Gradient Reverse Hölder Inequalities

• Gradient reverse Hölder inequalities are quantitative estimates that upgrade base energy bounds to higher gradient integrability in PDEs and variational settings.
• They use covering lemmas and Gehring iterations to achieve self-improving properties in both degenerate and uniformly elliptic systems.
• These inequalities support regularity theory and underpin applications to Riesz transforms, multi-phase integrals, and kinetic equations.

Gradient reverse Hölder inequalities are quantitative estimates controlling the higher integrability of gradient functionals in partial differential equations, variational problems, and abstract operator settings. They provide local or scale-invariant $L^p$ improvements relative to base energy bounds and underpin regularity theory for both degenerate and uniformly elliptic systems. These inequalities are self-improving by means of covering lemmas and Gehring-type iterations, and have become essential in the analysis of Riesz transforms, degenerate parabolic flows, multi-phase variational integrals, and equations with rough coefficients.

1. Foundational Principles and Prototypical Inequalities

The prototypical gradient reverse Hölder inequality (RHI) asserts that for a solution $u$ (or a suitable transformation of $u$), the $L^q$-norm of its gradient over a domain (or cylinder) can be controlled by its $L^p$-norm, for some $q>p$, typically with constants reflecting problem structure: $\left( \fint_{D} |\nabla u|^{q}\right)^{1/q} \leq C \left( \fint_{D'} |\nabla u|^{p}\right)^{1/p}, \quad D \subset D' \subset \Omega.$ Such inequalities are established in a broad range of settings, including degenerate/singular parabolic equations, divergence-form elliptic equations with non-smooth coefficients, and abstract accretive operators on metric-measure spaces. The core principle is that an initial energy bound ($L^2$ or other base regularity) can be upgraded to higher integrability, often strictly above the natural exponent (e.g., $L^2 \to L^{2+\varepsilon}$).

2. Gradient RHI for Elliptic and Parabolic PDEs

Classical results (e.g., Meyers reverse Hölder) apply to divergence form elliptic PDEs $-\mathrm{div}(A\nabla u) = 0$ where $A(x)$ is uniformly elliptic. The reverse Hölder gain states that for every cube $Q$,

$$\left( \frac{1}{|Q|} \int_{Q} |\nabla u|^{2+\varepsilon}\right)^{1/(2+\varepsilon)} \leq C \left( \frac{1}{|2Q|} \int_{2Q} |\nabla u|^{2}\right)^{1/2}$$

for some $\varepsilon>0$ depending on ellipticity and dimension (Saari et al., 7 Dec 2025). In the presence of Hölder continuous coefficients $A \in C^\alpha$, one obtains gradient bounds for arbitrary large exponents, up to $L^\infty$, exploiting Hardy-Hölder duality.

For parabolic flows such as degenerate porous medium equations and fast diffusion equations ($u_t - \mathrm{div} A(u, \nabla u) = f$), gradient RHI is proved on intrinsic cylinders adapted to the solution profile. For example, for $u$ a solution to the fast diffusion equation, the gradient of a power $u^m$ satisfies

$\left( \fint_{Q_{\theta r^2,\, r}} |D(u^m)|^{2p} \right)^{1/p} \leq C \left[ \fint_{Q_{2\theta r^2,\, 2r}} |f|^{2p}\right]^{1/p} + \text{lower-order terms}$

with explicit construction of intrinsic cylinder families and a sharp range for $m$ (Gianazza et al., 2018, Gianazza et al., 2016).

