Muckenhoupt weights

In mathematics, the class of Muckenhoupt weights A_p consists of those weights ω for which the Hardy–Littlewood maximal operator is bounded on L^p(dω). Specifically, we consider functions  f  on Rⁿ and their associated maximal functions M( f ) defined as

${\displaystyle M(f)(x)=\sup _{r>0}{\frac {1}{r^{n}}}\int _{B_{r}(x)}|f|,}$

where B_r(x) is the ball in Rⁿ with radius r and center at x. Let 1 ≤ p < ∞, we wish to characterise the functions ω : Rⁿ → [0, ∞) for which we have a bound

${\displaystyle \int |M(f)(x)|^{p}\,\omega (x)dx\leq C\int |f|^{p}\,\omega (x)\,dx,}$

where C depends only on p and ω. This was first done by Benjamin Muckenhoupt.^[1]