Capacitary Muckenhoupt Weights

• Capacitary Muckenhoupt weights are defined via non-additive capacities like Hausdorff content and Choquet integration, mirroring the role of classical Aₚ weights.
• They underpin weighted strong and weak type inequalities for maximal operators, and support reverse Hölder, self-improving, and factorization phenomena.
• These weights are crucial in developing extrapolation theorems and characterizing oscillation spaces such as BMO and BLO in non-linear capacitary settings.

Searching arXiv for papers on capacitary Muckenhoupt weights, Hausdorff content/capacities, and related extrapolation/BMO results. Capacitary Muckenhoupt weights are weighted classes defined relative to non-additive capacities—most prominently Hausdorff content—and Choquet integration, designed to play the role that classical $A_p$ weights play in Lebesgue measure theory. In the recent Hausdorff-content framework, the classes $A_{p,\beta}$ and $\mathcal A_{p,\delta}$ characterize weighted strong and weak type inequalities for capacitary Hardy–Littlewood maximal operators, support reverse Hölder and factorization phenomena, and furnish the weight-theoretic infrastructure for extrapolation theorems and BMO/BLO characterizations in non-linear capacitary function spaces (Huang et al., 28 Sep 2025, Deshmukh, 5 Oct 2025, Zhuo et al., 3 Nov 2025).

1. Capacitary setting and Choquet integration

The ambient framework replaces a measure by a capacity $C$, that is, a monotone set function $C:\mathcal P(\mathbb R^n)\to[0,\infty]$ satisfying countable subadditivity. For a non-negative function $f$, the Choquet integral with respect to $C$ is

$$\int_{\mathbb R^n} f\,dC:=\int_0^\infty C(\{x:f(x)>t\})\,dt.$$

For Hausdorff content, if $0<\beta\le n$,

$$H_\infty^\beta(E):=\inf\left\{\sum_j \omega_\beta r_j^\beta:E\subseteq \bigcup_j B_j\right\},$$

with an equivalent formulation using cubes. In the parallel notation used elsewhere, for $A_{p,\beta}$0,

$A_{p,\beta}$1

These are the basic capacities underlying the recent theory (Deshmukh, 5 Oct 2025, Zhuo et al., 3 Nov 2025).

The associated capacitary Lebesgue spaces are defined by

$A_{p,\beta}$2

Weights enter through the weighted Hausdorff capacity

$A_{p,\beta}$3

and analogously for $A_{p,\beta}$4. The persistent technical features of the theory are quasi-continuity, quasi-everywhere statements, and the non-linearity of the Choquet integral (Deshmukh, 5 Oct 2025).

2. Definitions and principal characterizations

The capacitary Hardy–Littlewood maximal operator attached to Hausdorff content is defined by

$A_{p,\beta}$5

or, in the $A_{p,\beta}$6-notation,

$A_{p,\beta}$7

For $A_{p,\beta}$8, the capacitary Muckenhoupt condition is

$A_{p,\beta}$9

for every cube $\mathcal A_{p,\delta}$0. This defines the class $\mathcal A_{p,\delta}$1. The endpoint class $\mathcal A_{p,\delta}$2 is given by

$\mathcal A_{p,\delta}$3

In the $\mathcal A_{p,\delta}$4 notation, the definition is formally identical with $\mathcal A_{p,\delta}$5 replaced by $\mathcal A_{p,\delta}$6, and $\mathcal A_{p,\delta}$7 is characterized by the maximal inequality $\mathcal A_{p,\delta}$8 (Deshmukh, 5 Oct 2025, Zhuo et al., 3 Nov 2025).

The defining significance of these classes is their maximal-operator characterization. For $\mathcal A_{p,\delta}$9, the weighted inequality

$C$0

holds if and only if $C$1. In the $C$2 formulation, for any $C$3 and $C$4, strong-type boundedness on $C$5, weak-type boundedness from $C$6 to $C$7, and membership in $C$8 are equivalent; the weak $C$9 inequality holds if and only if $C:\mathcal P(\mathbb R^n)\to[0,\infty]$0 (Huang et al., 28 Sep 2025).

When $C:\mathcal P(\mathbb R^n)\to[0,\infty]$1, Hausdorff content is equivalent to Lebesgue measure, so $C:\mathcal P(\mathbb R^n)\to[0,\infty]$2 and the capacitary classes recover the classical Muckenhoupt scale (Huang et al., 28 Sep 2025).

