Lorentz space

The $L^{1,\infty}$ norm of $f(x)=\frac{1}{|x-1|}$ is the area of the largest rectangle with sides parallel to the coordinate axes that can be inscribed in the graph.

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 ($p$) and the domain ($q$). The Lorentz norms, like the $L^{p}$ norms, are invariant under arbitrary rearrangements of the values of a function.

Definition

The Lorentz space on a measure space $(X, \mu)$ is the space of complex-valued measurable functions $f$ on X such that the following quasinorm is finite

$\|f\|_{L^{p,q}(X,\mu)} = p^{\frac{1}{q}} \left \|t\mu\{|f|\ge t\}^{\frac{1}{p}} \right \|_{L^q \left (\mathbf{R}^+, \frac{dt}{t} \right)}$

where $0 < p < \infty$ and $0 < q \leq \infty$. Thus, when $q < \infty$,

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

and, when $q = \infty$,

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

It is also conventional to set $L^{\infty,\infty}(X, \mu) = L^{\infty}(X, \mu)$.

Decreasing rearrangements

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, $(X, \mu)$, its decreasing rearrangement function, $f^{\ast}: [0, \infty) \to [0, \infty]$ can be defined as

$f^{\ast}(t) = \inf \{\alpha \in \mathbf{R}^{+}: d_f(\alpha) \leq t\}$

where $d_{f}$ is the so-called distribution function of $f$, given by

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

Here, for notational convenience, $\inf \varnothing$ is defined to be $\infty$.

The two functions $|f|$ and $f^{\ast}$ are equimeasurable, meaning that

$\mu \bigl( \{ x \in X : |f(x)| > \alpha\} \bigr) = \lambda \bigl( \{ t > 0 : f^{\ast}(t) > \alpha\} \bigr), \quad \alpha > 0,$

where $\lambda$ 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

$\mathbf{R} \ni t \mapsto \tfrac{1}{2} f^{\ast}(|t|).$

Given these definitions, for $0 < p < \infty$ and $0 < q \leq \infty$, the Lorentz quasinorms are given by

$\| f \|_{L^{p, q}} = \begin{cases} \left( \displaystyle \int_0^{\infty} \left (t^{\frac{1}{p}} f^{\ast}(t) \right )^q \, \frac{dt}{t} \right)^{\frac{1}{q}} & q \in (0, \infty), \\ \sup\limits_{t > 0} \, t^{\frac{1}{p}} f^{\ast}(t) & q = \infty. \end{cases}$

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, $L^{p, \infty}$ coincides with weak $L^{p}$. They are quasi-Banach spaces (that is, quasi-normed spaces which are also complete) and are normable for $1 < p < \infty$ and $1 \leq q \leq \infty$. 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 $L^{1,\infty}$, the weak $L^{1}$ space. As a concrete example that the triangle inequality fails in $L^{1,\infty}$, consider

$f(x) = \tfrac{1}{x} \chi_{(0,1)}(x)\quad \text{and} \quad g(x) = \tfrac{1}{1-x} \chi_{(0,1)}(x),$

whose $L^{1,\infty}$ 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 $L^{\infty}$.

See also

• Interpolation space
• Hardy–Littlewood inequality

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

1. G. Lorentz, "Some new function spaces", Annals of Mathematics 51 (1950), pp. 37-55.
2. G. Lorentz, "On the theory of spaces Λ", Pacific Journal of Mathematics 1 (1951), pp. 411-429.