Quantitative Unit-Scale Harnack Inequality

• Quantitative unit-scale Harnack inequality is a refinement of the classical Harnack principle, offering explicit, scale-sensitive bounds on solution oscillation.
• It rigorously links local behavior to global regularity for elliptic, parabolic, nonlocal, subelliptic, and nonlinear PDEs through geometry-dependent constants.
• Applications span potential theory, geometric flows, and stochastic control, with sharp constants controlling the influence of curvature, nonlinearity, and diffusion.

The quantitative unit-scale Harnack inequality generalizes the classical qualitative comparison principle for non-negative solutions of elliptic and parabolic PDEs by providing explicit, scale-aware estimates on the oscillation of solutions within domains of controlled geometry. These inequalities exhibit dependence of the Harnack constant on intrinsic geometric or analytic parameters—including spatial scale, equation structure, curvature, nonlinearity, and various PDE data—enabling rigorous control of local-to-global behavior in contexts ranging from classical harmonic functions to nonlocal, metric, subelliptic, and nonlinear flows.

1. Definition and Classical Quantitative Formulations

Let $u$ be a non-negative solution to a (possibly nonlinear or nonlocal) elliptic or parabolic PDE in a domain $G\subset\mathbb{R}^n$ (or more generally metric, subRiemannian, or measure spaces). The unit-scale Harnack inequality asserts that for any ball $B(x,r)\subset G$ and fixed $0 < s < 1$, there exists an explicit constant $C(s)$—the unit-scale Harnack constant—such that

$\sup_{B(x,sr)} u \leq C(s)\,\inf_{B(x,sr)} u.$

For harmonic functions, $C(s)=((1+s)/(1-s))^n$, which quantifies oscillation in relation to the ratio of inner to outer radii. In the general setting, constants depend on equation structure, dimension, geometric data, and nonlinearity (Kargar, 2023). Such quantitative forms enable precise local control and quantitative two-point oscillation bounds, underpinning refined potential-theoretic and analytic reasoning.

2. Fundamental Cases and Explicit Bounds

Harmonic and Classical Elliptic Equations

For positive harmonic functions in $\mathbb{R}^n$, the sharp local bound is (Kargar, 2023): $$\sup_{B(x,sr)} u \leq \left(\frac{1+s}{1-s}\right)^n \inf_{B(x,sr)} u.$$ This is optimal and the exponent $n$ cannot be lowered uniformly. In the unit disk, strengthened Harnack inequalities involve directional gradient terms, resulting in two-sided estimates (Svetlik, 16 Jan 2025): $$u(z) \in \left[\frac{1 - r^2 - 2c r}{1 + r^2 + 2c r},\, \frac{1 + r^2 + 2c r}{1 - r^2 - 2c r}\right] u(0),$$ where $c = |Vu(0)|/(2u(0))$ encodes gradient data at the origin. In potential theory this is connected to the "Harnack metric," $h_G(x,y) = \sup_{u>0} |\log(u(x)/u(y))|$ (Kargar, 2023), providing a pseudometric controlling local oscillation via $\log C(s)$.

Singular Parabolic Equations (p-Laplacian Range)

For one-dimensional singular parabolic equations ($1Düzgün et al., 2015): $$u(x,t) \geq \bar{\sigma} M \left(\frac{\bar{\rho}}{\rho}\right)^{p/(2-p)},$$ where $\bar{\rho}$ is the intrinsic length scale, and $\bar{\sigma}$ depends only on $p$. This quantifies "sidewise spreading of positivity"—the propagation of lower bounds outwards in space over unit time, capturing optimal decay rates in singular regimes.

Nonlocal Operators

For nonlocal Schrödinger operators, the unit-scale Harnack constant $C$ depends precisely on the structure of the diffusion matrix, the jump kernel, and potential regularity (Athreya et al., 2015). The result takes the scaling form: $$\sup_{B(x_0, R/2)} f \leq C\; \inf_{B(x_0, R/2)} f,$$ with $C$ determined by ellipticity, jump parameters, and Kato-class potential, and is fully adaptable under rescaling.

