---
tag: T86RSJ8M86
kind: concept
name: Muckenhoupt weights
msc: [42B25, 42B35]
sources:
  - url: https://en.wikipedia.org/wiki/Muckenhoupt_weights
same_as:
  wikidata: Q6931165
  wikipedia: Muckenhoupt weights
---

## Summary

A Muckenhoupt weight is a positive locally integrable function $w$ on $\R^n$ that satisfies the $A_p$ condition.
This condition is a uniform bound, over all cubes, on products of averages of $w$ and of a negative power of $w$.
The class $A_p$, for $1 < p < \infty$, is exactly the class of weights $w$ for which the [[Hardy–Littlewood maximal function|VX1A56P5KQ]] is bounded on $L^p(w \, dx)$.
The same weights make the Hilbert transform and other Calderón–Zygmund operators bounded on $L^p(w \, dx)$.
They are the central objects of weighted norm inequalities.

## Explanations

### L1 human

A weight changes how much each part of space counts in an $L^p$ norm.
The $A_p$ condition says that on every cube the weight is not too unbalanced.
It cannot be very large on average while also being very small on a large part of the cube.
Power weights $\abs{x}^a$ are the main examples; on the line, $\abs{x}^a$ is in $A_2$ exactly when $-1 < a < 1$.
For $a \le -1$ the weight is not even integrable near $0$, and for $a \ge 1$ its reciprocal is not.