Muckenhoupt's theorem

Remarks

Necessity in part 1 comes from testing on one cube. Let $\sigma = w^{-1/(p - 1)}$ and $f = \sigma \mathbf{1}_Q$. On $Q$ one has $Mf \ge \frac{1}{\abs{Q}} \int_Q \sigma$, and inserting this in the inequality gives the $A_p$ bound for $Q$, after a cut-off to make $f \in L^p(w)$. For sufficiency, one first shows the weak-type bound on $L^p(w)$ from the $A_p$ condition and the covering argument. Then one uses that an $A_p$ weight is also in $A_{p - \varepsilon}$ for some $\varepsilon > 0$, which follows from a reverse Hölder inequality, and interpolates with the Marcinkiewicz interpolation theorem.

Example: $w(x) = \abs{x}$ on $\R$ is not in $A_2$, and indeed the weighted inequality fails for $p = 2$. For $0 < \delta < 1$ let $f = \abs{x}^{-1} \mathbf{1}_{[\delta, 1]}$; then $\int f^2 w = \log(1/\delta)$. For $x \in [-1, 1]$ the interval of radius $2$ centred at $x$ contains $[\delta, 1]$, so $Mf(x) \ge \frac{1}{4} \log(1/\delta)$. Hence $\int (Mf)^2 w \ge \frac{1}{16} \log^2(1/\delta)$, and the ratio of the two sides tends to infinity as $\delta \to 0$.

Going deeper

For a student

Muckenhoupt proved the theorem in 1972. Shortly after, Hunt, Muckenhoupt and Wheeden showed that the same $A_p$ condition characterizes the weights for which the Hilbert transform is bounded on $L^p(w)$, $1 < p < \infty$. Coifman and Fefferman extended the sufficiency to general Calderón–Zygmund operators. The theorem is the starting point of weighted theory, including extrapolation and the sharp dependence of operator norms on $[w]_{A_p}$.