Super-Linear Reverse Hölder Inequalities

• Super-linear reverse Hölder inequalities are quantitative integrability bounds that extend the classical reverse Hölder property with explicit constants and self-improvement effects.
• They are applied across harmonic analysis, PDEs, high-dimensional probability, and operator theory to refine norm comparisons and establish maximal integrability.
• Recent advances include sharp constant formulations and extensions to variable exponent, Orlicz, and matrix contexts, underscoring their foundational role in analysis.

Super-linear reverse Hölder inequalities are quantitative integrability bounds that extend the classical reverse Hölder property into the regime of strictly super-linear (i.e., exponent $p>1$) averages, yielding self-improving integrability and sharp norm-comparisons for functions, measures, weights, operators, and distributions in functional analysis, harmonic analysis, partial differential equations, and high-dimensional probability. This article synthesizes core models, explicit constants, functional and geometric contexts, and rigorous sharpness results drawn from recent research.

1. Core Quantitative Model and Explicit Formulations

Super-linear reverse Hölder inequalities consistently exhibit the following structural form on a measure space $(X,\mu)$, for $p>1$: $\left( \fint_{Q} f^p\,d\mu \right)^{1/p} \leq C \fint_{Q} f\,d\mu\,,$ where $f\geq0$ is measurable, $Q$ is a cube (or ball, or rectangle), and $C$ depends on dimension, underlying quantitative weight characteristic, and other structural data.

Recent advances provide sharp constants and explicit dependence:

• Muckenhoupt $A_\infty$ and Strong $A^*_p$ Weights: Asymptotically sharp exponents and constants are established so that as the weight approaches uniformity, the integrability exponent blows up, e.g.,

$\left(\fint_Q w^r\right)^{1/r} \leq (w)_{A_\infty}^{1/r'} \left(\frac{r'-1}{r'-1-2^n\big[(w)_{A_\infty}-1\big]}\right)^{1/r} \fint_Q w,$

where $(w)_{A_\infty}$ is the Fujii–Wilson constant, $r < 1+1/[2^n((w)_{A_\infty}-1)]$ (Parissis et al., 2016), and analogously

$(\dashint_R w^{1+\varepsilon}d\mu)^{1/(1+\varepsilon)} \leq 2\,\dashint_R w\,d\mu, \quad \text{for } 0 < \varepsilon < 2^{-(p+2)}[w]_{A_p^*}^{-1}$

for strong $A_p^*$ weights, with no dimension dependence in the constant (Luque et al., 2015).

• Variable exponent weights $\mathcal{A}_{p(\cdot)}$: For Lebesgue spaces with variable exponent $p(x)\in [1,\infty)$, and $w\in A_{p(\cdot)}$, there exists $r>1$ such that

$$|Q|^{-1/r p_Q} \|w\chi_Q\|_{L^{rp(\cdot)}} \leq C_{p(\cdot)} |Q|^{-1/p_Q}\|w\chi_Q\|_{L^{p(\cdot)}},$$

with $$r = 1 + 1/[C_* [w]_{A_{p(\cdot)}}^{1+2C_\infty p^+/(p_\infty p^-)}]$$; the gain $\varepsilon = r-1$ is explicit (Cruz-Uribe et al., 2024).

• Extremal Sobolev Functions: Sharp reverse Hölder inequalities for extremals of Sobolev constants are quantified, e.g., for $u$ attaining the optimal Sobolev constant,

$\|u\|_{L^p} \geq K\,\|u\|_{L^q}, \quad q > p,$

with $K$ given explicitly in terms of geometric parameters and Sobolev constant (Carroll et al., 2014).

2. Self-Improvement and Structural Mechanisms

A recurring phenomenon in the super-linear regime is quantitative self-improvement: if a reverse Hölder inequality holds with exponent $p>1$, then there exists $\varepsilon>0$ depending only on the underlying constants (dimension, quantitative characteristic) such that the inequality holds for $p+\varepsilon$. This "Gehring-type" mechanism is now realized with explicit formulas:

• For flat $A_\infty$ weights, as $(w)_{A_\infty}\to 1^+$, the allowed exponent grows arbitrarily large, demonstrating maximal integrability for nearly constant weights (Parissis et al., 2016).
• In the dyadic setting, the reverse Hölder constant after rearrangement is sharpened via explicit formula $c' = 2^n c - 2^n + 1$, and maximal $L^q$ integrability is characterized as the sharp root $p_0(p,c')$ of a fixed algebraic equation (Nikolidakis et al., 2014).

A prototypical proof sequence employs:

• Covering or stopping-time arguments (Calderón–Zygmund decomposition, Riesz rising sun lemma) to localize high-mean sets (Luque et al., 2015, Nikolidakis et al., 2014).
• Level-set analysis or maximal function techniques to transfer local bounds to global integrability (Parissis et al., 2016).

3. Generalized Weight and Operator Settings

Super-linear reverse Hölder inequalities admit generalization in variable exponent, Orlicz, and operator contexts:

• Orlicz Scale: For Young functions $\Phi$ with super-linear growth, reverse Hölder inequalities and extrapolation theorems provide additional integrability on the class $RH_{\Phi}$, which is properly stronger than $A_\infty$ and allows refined norm inequalities for Calderón–Zygmund and multilinear singular integral operators (Anderson et al., 2016).
• Matrix and Operator Norms: In matrix analysis, reverse Hölder inequalities extend to Schatten quasi-norms (0<s<1) and super-linear powers, providing dual variational representations and characterizations of equality (Chayes, 2021). For positive operators in Hilbert spaces, explicit bounds for differences between $A^{t}x$ and $(Ax)^t$ are given in terms of lower moments (reverse Hölder–McCarthy) (Hajmohamadi et al., 2018).

