In mathematical analysis, Lorentz spaces, introduced by George Lorentz in the 1950s, are generalisations of the more familiar
L
p
spaces.
The Lorentz spaces are denoted by
L
p
,
q
. Like the
L
p
spaces, they are characterized by a norm (technically a quasinorm) that encodes information about the "size" of a function, just as the
L
p
norm does. The two basic qualitative notions of "size" of a function are: how tall is graph of the function, and how spread out is it. The Lorentz norms provide tighter control over both qualities than the
L
p
norms, by exponentially rescaling the measure in both the range (
p
) and the domain (
q
). The Lorentz norms, like the
L
p
norms, are invariant under arbitrary rearrangements of the values of a function.
The Lorentz space on a measure space
(
X
,
μ
)
is the space of complex-valued measurable functions
f
on X such that the following quasinorm is finite
∥
f
∥
L
p
,
q
(
X
,
μ
)
=
p
1
q
∥
t
μ
{
|
f
|
≥
t
}
1
p
∥
L
q
(
R
+
,
d
t
t
)
where
0
<
p
<
∞
and
0
<
q
≤
∞
. Thus, when
q
<
∞
,
∥
f
∥
L
p
,
q
(
X
,
μ
)
=
p
1
q
(
∫
0
∞
t
q
μ
{
x
:
|
f
(
x
)
|
≥
t
}
q
p
d
t
t
)
1
q
.
and, when
q
=
∞
,
∥
f
∥
L
p
,
∞
(
X
,
μ
)
p
=
sup
t
>
0
(
t
p
μ
{
x
:
|
f
(
x
)
|
>
t
}
)
.
It is also conventional to set
L
∞
,
∞
(
X
,
μ
)
=
L
∞
(
X
,
μ
)
.
The quasinorm is invariant under rearranging the values of the function
f
, essentially by definition. In particular, given a complex-valued measurable function
f
defined on a measure space,
(
X
,
μ
)
, its decreasing rearrangement function,
f
∗
:
[
0
,
∞
)
→
[
0
,
∞
]
can be defined as
f
∗
(
t
)
=
inf
{
α
∈
R
+
:
d
f
(
α
)
≤
t
}
where
d
f
is the so-called distribution function of
f
, given by
d
f
(
α
)
=
μ
(
{
x
∈
X
:
|
f
(
x
)
|
>
α
}
)
.
Here, for notational convenience,
inf
∅
is defined to be
∞
.
The two functions
|
f
|
and
f
∗
are equimeasurable, meaning that
μ
(
{
x
∈
X
:
|
f
(
x
)
|
>
α
}
)
=
λ
(
{
t
>
0
:
f
∗
(
t
)
>
α
}
)
,
α
>
0
,
where
λ
is the Lebesgue measure on the real line. The related symmetric decreasing rearrangement function, which is also equimeasurable with
f
, would be defined on the real line by
R
∋
t
↦
1
2
f
∗
(
|
t
|
)
.
Given these definitions, for
0
<
p
<
∞
and
0
<
q
≤
∞
, the Lorentz quasinorms are given by
∥
f
∥
L
p
,
q
=
{
(
∫
0
∞
(
t
1
p
f
∗
(
t
)
)
q
d
t
t
)
1
q
q
∈
(
0
,
∞
)
,
sup
t
>
0
t
1
p
f
∗
(
t
)
q
=
∞
.
When
(
X
,
μ
)
=
(
N
,
#
)
(the counting measure on
N
), the resulting Lorentz space is a sequence space. However, in this case it is convenient to use different notation.
for
(
a
n
)
n
=
1
∞
∈
R
N
(or
C
N
in the complex case), let
∥
(
a
n
)
n
=
1
∞
∥
p
=
(
∑
n
=
1
∞
|
a
n
|
p
)
1
/
p
denote the p-norm for
1
≤
p
<
∞
and
∥
(
a
n
)
n
=
1
∞
∥
∞
=
sup
|
a
n
|
the ∞-norm. Denote by
ℓ
p
the Banach space of all sequences with finite p-norm. Let
c
0
the Banach space of all sequences satisfying
lim
|
a
n
|
=
0
, endowed with the ∞-norm. Denote by
c
00
the normed space of all sequences with only finitely many nonzero entries. These spaces all play a role in the definition of the Lorentz sequence spaces
d
(
w
,
p
)
below.
Let
w
=
(
w
n
)
n
=
1
∞
∈
c
0
∖
ℓ
1
be a sequence of positive real numbers satisfying
1
=
w
1
≥
w
2
≥
w
3
⋯
, and define the norm
∥
(
a
n
)
∥
d
(
w
,
p
)
=
sup
σ
∈
Π
∥
(
a
σ
(
n
)
w
n
1
/
p
)
n
=
1
∞
∥
p
. The Lorentz sequence space
d
(
w
,
p
)
is defined as the Banach space of all sequences where this norm is finite. Equivalently, we can define
d
(
w
,
p
)
as the completion of
c
00
under
∥
⋅
∥
d
(
w
,
p
)
.
The Lorentz spaces are genuinely generalisations of the
L
p
spaces in the sense that, for any
p
,
L
p
,
p
=
L
p
, which follows from Cavalieri's principle. Further,
L
p
,
∞
coincides with weak
L
p
. They are quasi-Banach spaces (that is, quasi-normed spaces which are also complete) and are normable for
1
<
p
<
∞
and
1
≤
q
≤
∞
. When
p
=
1
,
L
1
,
1
=
L
1
is equipped with a norm, but it is not possible to define a norm equivalent to the quasinorm of
L
1
,
∞
, the weak
L
1
space. As a concrete example that the triangle inequality fails in
L
1
,
∞
, consider
f
(
x
)
=
1
x
χ
(
0
,
1
)
(
x
)
and
g
(
x
)
=
1
1
−
x
χ
(
0
,
1
)
(
x
)
,
whose
L
1
,
∞
quasi-norm equals one, whereas the quasi-norm of their sum
f
+
g
equals four.
The space
L
p
,
q
is contained in
L
p
,
r
whenever
q
<
r
. The Lorentz spaces are real interpolation spaces between
L
1
and
L
∞
.
