---
tag: 9534R6VMY0
kind: concept
name: Weak-type estimate
msc: [42B25, 47B38, "46E30"]
sources:
  - url: https://en.wikipedia.org/wiki/Lp_space
---

## Summary

A weak-type estimate bounds the measure of the set where $\abs{Tf}$ is larger than $\lambda$, instead of bounding a norm of $Tf$.
An operator $T$ is of weak type $(p, q)$ if this measure is at most $(C\norm{f}_p/\lambda)^q$.
Equivalently, $T$ maps $L^p$ boundedly into the [[weak Lp space|H4C888M5AZ]] $L^{q,\infty}$.
Such estimates are the natural substitute for $L^1$ bounds, which often fail, for maximal functions and singular integrals.
By the [[Marcinkiewicz interpolation theorem|R1PHMVBJ9J]], two weak-type estimates give strong $L^p$ bounds for all exponents in between.

## Explanations

### L1 human

Bounding $\int \abs{Tf}$ asks for control of the total size of $Tf$.
A weak-type bound asks for less: it only controls how much room the large values of $Tf$ take up.
For example, the function $1/x$ on $(0, \infty)$ has infinite integral, but the set where it exceeds $\lambda$ is $(0, 1/\lambda)$, of length exactly $1/\lambda$.
That is the typical shape of a weak $L^1$ function, and the Hilbert transform and the maximal function produce such tails.