---
tag: B7X9M15Q92
kind: definition
name: Lorentz space
msc: ["46E30", 42B35]
defines: 3240RP14MT
uses: [P15VQS5DJ9, YYNCJSR0PJ, HQVXKEMN5N, BWS042NC2P, RARTMA0ZEC]
sources:
  - url: https://en.wikipedia.org/wiki/Lorentz_space
---

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] and let $f$ be a real or complex [[measurable function|YYNCJSR0PJ]] on $X$.
Its [[distribution function|HQVXKEMN5N]] is $d_f(s) = \mu(\set{x : \abs{f(x)} > s})$ for $s \ge 0$.
Its [[decreasing rearrangement|BWS042NC2P]] is $f^*(t) = \inf \set{s \ge 0 : d_f(s) \le t}$ for $t \ge 0$.
Let $0 < p < \infty$ and $0 < q \le \infty$, and put
$$\norm{f}_{p,q} = \left( \int_0^\infty \left( t^{1/p} f^*(t) \right)^q \frac{dt}{t} \right)^{1/q} \text{ if } q < \infty, \qquad \norm{f}_{p,\infty} = \sup_{t > 0} t^{1/p} f^*(t).$$
The *Lorentz space* $L^{p,q}(\mu)$ is the set of measurable $f$ with $\norm{f}_{p,q} < \infty$, modulo equality [[almost everywhere|RARTMA0ZEC]].

## Remarks

Example: for $q = p$, $\norm{f}_{p,p}^p = \int_0^\infty f^*(t)^p \, dt = \int_X \abs{f}^p \, d\mu$, because $f^*$ and $\abs{f}$ have the same distribution function.
So $L^{p,p}(\mu) = L^p(\mu)$ with equal norms.

Non-example: on $(0, \infty)$ with Lebesgue measure, $f(x) = x^{-1/p}$ is not in $L^{p,q}$ for $q < \infty$.
Here $f^* = f$, so $t^{1/p} f^*(t) = 1$ and the integral $\int_0^\infty dt / t$ diverges.
It does lie in $L^{p,\infty}$, with $\norm{f}_{p,\infty} = 1$.

The space $L^{p,\infty}$ is the weak $L^p$ space, since $\sup_t t^{1/p} f^*(t) = \sup_s s \, d_f(s)^{1/p}$.
For fixed $p$, the spaces increase with $q$: $L^{p,q_1} \subseteq L^{p,q_2}$ when $q_1 < q_2$.
In general $\norm{\cdot}_{p,q}$ is only a quasinorm.