The Hardy-Littlewood maximal operator characterises A_p

Finite dyadic covering. For $a\in \{0,1\mathrm{/}3,2\mathrm{/}3{\}}^n$ use the grid ${\mathcal{D}}^a$ of half-open cubes $2^{-k}(m+(-1{)}^{k}a+(0,1{]}^n)$, $k\in \mathbb{Z}$, $m\in {\mathbb{Z}}^n$. The boundary offset of a parent, in child units, differs from the child offset by $-3(-1{)}^{k}a\in {\mathbb{Z}}^n$, so the grids are nested and partition each generation. Given an open cube $Q$ of side $\mathrm{\ell }$, choose a scale $L=2^{-k}$ with $3\mathrm{\ell }<L\le 6\mathrm{\ell }$. In each coordinate the boundary sets of the three offsets are separated by $L\mathrm{/}3$, so an interval of length $\mathrm{\ell }$ meets boundaries from at most one offset. Choose an offset avoiding its closure, coordinate by coordinate; then $Q$ is contained in one $R\in {\mathcal{D}}^a$ of side $L$, with $\mathrm{\mid }R\mathrm{\mid }\le 6^n\mathrm{\mid }Q\mathrm{\mid }$. Consequently $M_c^{\ast }f\le 6^n{\max }_a{M}_{{\mathcal{D}}^a}f$, where $M_{\mathcal{D}}f={\sup }_{R\in \mathcal{D},x\in R}\mathrm{\mid }R{\mathrm{\mid }}^{-1}{\int }_R\mathrm{\mid }f\mathrm{\mid }$.

Uniform weighted dyadic bounds. For any locally finite measure $\mu =vd\lambda$ with $v$ a weight, set $M_{\mathcal{D}}^{\mu }u(x)={\sup }_{R\in \mathcal{D},x\in R}\mu (R{)}^{-1}{\int }_R\mathrm{\mid }u\mathrm{\mid }d\mu$. Restrict first to generations $-N\le k\le N$. The maximal bad cubes at height $t>0$ exist because ancestor chains are finite; they are countable and disjoint, and cover the restricted superlevel set, so its $\mu$-measure is at most $t^{-1}\int \mathrm{\mid }u\mathrm{\mid }d\mu$. Letting $N\to \mathrm{\infty }$ gives weak $(1,1)$ with constant one. The operator is sublinear and bounded on $L^{\mathrm{\infty }}(\mu )$ with constant one, so [F2] gives $\mathrm{\parallel }{M}_{\mathcal{D}}^{\mu }u{\mathrm{\parallel }}_{L^q(\mu )}\le a_q\mathrm{\parallel }u{\mathrm{\parallel }}_{L^q(\mu )}$, independently of $v$ and its doubling constant. Measurability follows from the countable grid.

Sufficiency, pointwise estimate. Suppose $w\in A_p$, let $\sigma =w^{-1\mathrm{/}(p-1)}$, and fix one grid. Set $u=\mathrm{\mid }f\mathrm{\mid }{\sigma }^{-1}$, $F=M_{\mathcal{D}}^{\sigma }u$ and $h=F^{p-1}{w}^{-1}$, with powers assigned arbitrarily on the common null set where $w$ or $\sigma$ is not positive finite. For every grid cube $R$ and almost every $y\in R$, $F(y)\ge \sigma (R{)}^{-1}{\int }_R\mathrm{\mid }f\mathrm{\mid }$. Thus $⟨\mathrm{\mid }f\mathrm{\mid }{⟩}_R\le (\sigma (R)\mathrm{/}\mathrm{\mid }R\mathrm{\mid })F(y)$. Raise to $p-1$ and integrate with respect to Lebesgue measure over $R$ to get $(⟨\mathrm{\mid }f\mathrm{\mid }{⟩}_R{)}^{p-1}\le (\sigma (R)\mathrm{/}\mathrm{\mid }R\mathrm{\mid }{)}^{p-1}\mathrm{\mid }R{\mathrm{\mid }}^{-1}{\int }_R{F}^{p-1}\le [w{]}_{A_p}w(R{)}^{-1}{\int }_{R}hw$. The $A_p$ bound holds also for half-open cubes because their boundaries are null. Taking suprema at $x$ yields $M_{\mathcal{D}}f(x)\le [w{]}_{A_p}^{1\mathrm{/}(p-1)}(M_{\mathcal{D}}^{w}h(x){)}^{1\mathrm{/}(p-1)}$.

Sufficiency, norm estimate. We have $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p(\sigma )}=\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(w)}$ and $h^{p^{\mathrm{\prime }}}w=F^p\sigma$. Step 1.2 therefore ensures $F\in L^p(\sigma )$ and $h\in L^{p^{\mathrm{\prime }}}(w)$, and step 1.3 gives $\mathrm{\parallel }{M}_{\mathcal{D}}f{\mathrm{\parallel }}_{L^p(w)}\le [w{]}_{A_p}^{1\mathrm{/}(p-1)}{a}_{p^{\mathrm{\prime }}}^{1\mathrm{/}(p-1)}\mathrm{\parallel }h{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(w)}^{1\mathrm{/}(p-1)}\le [w{]}_{A_p}^{1\mathrm{/}(p-1)}{a}_{p^{\mathrm{\prime }}}^{1\mathrm{/}(p-1)}{a}_p\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(w)}$. By step 1.1, the maximum over the $3^n$ grids has norm at most $3^{n\mathrm{/}p}$ times this bound. The ball/cube comparisons [F4] give the displayed estimate for all the stated maximal functions, with a constant depending only on $n,p$.

Necessity. Suppose $\mathrm{\parallel }Mf{\mathrm{\parallel }}_{L^p(w)}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(w)}$ for locally integrable inputs in $L^p(w)$. By [F4], $M_c^{\ast }$ has bound $C_0=C_{n}C$. For a cube $Q$ put $f_{\epsilon }={\mathbf{1}}_Q(w+\epsilon {)}^{-1\mathrm{/}(p-1)}$, setting it to zero on the null set of exceptional weight values. It is bounded, compactly supported, and belongs to $L^p(w)$. On $Q$, $M_c^{\ast }{f}_{\epsilon }\ge ⟨f_{\epsilon }{⟩}_Q$. Since $f_{\epsilon }^{p}w\le f_{\epsilon }$, the norm bound implies $w(Q)⟨f_{\epsilon }{⟩}_Q^p\le C_0^p{\int }_Q{f}_{\epsilon }$, hence $⟨w{⟩}_Q⟨f_{\epsilon }{⟩}_Q^{p-1}\le C_0^p$. Let $\epsilon =1\mathrm{/}j\downarrow 0$ and use monotone convergence [F4] to obtain $⟨w{⟩}_Q⟨w^{-1\mathrm{/}(p-1)}{⟩}_Q^{p-1}\le C_0^p$ for every cube. Thus $w\in A_p$, completing both directions.