In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s,[1][2] are generalisations of the more familiar
spaces.
The Lorentz spaces are denoted by
. Like the
spaces, they are characterized by a norm (technically a quasinorm) that encodes information about the "size" of a function, just as the
norm does. The two basic qualitative notions of "size" of a function are: how tall is the graph of the function, and how spread out is it. The Lorentz norms provide tighter control over both qualities than the
norms, by exponentially rescaling the measure in both the range (
) and the domain (
). The Lorentz norms, like the
norms, are invariant under arbitrary rearrangements of the values of a function.
Definition
The Lorentz space on a measure space
is the space of complex-valued measurable functions
on X such that the following quasinorm is finite
![{\displaystyle \|f\|_{L^{p,q}(X,\mu )}=p^{\frac {1}{q}}\left\|t\mu \{|f|\geq t\}^{\frac {1}{p}}\right\|_{L^{q}\left(\mathbf {R} ^{+},{\frac {dt}{t}}\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/382b2fda2a490947749efa2f33a956bb7765ab14)
where
and
. Thus, when
,
![{\displaystyle \|f\|_{L^{p,q}(X,\mu )}=p^{\frac {1}{q}}\left(\int _{0}^{\infty }t^{q}\mu \left\{x:|f(x)|\geq t\right\}^{\frac {q}{p}}\,{\frac {dt}{t}}\right)^{\frac {1}{q}}=\left(\int _{0}^{\infty }{\bigl (}\tau \mu \left\{x:|f(x)|^{p}\geq \tau \right\}{\bigr )}^{\frac {q}{p}}\,{\frac {d\tau }{\tau }}\right)^{\frac {1}{q}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fce63475511be5469073652f3f4b15a502383504)
and, when
,
![{\displaystyle \|f\|_{L^{p,\infty }(X,\mu )}^{p}=\sup _{t>0}\left(t^{p}\mu \left\{x:|f(x)|>t\right\}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de85318d33a82adb40dbe8a5583c31e07dfe92a0)
It is also conventional to set
.
Decreasing rearrangements
The quasinorm is invariant under rearranging the values of the function
, essentially by definition. In particular, given a complex-valued measurable function
defined on a measure space,
, its decreasing rearrangement function,
can be defined as
![{\displaystyle f^{\ast }(t)=\inf\{\alpha \in \mathbf {R} ^{+}:d_{f}(\alpha )\leq t\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee61ba3c524ecf8ed3a1d7b1b13b79164a768545)
where
is the so-called distribution function of
, given by
![{\displaystyle d_{f}(\alpha )=\mu (\{x\in X:|f(x)|>\alpha \}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b45282058839bad497646be180380ef99f53a08d)
Here, for notational convenience,
is defined to be
.
The two functions
and
are equimeasurable, meaning that
![{\displaystyle \lambda {\bigl (}\{x\in X:|f(x)|>\alpha \}{\bigr )}=\lambda {\bigl (}\{t>0:f^{\ast }(t)>\alpha \}{\bigr )},\quad \alpha >0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86ab5b73d63f284278d10aeae0a4b4e31014ae6c)
where
is the Lebesgue measure on the real line. The related symmetric decreasing rearrangement function, which is also equimeasurable with
, would be defined on the real line by
![{\displaystyle \mathbf {R} \ni t\mapsto {\tfrac {1}{2}}f^{\ast }(|t|).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b161c3cad387c3837f9245d4c49c6c73d801b416)
Given these definitions, for
and
, the Lorentz quasinorms are given by
![{\displaystyle \|f\|_{L^{p,q}}={\begin{cases}\left(\displaystyle \int _{0}^{\infty }\left(t^{\frac {1}{p}}f^{\ast }(t)\right)^{q}\,{\frac {dt}{t}}\right)^{\frac {1}{q}}&q\in (0,\infty ),\\\sup \limits _{t>0}\,t^{\frac {1}{p}}f^{\ast }(t)&q=\infty .\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a31aab02841792e97be863e8fd99009e1349abc4)
Lorentz sequence spaces
When
(the counting measure on
), the resulting Lorentz space is a sequence space. However, in this case it is convenient to use different notation.
Definition.
For
(or
in the complex case), let
denote the p-norm for
and
the ∞-norm. Denote by
the Banach space of all sequences with finite p-norm. Let
the Banach space of all sequences satisfying
, endowed with the ∞-norm. Denote by
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
below.
Let
be a sequence of positive real numbers satisfying
, and define the norm
. The Lorentz sequence space
is defined as the Banach space of all sequences where this norm is finite. Equivalently, we can define
as the completion of
under
.
Properties
The Lorentz spaces are genuinely generalisations of the
spaces in the sense that, for any
,
, which follows from Cavalieri's principle. Further,
coincides with weak
. They are quasi-Banach spaces (that is, quasi-normed spaces which are also complete) and are normable for
and
. When
,
is equipped with a norm, but it is not possible to define a norm equivalent to the quasinorm of
, the weak
space. As a concrete example that the triangle inequality fails in
, consider
![{\displaystyle f(x)={\tfrac {1}{x}}\chi _{(0,1)}(x)\quad {\text{and}}\quad g(x)={\tfrac {1}{1-x}}\chi _{(0,1)}(x),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e087c20a314611ae74f7898f099f97bfcca3151)
whose
quasi-norm equals one, whereas the quasi-norm of their sum
equals four.
The space
is contained in
whenever
. The Lorentz spaces are real interpolation spaces between
and
.
Hölder's inequality
where
,
,
, and
.
Dual space
If
is a nonatomic σ-finite measure space, then
(i)
for
, or
;
(ii)
for
, or
;
(iii)
for
. Here
for
,
for
, and
.
Atomic decomposition
The following are equivalent for
.
(i)
.
(ii)
where
has disjoint support, with measure
, on which
almost everywhere, and
.
(iii)
almost everywhere, where
and
.
(iv)
where
has disjoint support
, with nonzero measure, on which
almost everywhere,
are positive constants, and
.
(v)
almost everywhere, where
.
See also
References
- Grafakos, Loukas (2008), Classical Fourier analysis, Graduate Texts in Mathematics, vol. 249 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-09432-8, ISBN 978-0-387-09431-1, MR 2445437.
Notes
- ^ G. Lorentz, "Some new function spaces", Annals of Mathematics 51 (1950), pp. 37-55.
- ^ G. Lorentz, "On the theory of spaces Λ", Pacific Journal of Mathematics 1 (1951), pp. 411-429.
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