Muckenhoupt A_p and A_1 weights

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

Fix $n\ge 1$. All averages and maximal functions below are the cube-based ones of Axis-parallel cubes, their averages, and cube maximal functions. Nonnegative measurable averages are extended integrals in $[0,\mathrm{\infty }]$; in particular the reciprocal-weight average may be infinite before $A_p$ membership is established. Use the positive finite representative of Weights, their associated measures, and the spaces L^p(w) for reciprocal powers.

The class $A_p$, $1<p<\mathrm{\infty }$. A weight $w$ (Weights, their associated measures, and the spaces L^p(w)) belongs to $A_p$ when $[w{]}_{A_p}:={\sup }_Q⟨w{⟩}_Q⟨w^{-1\mathrm{/}(p-1)}{⟩}_Q^{p-1}<\mathrm{\infty },$ the supremum over all axis-parallel cubes $Q$, and $[w{]}_{A_p}$ is the $A_p$ characteristic of $w$. The supremum over all Euclidean balls instead differs from the cube supremum by at most a dimensional factor: the two sandwiches of Ball and cube maximal functions are pointwise comparable compare $w$- and $w^{-1\mathrm{/}(p-1)}$-averages over nested balls and cubes, so ${\sup }_B⟨w{⟩}_B⟨w^{-1\mathrm{/}(p-1)}{⟩}_B^{p-1}$ is finite exactly when $[w{]}_{A_p}$ is, and the two numbers are bounded by dimensional powers of one another. If $[w{]}_{A_p}<\mathrm{\infty }$, then $⟨w^{-1\mathrm{/}(p-1)}{⟩}_Q<\mathrm{\infty }$ and $⟨w{⟩}_Q>0$ for every cube $Q$ (both are finite or nonzero because $w>0$ a.e. and $[w{]}_{A_p}$ is finite), so $w^{-1\mathrm{/}(p-1)}\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ and every cube satisfies $0<w(Q)<\mathrm{\infty }$ and $0<{\int }_Q{w}^{-1\mathrm{/}(p-1)}d\lambda <\mathrm{\infty }$.

The class $A_1$. A weight $w$ belongs to $A_1$ when $M^{\ast }w\le Cw$ almost everywhere for some constant $C<\mathrm{\infty }$, where $M^{\ast }$ is the uncentred ball maximal function of The centered and uncentered Hardy-Littlewood maximal functions; by the ball-cube comparison this is equivalent to the same pointwise bound for the uncentred cube maximal function $M_c^{\ast }$, and $M_{c}w\le M_c^{\ast }w\le 2^n{M}_{c}w$ pointwise (an uncentred cube $Q\ni x$ of side $\mathrm{\ell }$ is contained in the centred cube $Q(x,\mathrm{\ell })$ of $2^n$-fold volume), so the centred form is equivalent as well. For an $A_1$ weight the infimum $a$ of admissible constants is attained: choose admissible $C_j<a+1\mathrm{/}(j+1)$, discard their countably many exceptional null sets, and pass to the limit in $M^{\ast }w\le C_{j}w$. Thus the $A_1$ characteristic $[w{]}_{A_1}$ is defined as the least such $C$, and $M^{\ast }w\le [w{]}_{A_1}w$ holds almost everywhere. The class $A_1$ is not obtained by substituting $p=1$ into the $A_p$ formula: the reciprocal power $w^{-1\mathrm{/}(p-1)}$ has no finite-exponent analogue, and the equivalent cube-average/essential-infimum condition $⟨w{⟩}_Q\le c_n[w{]}_{A_1}{\mathrm{ess\hspace{0.17em}inf}}_{Q}w$, with a dimensional factor $c_n$ is a separate lemma on this page. Here ${\mathrm{ess\hspace{0.17em}inf}}_{Q}w:=\sup \{a\ge 0:w\ge a\text{ a.e. on }Q\}$, the lower-bound analogue of The essential supremum of a measurable function with respect to a measure. It is finite because $w$ is integrable on $Q$, and $w$ is at least this supremum a.e.: take a sequence of admissible bounds tending to it and discard their countable union of null exceptional sets.

Invariance. Translations, positive isotropic dilations and positive scalar multiples preserve both classes with the same characteristic: for ${\tau }_{z}w(x):=w(x-z)$ one has $⟨{\tau }_{z}w{⟩}_Q=⟨w{⟩}_{Q-z}$ and, with $\sigma =w^{-1\mathrm{/}(p-1)}$, $⟨{\tau }_z\sigma {⟩}_Q^{p-1}=⟨\sigma {⟩}_{Q-z}^{p-1}$ by the $C^1$ change-of-variables theorem applied to the translation $x\mapsto x-z$ (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions), which carries cubes to cubes of the same side length; for ${\delta }_{\lambda }w(x):=w(\lambda x)$ with $\lambda >0$ the determinant ${\lambda }^n$ introduced by the substitution $x\mapsto \lambda x$ cancels between the two averages, since ${\delta }_{\lambda }\sigma =({\delta }_{\lambda }w{)}^{-1\mathrm{/}(p-1)}$; and $⟨cw{⟩}_Q⟨(cw{)}^{-1\mathrm{/}(p-1)}{⟩}_Q^{p-1}=⟨w{⟩}_Q⟨w^{-1\mathrm{/}(p-1)}{⟩}_Q^{p-1}$ for $c>0$. The same computations apply verbatim to the condition $M^{\ast }w\le Cw$ a.e.