3. Structural properties and internal calculus

A central feature of capacitary Muckenhoupt classes is that they reproduce much of the internal algebra of the classical theory. For $C:\mathcal P(\mathbb R^n)\to[0,\infty]$3, the reverse-exponent symmetry

$C:\mathcal P(\mathbb R^n)\to[0,\infty]$4

holds. Under the additional assumption that $C:\mathcal P(\mathbb R^n)\to[0,\infty]$5 is quasi-continuous, the classes are self-improving in two distinct senses: if $C:\mathcal P(\mathbb R^n)\to[0,\infty]$6, then $C:\mathcal P(\mathbb R^n)\to[0,\infty]$7 for some $C:\mathcal P(\mathbb R^n)\to[0,\infty]$8; and if $C:\mathcal P(\mathbb R^n)\to[0,\infty]$9, then $f$0 for some $f$1. The capacitary Jones factorization theorem states that

$f$2

These statements are explicitly described as analogues of reverse Hölder, openness, and factorization phenomena in the classical $f$3 theory (Deshmukh, 5 Oct 2025).

A technically important construction is the generation of $f$4 weights from the maximal operator. If $f$5 and $f$6 quasi-everywhere, then for $f$7,

$f$8

The proof uses decomposition of $f$9, sublinearity, weak-type estimates, and covering arguments adapted to the non-additive Choquet integral and the strong subadditivity of dyadic Hausdorff content (Deshmukh, 5 Oct 2025).

In the $C$0 notation, the theory also exhibits monotonicity in $C$1 and strict monotonicity in the Hausdorff-content dimension: $C$2 Reverse Hölder and self-improving properties, as well as Jones factorization, are established there by methods adapted to sparse coverings, packing conditions, and substitutes for linearity and Fubini (Huang et al., 28 Sep 2025).

4. Model families and geometric criteria

The capacitary classes admit explicit model weights. A basic example is the radial power weight

$C$3

which belongs to $C$4 if and only if

$C$5

This is the direct Hausdorff-content analogue of the classical Euclidean power-weight range (Huang et al., 28 Sep 2025).

Related work on classical Muckenhoupt distance weights supplies a geometric taxonomy of singular behavior near lower-dimensional sets. On an $C$6-Ahlfors metric measure space $C$7, if $C$8 is closed and $C$9 is $$\int_{\mathbb R^n} f\,dC:=\int_0^\infty C(\{x:f(x)>t\})\,dt.$$0-Ahlfors with $$\int_{\mathbb R^n} f\,dC:=\int_0^\infty C(\{x:f(x)>t\})\,dt.$$1, then

$$\int_{\mathbb R^n} f\,dC:=\int_0^\infty C(\{x:f(x)>t\})\,dt.$$2

and these weights are actually $$\int_{\mathbb R^n} f\,dC:=\int_0^\infty C(\{x:f(x)>t\})\,dt.$$3 when $$\int_{\mathbb R^n} f\,dC:=\int_0^\infty C(\{x:f(x)>t\})\,dt.$$4. The dependence on the codimension $$\int_{\mathbb R^n} f\,dC:=\int_0^\infty C(\{x:f(x)>t\})\,dt.$$5 is explicit, and the paper gives fractal illustrations such as the Sierpiński gasket (Aimar et al., 2013).

In $$\int_{\mathbb R^n} f\,dC:=\int_0^\infty C(\{x:f(x)>t\})\,dt.$$6-regular or doubling metric settings, the sharp criterion can be expressed through Assouad codimension. For a closed set $$\int_{\mathbb R^n} f\,dC:=\int_0^\infty C(\{x:f(x)>t\})\,dt.$$7 and

$$\int_{\mathbb R^n} f\,dC:=\int_0^\infty C(\{x:f(x)>t\})\,dt.$$8

one has

$$\int_{\mathbb R^n} f\,dC:=\int_0^\infty C(\{x:f(x)>t\})\,dt.$$9

and

$0<\beta\le n$0

These bounds are stated as sharp, with consequences for Hardy–Sobolev and fractional Hardy–Sobolev inequalities (Dyda et al., 2017).

For the endpoint $0<\beta\le n$1 regime in spaces of homogeneous type, there is a complete geometric characterization: $0<\beta\le n$2 for some $0<\beta\le n$3 if and only if $0<\beta\le n$4 is weakly porous and the maximal $0<\beta\le n$5-free hole function $0<\beta\le n$6 is doubling (Aimar et al., 2024). For $0<\beta\le n$7 with $0<\beta\le n$8, the admissible sets are strictly more general: existence of a nontrivial distance weight $0<\beta\le n$9 is equivalent to a multiscale distribution condition controlling the ratio between the largest and smallest pores that account for a fixed proportion of the complement inside every cube. The paper states that this condition is more general than weak porosity and allows a balance of small-scale and large-scale pores rather than a rigid large-pore condition (Vargas, 24 Jul 2025).

