
n
)
{\displaystyle L^{q}(\mathbb {R} ^{n})}
q > p . More specifically, when does
‖
D
u
‖
L
p
(
R
n
)
{\displaystyle \|Du\|_{L^{p}(\mathbb {R} ^{n})}}
‖
u
‖
L
q
(
R
n
)
{\displaystyle \|u\|_{L^{q}(\mathbb {R} ^{n})}}
‖
u
‖
L
q
(
R
n
)
≤
C
(
p
,
q
)
‖
D
u
‖
L
p
(
R
n
)
(
∗
)
{\displaystyle \|u\|_{L^{q}(\mathbb {R} ^{n})}\leq C(p,q)\|Du\|_{L^{p}(\mathbb {R} ^{n})}\qquad \qquad (*)}
can not be true for arbitrary q . Consider
u
(
x
)
∈
C
c
∞
(
R
n
)
{\displaystyle u(x)\in C_{c}^{\infty }(\mathbb {R} ^{n})}
u
λ
(
x
)
:=
u
(
λ
x
)
{\displaystyle u_{\lambda }(x):=u(\lambda x)}
‖
u
λ
‖
L
q
(
R
n
)
q
=
∫
R
n
|
u
(
λ
x
)
|
q
d
x
=
1
λ
n
∫
R
n
|
u
(
y
)
|
q
d
y
=
λ
−
n
‖
u
‖
L
q
(
R
n
)
q
‖
D
u
λ
‖
L
p
(
R
n
)
p
=
∫
R
n
|
λ
D
u
(
λ
x
)
|
p
d
x
=
λ
p
λ
n
∫
R
n
|
D
u
(
y
)
|
p
d
y
=
λ
p
−
n
‖
D
u
‖
L
p
(
R
n
)
p
{\displaystyle {\begin{aligned}\|u_{\lambda }\|_{L^{q}(\mathbb {R} ^{n})}^{q}&=\int _{\mathbb {R} ^{n}}|u(\lambda x)|^{q}dx={\frac {1}{\lambda ^{n}}}\int _{\mathbb {R} ^{n}}|u(y)|^{q}dy=\lambda ^{-n}\|u\|_{L^{q}(\mathbb {R} ^{n})}^{q}\\\|Du_{\lambda }\|_{L^{p}(\mathbb {R} ^{n})}^{p}&=\int _{\mathbb {R} ^{n}}|\lambda Du(\lambda x)|^{p}dx={\frac {\lambda ^{p}}{\lambda ^{n}}}\int _{\mathbb {R} ^{n}}|Du(y)|^{p}dy=\lambda ^{p-n}\|Du\|_{L^{p}(\mathbb {R} ^{n})}^{p}\end{aligned}}}
The inequality (*) for
u
λ
{\displaystyle u_{\lambda }}
u
{\displaystyle u}
‖
u
‖
L
q
(
R
n
)
≤
λ
1
−
n
p
+
n
q
C
(
p
,
q
)
‖
D
u
‖
L
p
(
R
n
)
{\displaystyle \|u\|_{L^{q}(\mathbb {R} ^{n})}\leq \lambda ^{1-{\frac {n}{p}}+{\frac {n}{q}}}C(p,q)\|Du\|_{L^{p}(\mathbb {R} ^{n})}}
If
1
−
n
p
+
n
q
≠
0
,
{\displaystyle 1-{\frac {n}{p}}+{\frac {n}{q}}\neq 0,}
λ
{\displaystyle \lambda }
q
=
p
n
n
−
p
{\displaystyle q={\frac {pn}{n-p}}}
which is the Sobolev conjugate.
