In mathematics, the Morrey–Campanato spaces (named after Charles B. Morrey, Jr. and Sergio Campanato)
L
λ
,
p
(
Ω
)
are Banach spaces which extend the notion of functions of bounded mean oscillation, describing situations where the oscillation of the function in a ball is proportional to some power of the radius other than the dimension. They are used in the theory of elliptic partial differential equations, since for certain values of
λ
, elements of the space
L
λ
,
p
(
Ω
)
are Hölder continuous functions over the domain
Ω
.
The seminorm of the Morrey spaces is given by
[
u
]
λ
,
p
p
=
sup
0
<
r
<
diam
(
Ω
)
,
x
0
∈
Ω
1
r
λ
∫
B
r
(
x
0
)
∩
Ω
|
u
(
y
)
|
p
d
y
.
When
λ
=
0
, the Morrey space is the same as the usual
L
p
space. When
λ
=
n
, the spatial dimension, the Morrey space is equivalent to
L
∞
, due to the Lebesgue differentiation theorem. When
λ
>
n
, the space contains only the 0 function.
The seminorm of the Campanato space is given by
[
u
]
λ
,
p
p
=
sup
0
<
r
<
diam
(
Ω
)
,
x
0
∈
Ω
1
r
λ
∫
B
r
(
x
0
)
∩
Ω
|
u
(
y
)
−
u
r
,
x
0
|
p
d
y
where
u
r
,
x
0
=
1
|
B
r
(
x
0
)
∩
Ω
|
∫
B
r
(
x
0
)
∩
Ω
u
(
y
)
d
y
.
It is known that the Morrey spaces with
0
≤
λ
<
n
are equivalent to the Campanato spaces with the same value of
λ
when
Ω
is a sufficiently regular domain, that is to say, when there is a constant A such that
|
Ω
∩
B
r
(
x
0
)
|
>
A
r
n
for every
x
0
∈
Ω
and
r
<
diam
(
Ω
)
.
When
n
=
λ
, the Campanato space is the space of functions of bounded mean oscillation. When
n
<
λ
≤
n
+
p
, the Campanato space is the space of Hölder continuous functions
C
α
(
Ω
)
with
α
=
λ
−
n
p
. For
λ
>
n
+
p
, the space contains only constant functions.
