---
tag: H4C888M5AZ
kind: concept
name: Weak Lp space
msc: ["46E30", 42B35]
specializes: [3240RP14MT]
sources:
  - url: https://en.wikipedia.org/wiki/Lorentz_space
---

## Summary

The weak $L^p$ space $L^{p,\infty}(\mu)$ contains the measurable functions whose [[distribution function|HQVXKEMN5N]] decays at least as fast as $t^{-p}$.
It contains [[$L^p(\mu)$|SASP14MA1Z]], by Chebyshev's inequality.
It is the natural target space for operators that fail to be bounded on $L^1$, such as the [[Hardy–Littlewood maximal function|VX1A56P5KQ]].
It is the [[Lorentz space|3240RP14MT]] $L^{p,q}$ with $q = \infty$.

## Explanations

### L1 human

A function is in $L^p$ when $\abs{f}^p$ has finite integral.
Weak $L^p$ asks only that the set where $\abs{f}$ exceeds $t$ has measure at most $C t^{-p}$.
This allows functions that just barely fail to be in $L^p$.
On $\R$, the function $1/\abs{x}$ is not integrable near $0$, but the set where it exceeds $t$ is an interval of length $2/t$, so it lies in weak $L^1$.