3. Reverse Hölder for Abstract Gradient and Accretive Operators

In doubling metric measure spaces $(X, d, \mu)$ with a sublinear local gradient $\Gamma$ and an injective $\omega$-accretive operator $L$ satisfying Davies-Gaffney estimates, one defines the Riesz transform $R = \Gamma L^{-1/2}$ and establishes boundedness in $L^p$ for $p$ slightly above $2$ via gradient reverse Hölder: $(RH_{p_0}) \quad \left(\fint_B |\Gamma u|^{p_0} \right)^{1/p_0} \leq C \left(\fint_{2B} |\Gamma u|^2 \right)^{1/2}$ for $L$-harmonic $u$ (Bernicot et al., 2015). These conditions, in conjunction with semigroup gradient bounds,

$$(G_p) \quad \sup_{t>0} t^{1/2} \|\Gamma e^{-tL} f\|_{L^p} \leq C \|f\|_{L^p}$$

allow extension of $R$ to $L^q$ for $q > 2$. The key innovation is that reverse Hölder for the gradient replaces, or synergizes with, Poincaré inequality assumptions, facilitating abstract interpolation and extrapolation techniques.

4. Variational and Multi-Phase Integral Settings

For minimizers of multi-phase variational integrals with non-uniform phase coefficients and exponents,

$H(x, z) = |z|^p + \sum_{v=1}^k a_v(x) |z|^{p_v},$

gradient RHI distinguishes degenerate and non-degenerate regimes (based on size relations among $a_v$ and their semi-norms): $\begin{array}{rl} \text{Degenerate:} & \left(\dashint_{B_{R/2}} |\nabla u|^d \right)^{1/d} \leq c d^T \left(\dashint_{B_R} H(x, \nabla u) \right)^{1/p} \ \text{Non-degenerate/mixed:} & \left(\dashint_{B_{R/2}} |\nabla u|^{\varepsilon d}\right)^{1/(\varepsilon d)} \leq c \varepsilon^{-\lambda} \left(\dashint_{B_R} H(x, \nabla u)\right)^{1/p} \end{array}$ yielding higher $L^q$ integrability for $\nabla u$, with constants depending on the phase data and covering parameters (Filippis, 2021). These results extend to calibrated Calderón–Zygmund-type estimates for non-homogeneous problems.

5. Covering Arguments, Self-Improvement, and Gehring Lemma

Gradient RHI is self-improving: Gehring’s lemma and sophisticated covering strategies (e.g., Vitali, Calderón–Zygmund) enable bootstrapping of initial reverse Hölder bounds for the gradient (or its powers) to strictly stronger $L^q$ estimates. For equations admitting an intrinsic scaling (via time, space, or velocity variables), cylinders or balls are constructed so the energy scale matches the degeneracy or nonhomogeneous growth. Iterative application of RHI on these covers yields global higher integrability, even under singular or rough coefficient regimes (Gianazza et al., 2018, Saari et al., 2019, Guerand et al., 7 Oct 2024).

In kinetic equations with hypoelliptic scaling, the reverse Hölder estimate for the velocity gradient in kinetic cylinders,

$\left(\fint_{Q_r(z_0)} |\nabla_v u|^{2+\varepsilon} \right)^{1/(2+\varepsilon)} \leq C \left(\fint_{Q_{2r}(z_0)} |\nabla_v u|^2 \right)^{1/2} + C \left(\fint_{Q_{2r}(z_0)} |S|^{2+\varepsilon} \right)^{1/(2+\varepsilon)}$

is established with explicit dependence on the uniform ellipticity, drift, and source integrability (Guerand et al., 7 Oct 2024).

6. Extensions to Higher-Order Systems and Non-Uniform Contexts

Reverse Hölder theory generalizes to linear elliptic systems of arbitrary order $2m$ with nontrivial lower-order terms: $$L\vec u = \sum_{j,k}\sum_{|\alpha|,|\beta|\le m} \partial^\alpha(A^{j,k}_{\alpha,\beta}(x)\partial^\beta u_k)$$ where each vector gradient is estimated via higher-order Caccioppoli inequalities and analytic covering arguments: $$\|\nabla^j \vec u\|_{L^p(Q)} \leq C \|L\vec u\|_{Y^{-m,p}(\theta Q)} + C \|\nabla^\varpi \vec u\|_{L^q(\theta Q \setminus Q)}$$ for appropriate $p>2$ and companion regularity assumptions (Barton et al., 2022). These results achieve optimal gains in gradient integrability, up to the limits dictated by criticality of lower-order coefficients.

