Lorentz space

In mathematical analysis, Lorentz spaces, introduced by George Lorentz in the 1950s^[1]^[2], 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 (the p) and the domain (the q). The Lorentz norms, like the L^p norms, are invariant under arbitrary rearrangements of the values of a function.

1 Definition
2 Decreasing rearrangements
3 Properties
4 References
5 Notes

Definition

The Lorentz space on a measure space (X,μ) is the space of complex-valued measurable functions ƒ on X such that the following quasinorm is finite

$\|f\|_{L^{p,q}(X,\mu)} = p^{1/q}\|t\mu\{|f|\ge t\}^{1/p}\|_{L^q(\mathbb{R}^%2B,\frac{dt}{t})}$

where 0 < p < ∞ and 0 < q ≤ ∞. Thus, when q < ∞,

$\|f\|_{L^{p,q}(X,\mu)}=p^{1/q}\left(\int_0^\infty t^q \mu\left\{x\mid |f(x)| \ge t\right\}^{q/p}\,\frac{dt}{t}\right)^{1/q}.$

and when q = ∞,

$\|f\|_{L^{p,\infty}(X,\mu)}^p = \sup_{t>0}\left(t^p\mu\left\{x\mid |f(x)|>t\right\}\right).$

It is also conventional to set L^∞,∞(X,μ) = L^∞(X,μ).

Decreasing rearrangements

The quasinorm is invariant under rearranging the values of the function ƒ, essentially by definition. In particular, given a complex-valued measurable function ƒ defined on a measure space, (X, μ), its decreasing rearrangement function, $f^{*}: [0, \infty) \rightarrow [0, \infty]$ can be defined as

$f^{*}(t) = \inf\{\alpha \in \mathbb{R}^{%2B}: d_f(\alpha) \leq t\}$

where d_ƒ is the so-called distribution function of ƒ, given by

$d_f(\alpha) = \mu(\{x \in X�: |f(x)| > \alpha\}).$

Here, for notational convenience, $\inf \emptyset$ is defined to be ∞.

Given these definitions, for p, q ∈ (0, ∞), the Lorentz norms are given by

$\| f \|_{L^{p, q}} = \left\{ \begin{array}{l l} \left( \int_0^{\infty} (t^{\frac{1}{p}} f^{*}(t))^q \, \frac{dt}{t} \right)^{\frac{1}{q}} & q \in (0, \infty),\\ \displaystyle \sup_{t > 0} t^{\frac{1}{p}} f^{*}(t) & q = \infty. \end{array} \right.$

Properties

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 p ∈ (1, ∞), q ∈ [1, ∞]. L^p,q is a subspace of L^p,r whenever q < r.

References

Grafakos, Loukas (2008), Classical Fourier analysis, Graduate Texts in Mathematics, 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.