Reverse Hölder Inequality

• Reverse Hölder Inequality is a foundational concept that defines conditions under which the mean of a function raised to a higher exponent is controlled by its mean at a smaller exponent.
• It extends classical integral inequalities by incorporating sharp quantitative constants and self-improving properties in settings such as multilinear, Orlicz, and variable exponent frameworks.
• Applications of the Reverse Hölder Inequality span operator theory, nonlinear PDE regularity, geometric rigidity, and probabilistic models, offering robust tools for integral and norm estimates.

The reverse Hölder inequality is a fundamental tool in harmonic analysis, partial differential equations, and probability, serving as a self-improving regularity principle for weighted integrability, nonlinear structures, and stochastic models. It quantifies when the mean of a function or a weight under a larger exponent is controlled by its mean under a smaller exponent—often in sharp quantitative form depending on structural constants or weight classes. The inequality, classical or generalized, often underpins higher integrability, dimension-free regularity, operator bounds, and geometric rigidity phenomena across analysis and geometry.

1. Classical and Weighted Reverse Hölder Inequalities

The most prevalent form asserts: if $w \geq 0$ is locally integrable on %%%%1%%%% and for $s > 1$,

$$[w]_{RH_s} = \sup_Q \left(\frac{1}{|Q|} \int_Q w^s \right)^{1/s} \left(\frac{1}{|Q|}\int_Q w \right)^{-1} < \infty,$$

then for every cube $Q$,

$$\left(\frac{1}{|Q|} \int_Q w^s \right)^{1/s} \leq C \frac{1}{|Q|}\int_Q w.$$

This encapsulates the standard (linear) reverse Hölder property for weights, with $C$ depending on $[w]_{RH_s}$ (Cruz-Uribe et al., 2017).

For Muckenhoupt classes $A_p$, $1 < p < \infty$, one similarly has reverse Hölder self-improvement: there exists $r > 1$ and $C$ depending on $[w]_{A_p}$ so that

$$\left( \frac{1}{|Q|} \int_Q w^r \right)^{1/r} \leq C \frac{1}{|Q|}\int_Q w.$$

In endpoint and limiting regimes, sharp constants and dependence on structural invariants (e.g., Fujii–Wilson or Wilson's $A_{\infty}$ constant) emerge (Ortiz-Caraballo et al., 2012, Parissis et al., 2016).

2. Sharp Quantitative Versions and Optimal Constants

Recent works have focused on optimal exponent improvements and sharp constants, essential for rigorous operator estimates and fine regularity. For $A_{\infty}$ weights in $\mathbb{R}^n$, let

$$[w]_{A_{\infty}} := \sup_Q \frac{1}{w(Q)} \int_Q M(w \chi_Q),$$

where $M$ is the Hardy-Littlewood maximal operator. Then for $\tau_n = 2^{n+1}$ and $r_w = 1 + 1 / (\tau_n [w]_{A_{\infty}})$,

$$\left( \frac{1}{|Q|} \int_Q w^{r_w} \right)^{1/r_w} \leq 2 \frac{1}{|Q|}\int_Q w$$

holds for all cubes $Q$, and this exponent is optimal: for any $(r, C)$ such that the RHI holds, necessarily $[w]_{A_{\infty}} \leq \tau_n C (r - 1)^{-1}$ (Ortiz-Caraballo et al., 2012).

For strong $A_p^*$ weights measured by rectangles and general Radon measures,

$$\frac{1}{\mu(R)}\int_R w^{1+\epsilon} d\mu \leq 2 \left( \frac{1}{\mu(R)} \int_R w d\mu \right)^{1+\epsilon}$$

with $0 < \epsilon \leq 1 / (2^{p+2}[w]_{A_p^*} - 1)$, and the constant $2$ is independent of the dimension (Luque et al., 2015).

For "flat" $A_\infty$ weights (Fujii–Wilson constant close to 1), the admissible exponent blows up as the constant approaches 1, interpolating between weighted and unweighted regimes: $$1 < r < 1 + \frac{1}{2^n(\delta-1)},\;\; C(w, r) = \delta \frac{r' - 1}{r' - 1 - 2^n(\delta - 1)} \quad \text{with} \quad r' = \frac{r}{r-1},\;\;(w)_{A_\infty} < \delta$$ (Parissis et al., 2016).

