--- title: Reverse Hölder Inequality url: https://www.emergentmind.com/topics/reverse-holder-inequality type: topic --- # Reverse Hölder Inequality 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 $\mathbb{R}^n$ 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 $[w]_{RH_s}$ [1701.07800]. 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 [1204.1667, 1612.01932]. ## 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 $[w]_{A_{\infty}} \leq \tau_n C (r - 1)^{-1}$ [1204.1667]. 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 [1512.01112]. 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: \[ 1 < r < 1 + \frac{1}{2^n(\delta-1)},\;\; C(w, r) = \delta \frac{r' - 1}{r' - 1 - 2^n(\delta - 1)} \quad \text{with} \quad r' = \frac{r}{r-1},\;\;(w)_{A_\infty} < \delta \] [1612.01932]. ## 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 [1701.07800]. - **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 [2107.05022]. - **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 [1605.00922]. - **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 [2411.12849]. ## 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 [1910.10498]. - **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 [2110.00292]. - **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 [2209.13442]. - **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 [2309.16278]. ## 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 [1811.05209]. 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 [1701.07800]. ## 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 $\beta \to \infty$ [1810.05693]. 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 [2107.05022]. ## 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 $2$ [1204.1667] | | 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 [2107.05022]| | 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 $[w]_{\mathcal{A}_{p(\cdot)}}$ [2411.12849] | ## References - [1204.1667] Improving bounds for singular operators via Sharp Reverse Hölder Inequality for $A_{\infty}$ - [1512.01112] Reverse Hölder Property for strong weights and general measures - [1612.01932] Asymptotically sharp reverse Hölder inequalities for flat Muckenhoupt weights - [1701.07800] A multilinear reverse Hölder inequality with applications to multilinear weighted norm inequalities - [1810.05693] On the extension of the Reverse Hölder Inequality for power functions on the real axis - [1811.05209] Sharp reverse Hölder inequality for $C_p$ weights and applications - [1910.10498] A reverse Hölder inequality for the gradient of solutions to Trudinger's equation - [2107.05022] Characterizations of weak reverse Hölder inequalities on metric measure spaces - [2209.13442] Hölder's inequality and its reverse-a probabilistic point of view - [2309.16278] Reverse Hölder inequalities on the space of Kähler metrics of a Fano variety and effective openness - [2411.12849] The reverse Hölder inequality for $\mathcal{A}_{p(\cdot)}$ weights with applications to matrix weights --- 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. Source: https://www.emergentmind.com/topics/reverse-holder-inequality