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 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.