3. Multilinear, Weak, Orlicz, and Variable Exponent Extensions

Reverse Hölder inequalities extend beyond classical $L^p$ to:

• Multilinear setting: If $s_1, \ldots, s_m > 1$, $\sum 1/s_i = 1$, and $w_i \in RH_{s_i}$,

$$\prod_{i=1}^m \left(\frac{1}{|Q|}\int_Q w_i^{s_i}\right)^{1/s_i} \leq C \prod_{i=1}^m \frac{1}{|Q|}\int_Q w_i$$

which underpins factorization and norm bounds for multilinear maximal operators and weights (Cruz-Uribe et al., 2017).

• Weak RHIs on metric measure spaces: For a weight $w$ on $(X, d, \mu)$, $w$ satisfies a weak RHI if

$$\left( \frac{1}{\mu(B)}\int_B w^p d\mu \right)^{1/p} \leq C \frac{1}{\mu(2B)} \int_{2B} w d\mu$$

with $p > 1, C > 0$, allowing nondoubling weights and various characterizations equivalent to generalized (weak) $A_\infty$ properties (Kinnunen et al., 2021).

• Orlicz scale: For Young functions $\Psi$,

$$[RH_\Psi] := \sup_Q \frac{\|w\|_{\Psi, Q}}{\langle w \rangle_Q} < \infty$$

generalizes the classical reverse Hölder to non-power growth, enabling total extrapolation of operator bounds in the Orlicz scale (Anderson et al., 2016).

• Variable exponent ($p(\cdot)$) and matrix weights: For $w \in \mathcal{A}_{p(\cdot)}$, there exists $r > 1$,

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

with explicit dependence of $r$ and $C$ on $[w]_{\mathcal{A}_{p(\cdot)}}$, log-Hölder constants, and dimension (Cruz-Uribe et al., 2024).

4. Analytical, Geometric, and Probabilistic Generalizations

• Nonlinear PDE regularity: The RHI for gradients is crucial for scalar solutions to degenerate parabolic equations, e.g., Trudinger’s equation. For $u \ge 0$, $p \ge 2$,

$$\left(\int_{Q_{r,r^p}} |\nabla u|^{p(1+\epsilon)} \right)^{1/(p(1+\epsilon))} \leq C \left(\int_{Q_{2r, (2r)^p}} |\nabla u|^p \right)^{1/p} (1 + \text{lower order terms})^{\epsilon/p}$$

with $\epsilon, C$ depending only on dimension, $p$, and structural constants (Saari et al., 2019).

• Reverse Hölder for first Dirichlet Laplacian eigenfunctions: In RCD$(K,N)$ spaces, for $0 < p < q$,

$$\|u\|_{L^q(\Omega)} \geq \frac{\|z\|_{L^q}}{\|z\|_{L^p}} \|u\|_{L^p(\Omega)}$$

where $z$ is the comparator eigenfunction on the spherical suspension model, producing rigidity and stability consequences (Gunes et al., 2021).

• Probabilistic versions: If $(X, Y)$ are independent random vectors, e.g., uniformly distributed on $\ell_p^n$ balls,

$$\langle X, Y \rangle \geq (m_{p,q} - \varepsilon) \|X\|_p \|Y\|_q$$

holds with high probability for large $n$, with explicit $m_{p,q}$ given by Gamma functions, quantifying probabilistic reversals of Hölder's inequality (Frühwirth et al., 2022).

• Kähler geometry: For Kähler metrics on Fano varieties or their singular analogues, under uniform Ricci potential bounds there is a reverse Hölder-type control for Darvas $L^p$ Finsler metrics: $$d_p(u, 0) \leq A_{p, n} e^{2R/p} d_1(u, 0) + B_{p, \gamma} e^R$$ with $A_{p, n}, B_{p, \gamma}$ universal (Berman, 2023).

5. Operator Theory, Self-Improvement, and Applications

Sharp reverse Hölder constants yield best-possible weighted norm inequalities for Calderón–Zygmund operators, their commutators, and maximal functions. For $A_\infty$ weights,

$$\|Tf\|_{L^p(w)} \leq C p' [w]_{A_\infty} \|Mf\|_{L^p(w)}$$

and for Cp weights,

$$\|Tf\|_{L^p(w)} \lesssim [w]_{C_q}(1 + \log^+[w]_{C_q}) \|Mf\|_{L^p(w)},\; w \in C_q,\, q > p > 1$$

with logarithmic correction quantifying the sufficiency of Cp in Sawyer's theorem (Canto, 2018).

