The Muckenhoupt A_infinity class

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

The Muckenhoupt $A_{\mathrm{\infty }}$ class is the union of the finite-exponent classes: $A_{\mathrm{\infty }}:={\bigcup }_{1\le p<\mathrm{\infty }}{A}_p,$ where $A_p$ and $A_1$ are the classes of Muckenhoupt A_p and A_1 weights. Thus a weight $w$ belongs to $A_{\mathrm{\infty }}$ exactly when $w\in A_p$ for some finite $p$, and by the nesting property of Duality and nesting of the A_p classes the witnessing exponent may be replaced by any larger one: if $w\in A_p$ then $w\in A_q$ for every $q>p$. The class is not defined by a single limit formula in $p$; its equivalence with the power-decay condition $w(E)\mathrm{/}w(Q)\le C(\mathrm{\mid }E\mathrm{\mid }\mathrm{/}\mathrm{\mid }Q\mathrm{\mid }{)}^{\delta }$ and with the reverse Hölder property is a theorem proved separately on this page.

Every $A_{\mathrm{\infty }}$ weight is doubling: choose a witnessing exponent $p>1$ using the nesting lemma. Then then A_p weights are doubling gives $w(\lambda Q)\le {\lambda }^{np}[w{]}_{A_p}w(Q)$ for every cube and $\lambda >1$, and the corresponding ball bound with a constant depending only on the indicated data. No new choice principle is used in the definition itself.