In mathematics, the class of Muckenhoupt weights Ap consists of those weights ω for which the Hardy–Littlewood maximal operator is bounded on Lp(dω). Specifically, we consider functions f on Rn and their associated maximal functions M( f ) defined as
M ( f ) ( x ) = sup r > 0 1 r n ∫ B r ( x ) | f | , {\displaystyle M(f)(x)=\sup _{r>0}{\frac {1}{r^{n}}}\int _{B_{r}(x)}|f|,}
where Br(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
∫ | M ( f ) ( x ) | p ω ( x ) d x ≤ C ∫ | f | p ω ( x ) d x , {\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.
Definition For a fixed 1 < p < ∞, we say that a weight ω : Rn → [0, ∞) belongs to Ap if ω is locally integrable and there is a constant C such that, for all balls B in Rn, we have
( 1 | B | ∫ B ω ( x ) d x ) ( 1 | B | ∫ B ω ( x ) − q p d x ) p q ≤ C < ∞ , {\displaystyle \left({\frac {1}{|B|}}\int _{B}\omega (x)\,dx\right)\left({\frac {1}{|B|}}\int _{B}\omega (x)^{-{\frac {q}{p}}}\,dx\right)^{\frac {p}{q}}\leq C<\infty ,}
where |B| is the Lebesgue measure of B, and q is a real number such that: 1/p + 1/q = 1. We say ω : Rn → [0, ∞) belongs to A1 if there exists some C such that
1 | B | ∫ B ω ( y ) d y ≤ C ω ( x ) , {\displaystyle {\frac {1}{|B|}}\int _{B}\omega (y)\,dy\leq C\omega (x),}
for almost every x ∈ B and all balls B.
Equivalent characterizations This following result is a fundamental result in the study of Muckenhoupt weights.
Theorem. Let 1 < p < ∞. A weight ω is in Ap if and only if any one of the following hold. (a) The Hardy–Littlewood maximal function is bounded on Lp(ω(x)dx), that is
∫ | M ( f ) ( x ) | p ω ( x ) d x ≤ C ∫ | f | p ω ( x ) d x , {\displaystyle \int |M(f)(x)|^{p}\,\omega (x)\,dx\leq C\int |f|^{p}\,\omega (x)\,dx,}
for some C which only depends on p and the constant A in the above definition. (b) There is a constant c such that for any locally integrable function f on Rn, and all balls B:
( f B ) p ≤ c ω ( B ) ∫ B f ( x ) p ω ( x ) d x , {\displaystyle (f_{B})^{p}\leq {\frac {c}{\omega (B)}}\int _{B}f(x)^{p}\,\omega (x)\,dx,}
where:
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