The reverse Hölder inequality is pivotal in the self-improvement of weight classes ($A_p$ to $RH_s$), factorization results for multilinear weights, the structure theorem for $A_{\vec{p}}$ weights, and the streamlined sufficient conditions in two-weight norm inequalities for maximal operators (Cruz-Uribe et al., 2017).

6. Geometry of Extension, Weak and Non-Doubling Regimes

RHI extends to even extensions of functions and non-doubling measures. For $f: \mathbb{R}_+ \to \mathbb{R}_+$, one can bound

$$R_{\alpha, \beta}(\tilde{f}) \leq 2^{1/\alpha} P_{\alpha, \beta}(f)$$

with $P_{\alpha, \beta}$ the best RHI constant on $\mathbb{R}_+$ and $R_{\alpha, \beta}$ its counterpart on $\mathbb{R}$, achieving asymptotic sharpness as $\beta \to \infty$ (Shalukhina, 2018).

Weak RHIs characterize weak $A_\infty$ weights—generalizing Muckenhoupt’s $A_\infty$ to non-doubling and vanishing settings—with ten equivalent conditions spanning set-decay, log-bump, and BMO-pairing properties (Kinnunen et al., 2021).

7. Summary Table: Sharp Reverse Hölder Inequality Regimes

Weight Class or Setting	RHI Form / Result	Optimal Constant/Exponent
$A_p$ ($1 < p < \infty$), cubes	$$\left(\frac{1}{|Q|}\int_Q w^r\right)^{1/r} \leq C\frac{1}{|Q|}\int_Q w$$	$r - 1 \approx 1/[w]_{A_p}$
$A_\infty$, Wilson	$r_w = 1 + 1/(2^{n+1}[w]_{A_\infty})$	Factor $2$ (Ortiz-Caraballo et al., 2012)
Strong $A_p^*$, rectangles	$$(1/\mu(R))\int_R w^{1+\epsilon} d\mu \leq 2 (\dots)$$	$\epsilon \leq 1/(2^{p+2}[w]_{A_p^*})$
Weak $A_\infty$	$$\left( \frac{1}{\mu(B)}\int_B w^p \right)^{1/p} \leq C \frac{1}{\mu(2B)}\int_{2B} w$$	$p > 1$, $C$ via covering constants (Kinnunen et al., 2021)
Variable exponent $\mathcal{A}_{p(\cdot)}$	$$|Q|^{-1/(r p_Q)}\|w\|_{L^{r p(\cdot)}(Q)} \leq C |Q|^{-1/p_Q} \|w\|_{L^{p(\cdot)}(Q)}$$	Exponent, constant depend on $[w]_{\mathcal{A}_{p(\cdot)}}$ (Cruz-Uribe et al., 2024)

• (Ortiz-Caraballo et al., 2012) Improving bounds for singular operators via Sharp Reverse Hölder Inequality for $A_{\infty}$
• (Luque et al., 2015) Reverse Hölder Property for strong weights and general measures
• (Parissis et al., 2016) Asymptotically sharp reverse Hölder inequalities for flat Muckenhoupt weights
• (Cruz-Uribe et al., 2017) A multilinear reverse Hölder inequality with applications to multilinear weighted norm inequalities
• (Shalukhina, 2018) On the extension of the Reverse Hölder Inequality for power functions on the real axis
• (Canto, 2018) Sharp reverse Hölder inequality for $C_p$ weights and applications
• (Saari et al., 2019) A reverse Hölder inequality for the gradient of solutions to Trudinger's equation
• (Kinnunen et al., 2021) Characterizations of weak reverse Hölder inequalities on metric measure spaces
• (Frühwirth et al., 2022) Hölder's inequality and its reverse-a probabilistic point of view
• (Berman, 2023) Reverse Hölder inequalities on the space of Kähler metrics of a Fano variety and effective openness
• (Cruz-Uribe et al., 2024) The reverse Hölder inequality for $\mathcal{A}_{p(\cdot)}$ weights with applications to matrix weights

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The reverse Hölder inequality constitutes both a deep structural regularity principle and a central constructive device for sharp bounds and rigidity in weight theory, geometric analysis, PDE regularity, and operator theory, with extensive ramifications in quantitative estimates and self-improvement phenomena.