7. Interplay with Duality, Sparse Forms, and Regularity Theory

Advanced proofs leverage duality between local Hardy and Campanato–Hölder spaces, especially in the context of Schauder theory: one can transfer gradient RHI from constant coefficient cases to rough or variable coefficients by decomposing error terms and applying iterative geometric decay (Saari et al., 7 Dec 2025). Sparse bounds, covering arguments, and abstract functional analysis feed into sharp estimates, which further yield regularity results such as Hölder continuity of the gradient and precise sup-norm bounds.

Summary Table of Canonical Gradient Reverse Hölder Settings

Equation/Class	Gradient RHI Form	Key Technique/Reference
Elliptic, $-\operatorname{div}(A\nabla u)$	$$\|\nabla u\|_{L^{2+\varepsilon}(Q)} \leq C\|\nabla u\|_{L^2(2Q)}$$	Meyers/duality (Saari et al., 7 Dec 2025)
Fast diffusion/porous medium	$\|D(u^m)\|_{L^{2p}(Q)} \leq C\|f\|_{L^{2p}(2Q)}$	Intrinsic scaling (Gianazza et al., 2018)
Trudinger’s eqn.	$$\|\nabla u\|_{L^{p(1+\varepsilon)}(Q)} \leq C\|\nabla u\|_{L^p(Q')}$$	Cylinder stopping (Saari et al., 2019)
Accretive operator/Riesz transform	$\left(\fint_{B}|\Gamma u|^{p_0}\right)^{1/p_0} \leq C \left(\fint_{2B}|\Gamma u|^2\right)^{1/2}$	Semigroup/abstract (Bernicot et al., 2015)
Multi-phase variational	$$\|\nabla u\|_{L^q(B_{R/2})} \leq C \|\nabla u\|_{L^p(B_R)}$$	Interpolation/covering (Filippis, 2021)
Kinetic Fokker–Planck	$$\|\nabla_v u\|_{L^{2+\varepsilon}(Q_r)} \leq C\|\nabla_v u\|_{L^2(Q_{2r})}$$	Kinetic Gehring (Guerand et al., 7 Oct 2024)
Higher-order elliptic system	$$\|\nabla^j u\|_{L^p(Q)} \leq C\|\nabla^j u\|_{L^2(2Q)}$$	High-order Caccioppoli (Barton et al., 2022)

All constants and gain exponents depend explicitly on the structural parameters: ellipticity, domain geometry, regularity and modulus of the coefficients, and growth properties of the nonlinearity or lower-order terms.

A plausible implication is that gradient RHI is universally applicable across scales and regularity contexts, provided the underlying energy and structure bounds are in place. The self-improving integrability phenomena have direct impact on the development of sharp Calderón–Zygmund theory, regularity results, and the extension of boundedness theorems for fundamental operators such as the Riesz transform.

• "Riesz transforms through reverse Hölder and Poincaré inequalities" (Bernicot et al., 2015)
• "A reverse Hölder inequality for the gradient of solutions to Trudinger's equation" (Saari et al., 2019)
• "Optimal gradient estimates for multi-phase integrals" (Filippis, 2021)
• "Self-improving property of the fast diffusion equation" (Gianazza et al., 2018)
• "Self-improving property of degenerate parabolic equations of porous medium-type" (Gianazza et al., 2016)
• "Gehring's Lemma for kinetic Fokker-Planck equations" (Guerand et al., 7 Oct 2024)
• "Gradient estimates and the fundamental solution for higher-order elliptic systems with lower-order terms" (Barton et al., 2022)
• "A duality approach to gradient Hölder estimates for linear divergence form elliptic equations" (Saari et al., 7 Dec 2025)