4. Nonlinear PDEs and Harmonic Analysis Applications

Super-linear reverse Hölder inequalities are fundamental in establishing higher integrability for gradients in nonlinear PDEs:

• Trudinger Equation: For solutions $u\geq0$ of the doubly-nonlinear equation, a covering and reverse Hölder iteration yields

$$\|\nabla u\|_{L^{p(1+\varepsilon)}(Q)} \leq C\,\|\nabla u\|_{L^p(2Q)},$$

where $\varepsilon$ and $C$ depend only on structure constants, without upper bounds on $p$ (Saari et al., 2019).

• Riesz Transforms and Metric Spaces: On doubling metric measure spaces, super-linear reverse Hölder inequalities ($q>1$) for weights or gradients are central to extending $L^p$-boundedness for Riesz transforms and related operators, with explicit quantitative control via the self-improvement mechanism (Gehring lemma) (Bernicot et al., 2015).

5. Probability, Gaussian Measures, and Extremal Distributions

Super-linear reverse Hölder phenomena govern discrete and continuous moment inequalities in high-dimensional probability:

• Log-Concave Distributions: For centred log-concave $X$, sharp $L_p$-$L_1$ and mixed $L_p$-$L_q$ reverse Hölder inequalities are determined by a universal family of two-sided exponential laws, with a phase transition at $p_0\approx2.9414$ (Melbourne et al., 2 May 2025).

$$\|X\|_p \geq \frac{\Gamma(p+1)^{1/p}}{\Gamma(q+1)^{1/q}}\|X\|_q$$

for $-1<p\leq1\leq q\leq p_0$, with sharp attainment on the Laplace and exponential distributions.

• Correlated Gaussians and Functional Inequalities: For block covariance matrices $T$ and exponents $p_i>1$, a matrix criterion $T\succeq P$ yields

$$\mathbb{E}\prod_{i=1}^m f_i(X_i) \geq \prod_{i=1}^m (\mathbb{E}f_i(X_i)^{p_i})^{1/p_i}$$

with equality for exponential-linear test functions. This unifies reverse hypercontractivity, Prekopa–Leindler, and Young inequalities (Chen et al., 2013).

6. Sharpness, Optimality, and Limiting Cases

Recent advances emphasize sharp constants, characterization of equality, and limiting cases:

• For $A_\infty$ and $A_1$ weights, both the range of allowed exponents and multiplicative constants have been shown to be best possible via explicit extremizers (Parissis et al., 2016, Luque et al., 2015).
• The continuous improvement as the underlying weight or operator becomes "flatter" or more regular is quantified; in particular, as the characteristic tends to its minimal value, the integrability exponent diverges.
• Reverse Hölder inequalities for extremal Sobolev functions attain equality only for balls (Carroll et al., 2014), and in probabilistic and operator settings, extremizers are explicit.

7. Summary Table: Explicit Super-linear RHI Models

Context/class	Inequality (core model)	Quantitative range/constant
Classical $A_p$/$A_\infty$	$\left(\fint_Q w^r\right)^{1/r} \leq C\,\fint_Q w$	$r < 1+1/[2^n((w)_{A_\infty}-1)]$ (Parissis et al., 2016)
Dyadic cubes in $[0,1]^n$	$$\frac{1}{|Q|}\int_Q \varphi^p \leq c\,(\frac{1}{|Q|}\int_Q \varphi)^p$$	$c' = 2^n c - 2^n + 1$, explicit $p_0(p,c')$ (Nikolidakis et al., 2014)
Variable exponent weights	$$|Q|^{-1/rp_Q}\|w\chi_Q\|_{L^{rp(\cdot)}} \leq C\,|Q|^{-1/p_Q}\|w\chi_Q\|_{L^{p(\cdot)}}$$	$r = 1 + 1/[\text{quantitative characteristic}]$ (Cruz-Uribe et al., 2024)
Sobolev extremals	$\|u\|_{L^p} \geq K\,\|u\|_{L^q}$	$K$ explicit from geometry, only balls attain equality (Carroll et al., 2014)
Log-concave $X$	$$\|X\|_p \geq [\Gamma(p+1)^{1/p}/\Gamma(q+1)^{1/q}]\|X\|_q$$	$p\leq1\leq q\leq p_0 \approx 2.94$ (Melbourne et al., 2 May 2025)

8. Further Directions and Open Problems

Current research continues to refine constants and exponents, extend reverse Hölder theory to nonlinear growth (Orlicz, $p(\cdot)$), matrix-valued weights, non-doubling and product measures, and to applications in sharp weighted norm inequalities for singular integrals. Optimizing constants in variable exponent reverse Hölder remains open (Cruz-Uribe et al., 2024). Extending sharp RHI to sign-changing or vector-valued solutions in nonlinear PDE theory is also unresolved (Saari et al., 2019).

Super-linear reverse Hölder inequalities thus constitute a foundational, highly quantitative framework with far-reaching consequences in analysis, probability, and operator theory.