In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the
L
2
→
L
2
operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem).
Here is one version. Let
X
,
Y
be two measurable spaces (such as
R
n
). Let
T
be an integral operator with the non-negative Schwartz kernel
K
(
x
,
y
)
,
x
∈
X
,
y
∈
Y
:
T
f
(
x
)
=
∫
Y
K
(
x
,
y
)
f
(
y
)
d
y
.
If there exist functions
p
(
x
)
>
0
and
q
(
x
)
>
0
and numbers
α
,
β
>
0
such that
(
1
)
∫
Y
K
(
x
,
y
)
q
(
y
)
d
y
≤
α
p
(
x
)
for almost all
x
and
(
2
)
∫
X
p
(
x
)
K
(
x
,
y
)
d
x
≤
β
q
(
y
)
for almost all
y
, then
T
extends to a continuous operator
T
:
L
2
→
L
2
with the operator norm
∥
T
∥
L
2
→
L
2
≤
α
β
.
Such functions
p
(
x
)
,
q
(
x
)
are called the Schur test functions.
In the original version,
T
is a matrix and
α
=
β
=
1
.
Common usage and Young's inequality
A common usage of the Schur test is to take
p
(
x
)
=
q
(
x
)
=
1.
Then we get:
∥
T
∥
L
2
→
L
2
2
≤
sup
x
∈
X
∫
Y
|
K
(
x
,
y
)
|
d
y
⋅
sup
y
∈
Y
∫
X
|
K
(
x
,
y
)
|
d
x
.
This inequality is valid no matter whether the Schwartz kernel
K
(
x
,
y
)
is non-negative or not.
A similar statement about
L
p
→
L
q
operator norms is known as Young's inequality:
if
sup
x
(
∫
Y
|
K
(
x
,
y
)
|
r
d
y
)
1
/
r
+
sup
y
(
∫
X
|
K
(
x
,
y
)
|
r
d
x
)
1
/
r
≤
C
,
where
r
satisfies
1
r
=
1
−
(
1
p
−
1
q
)
, for some
1
≤
p
≤
q
≤
∞
, then the operator
T
f
(
x
)
=
∫
Y
K
(
x
,
y
)
f
(
y
)
d
y
extends to a continuous operator
T
:
L
p
(
Y
)
→
L
q
(
X
)
, with
∥
T
∥
L
p
→
L
q
≤
C
.
Using the Cauchy–Schwarz inequality and the inequality (1), we get:
|
T
f
(
x
)
|
2
=
|
∫
Y
K
(
x
,
y
)
f
(
y
)
d
y
|
2
≤
(
∫
Y
K
(
x
,
y
)
q
(
y
)
d
y
)
(
∫
Y
K
(
x
,
y
)
f
(
y
)
2
q
(
y
)
d
y
)
≤
α
p
(
x
)
∫
Y
K
(
x
,
y
)
f
(
y
)
2
q
(
y
)
d
y
.
Integrating the above relation in
x
, using Fubini's Theorem, and applying the inequality (2), we get:
∥
T
f
∥
L
2
2
≤
α
∫
Y
(
∫
X
p
(
x
)
K
(
x
,
y
)
d
x
)
f
(
y
)
2
q
(
y
)
d
y
≤
α
β
∫
Y
f
(
y
)
2
d
y
=
α
β
∥
f
∥
L
2
2
.
It follows that
∥
T
f
∥
L
2
≤
α
β
∥
f
∥
L
2
for any
f
∈
L
2
(
Y
)
.
