---
tag: 3240RP14MT
kind: concept
name: Lorentz space
msc: ["46E30", 42B35]
sources:
  - url: https://en.wikipedia.org/wiki/Lorentz_space
same_as:
  wikidata: Q6680542
  wikipedia: Lorentz space
---

## Summary

The Lorentz spaces $L^{p,q}(\mu)$ form a two-parameter family of function spaces that refines the scale of [[Lp spaces|SASP14MA1Z]].
They are defined with the [[decreasing rearrangement|BWS042NC2P]] of a function, so they only see the sizes of the level sets of the function.
The first index $p$ gives the main size and the second index $q$ gives a fine adjustment.
They contain $L^p = L^{p,p}$ and the [[weak Lp space|H4C888M5AZ]] $L^{p,\infty}$, and they appear in interpolation theory and in sharp forms of Sobolev inequalities.

## Explanations

### L1 human

To measure the size of a function, rearrange its values in decreasing order, like sorting a pile of sand by height.
The Lorentz norm then weighs this sorted profile in a way governed by two numbers $p$ and $q$.
With $q = p$ one gets the usual $L^p$ norm.
Letting $q$ grow makes the space larger, so for $p \ge 1$ one has $L^{p,1} \subseteq L^{p,p} \subseteq L^{p,\infty}$.
For example, $\abs{x}^{-1}$ on $\R$ is in $L^{1,\infty}$ but not in $L^{1,1} = L^1$.