The Sobolev conjugate of p for
1
≤
p
<
n
, where n is space dimensionality, is
p
∗
=
p
n
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
)
belongs to
L
q
(
R
n
)
for some q>p. More specifically, when does
∥
D
u
∥
L
p
(
R
n
)
control
∥
u
∥
L
q
(
R
n
)
? It is easy to check that the following inequality
∥
u
∥
L
q
(
R
n
)
≤
C
(
p
,
q
)
∥
D
u
∥
L
p
(
R
n
)
(*)
can not be true for arbitrary q. Consider
u
(
x
)
∈
C
c
∞
(
R
n
)
, infinitely differentiable function with compact support. Introduce
u
λ
(
x
)
:=
u
(
λ
x
)
. We have that
∥
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
The inequality (*) for
u
λ
results in the following inequality for
u
∥
u
∥
L
q
(
R
n
)
≤
λ
1
−
n
/
p
+
n
/
q
C
(
p
,
q
)
∥
D
u
∥
L
p
(
R
n
)
If
1
−
n
/
p
+
n
/
q
≠
0
, then by letting
λ
going to zero or infinity we obtain a contradiction. Thus the inequality (*) could only be true for
q
=
p
n
n
−
p
,
which is the Sobolev conjugate.