5. Maximal inequalities, extrapolation, and oscillation spaces

The primary analytical role of capacitary Muckenhoupt weights is the exact control of weighted norm inequalities for maximal operators. In the Hausdorff-content setting, for $$H_\infty^\beta(E):=\inf\left\{\sum_j \omega_\beta r_j^\beta:E\subseteq \bigcup_j B_j\right\},$$0, boundedness of the capacitary Hardy–Littlewood maximal operator on $$H_\infty^\beta(E):=\inf\left\{\sum_j \omega_\beta r_j^\beta:E\subseteq \bigcup_j B_j\right\},$$1, boundedness on the weak weighted Choquet-Lebesgue space $$H_\infty^\beta(E):=\inf\left\{\sum_j \omega_\beta r_j^\beta:E\subseteq \bigcup_j B_j\right\},$$2, and membership in $$H_\infty^\beta(E):=\inf\left\{\sum_j \omega_\beta r_j^\beta:E\subseteq \bigcup_j B_j\right\},$$3 are equivalent; the weighted weak $$H_\infty^\beta(E):=\inf\left\{\sum_j \omega_\beta r_j^\beta:E\subseteq \bigcup_j B_j\right\},$$4 inequality is equivalent to $$H_\infty^\beta(E):=\inf\left\{\sum_j \omega_\beta r_j^\beta:E\subseteq \bigcup_j B_j\right\},$$5 (Huang et al., 28 Sep 2025).

A major recent application is extrapolation. If $$H_\infty^\beta(E):=\inf\left\{\sum_j \omega_\beta r_j^\beta:E\subseteq \bigcup_j B_j\right\},$$6 and an operator $$H_\infty^\beta(E):=\inf\left\{\sum_j \omega_\beta r_j^\beta:E\subseteq \bigcup_j B_j\right\},$$7 is bounded on

$$H_\infty^\beta(E):=\inf\left\{\sum_j \omega_\beta r_j^\beta:E\subseteq \bigcup_j B_j\right\},$$8

for all $$H_\infty^\beta(E):=\inf\left\{\sum_j \omega_\beta r_j^\beta:E\subseteq \bigcup_j B_j\right\},$$9, with norm depending only on the $A_{p,\beta}$00-constant, then two conclusions follow. First, for all $A_{p,\beta}$01 and $A_{p,\beta}$02, the operator $A_{p,\beta}$03 is bounded on $A_{p,\beta}$04. Second, for all $A_{p,\beta}$05 and all quasi-continuous $A_{p,\beta}$06, the operator $A_{p,\beta}$07 is bounded on the same weighted capacitary $A_{p,\beta}$08 space. The proof is described as relying on Jones factorization, self-improvement, duality, and the new $A_{p,\beta}$09-construction from maximal functions (Deshmukh, 5 Oct 2025).

Capacitary Muckenhoupt weights also control oscillation spaces defined with Hausdorff content. For every $A_{p,\beta}$10,

$A_{p,\beta}$11

and

$A_{p,\beta}$12

The same work proves a John–Nirenberg inequality for BLO, a capacitary weighted John–Nirenberg inequality, the coincidence of weighted and unweighted BMO spaces for $A_{p,\beta}$13, and factorization theorems of BMO/BLO via Hardy–Littlewood maximal operators (Zhuo et al., 3 Nov 2025).

A common misconception is that these results are formal transcriptions of the classical measure-theoretic theory. The recent extrapolation work explicitly states that the theorem is not merely a translation from the classical case: quasi-continuity, quasi-everywhere formulations, Choquet integration, and recent structural results on capacitary weights are essential to the argument (Deshmukh, 5 Oct 2025).

6. Adjacent weighted-capacity frameworks and conceptual distinctions

Capacitary Muckenhoupt weights sit inside a broader ecosystem of weighted potential theory, but they are not identical to every weight framework attached to capacities. One related theory concerns local Muckenhoupt weights $A_{p,\beta}$14, defined by restricting the $A_{p,\beta}$15 supremum to cubes of bounded sidelength. In that setting, one obtains weighted Bessel and local Riesz capacities, local Muckenhoupt–Wheeden inequalities, capacitary strong type inequalities for nonlinear potentials, boundedness of the local maximal function on Choquet-Lorentz spaces, and a Kellogg property stating that the thin points of a Borel set form a set of weighted capacity zero (Ooi, 2024).

Another adjacent framework uses $A_{p,\beta}$16-admissible weights, meaning weights $A_{p,\beta}$17 for which $A_{p,\beta}$18 is doubling and satisfies a $A_{p,\beta}$19-Poincaré inequality. Every classical $A_{p,\beta}$20 weight is $A_{p,\beta}$21-admissible, but in dimensions $A_{p,\beta}$22 the class of $A_{p,\beta}$23-admissible weights is strictly larger than $A_{p,\beta}$24. For such weights, removable-set results are proved via geometric porosity and cube coverings rather than via a new capacitary Muckenhoupt criterion; for $A_{p,\beta}$25, the classical equivalence between removability and capacity remains available because $A_{p,\beta}$26 in that case (Esmayli et al., 26 May 2025).

These distinctions are substantive. Capacitary Muckenhoupt weights in the Hausdorff-content sense are the precise scale governing weighted maximal inequalities, extrapolation, and BMO/BLO theory for Choquet-integral spaces. Local Muckenhoupt weights and $A_{p,\beta}$27-admissible weights govern related but different capacity problems, with different localization, geometry, and functional-analytic consequences.