3. Metric, Subelliptic, and Nonlinear Contexts

Subelliptic PDEs and Carnot-Carathéodory Geometry

In subelliptic nondivergence form equations, e.g., the Grushin model (Montanari, 2014), unit-scale Harnack holds on Carnot-Carathéodory balls, with constants $C(n,\kappa)$ explicit in the ellipticity ratio $\kappa$. The axiomatic approach—incorporating weighted Aleksandrov-Bakelman-Pucci estimates, critical density, double-ball properties, and power decay—yields scale-invariant bounds

$$\sup_{B_{cc}(z,r)} u \leq C(\kappa) \inf_{B_{cc}(z,r)} u,$$

with sharp dependence and extension to fully nonlinear contexts provided sufficient barrier estimates exist.

Metric Measure and Doubling-Poincaré Spaces

For parabolic minimizers in geodesic metric measure spaces with doubling and Poincaré inequalities, the location/scale invariant Harnack (Marola et al., 2013) is

$\esssup_{\delta Q^-(x_0,t_0;1)} u \leq C \essinf_{\delta Q^+(x_0,t_0;1)} u,$

where $C$ is strictly determined by variational and structural data, and rescaling preserves $C$. This establishes full sufficiency for Grigor'yan–Saloff-Coste-type theorems in nonlinear ($p > 1$) settings.

Porosity, Comparison, and Quasi-Harnack Frameworks

Generalizations where PDE solutions only satisfy comparison principles at two scales—using paraboloid support and weak measure-theoretic oscillation—yield quasi-Harnack inequalities (Silva et al., 2018): $\sup_{B_{1/2}} u \leq C_* \inf_{B_{1/2}} u,$ with $C_*$ bounded by uniform oscillation at small scales, and invariant under $r \to 0$.

4. Geometric, Curvature, and Measure-Theoretic Extensions

Riemannian and Curvature-Dimension Spaces

On Riemannian metric measure spaces with Bakry-Émery Ricci lower bounds, quantitative unit-ball Harnack inequalities are established purely via Alexandrov–Bakelman–Pucci measure estimates and covering arguments (Wang et al., 2011): $\sup_{B_1} u \leq C_2 \left( \inf_{B_1} u + \left( \fint_{B_2} |f|^{N\eta} \right)^{1/(N\eta)} \right),$ with constants $C_2$ and exponents fully determined by curvature and effective dimension. These bounds extend to fully nonlinear operators (Pucci extremals), with error terms controlled by sectional curvature.

In $RCD^*(K,N)$ spaces, the scale-invariant Harnack for the heat flow (Garofalo et al., 2013) reads: $$u(2, y) \geq u(1, x) \cdot \exp\left(-\frac{d^2(x,y)}{4 e^{2K}}\right) \cdot \left(\frac{1-e^{-2K}}{1-e^{-4K}}\right)^{N/2}.$$ Curvature and dimension enter directly; as $K\to 0$, classical Gaussian bounds are recovered.

5. Nonlinear "sup+inf"-Type Inequalities and Blow-Up Analysis

For quasilinear equations with exponential nonlinearity (e.g., $n$-Liouville), sharp "sup + inf" inequalities are achieved via blow-up methods (Esposito et al., 2021): $$\sup_{B_1(x_0)} u + C_1 \inf_{B_1(x_0)} u \leq C_2,$$ with $C_1 > n-1$ and $C_2 = C_2(n, a, b, \Lambda, C_1)$ dependent only on dimension and weak domain parameters. Scaling yields

$$\sup_{B_R(x_0)} u + C_1 \inf_{B_R(x_0)} u \leq C_2 - (1+C_1)n \ln R.$$

The proof leverages construction of bubbles via radial blow-up and precise classification of limiting profiles, showing sharpness of $C_1 > n-1$.

6. Discrete and Evolutionary Quantitative Harnack Schemes

Jordan-Kinderlehrer-Otto (JKO) schemes for discretized parabolic flows demonstrate unit-scale matrix and scalar Harnack bounds uniform in mesh size $N$ (Lee, 2015): $$u^N(t_1, x) \leq \left( \frac{t_2 + K/N}{t_1} \right)^n \exp\left( \frac{d^2(x, y)}{2 (t_2 - t_1 - K/N)} \right) u^N(t_2, y)$$ Convergence recovers Li-Yau/Hamilton bounds for the continuous heat equation. Uniform quantitative estimates hold for time- and space-scales within the fixed macroscopic domain.

7. Sharpness, Scaling, and Applications

Every unit-scale Harnack inequality cited is sharp within the constraints of the respective PDE, geometric structure, and analytical apparatus. Constants decay exactly as predicted when scale collapses ($s\to 0$, $r\to 0$) and blow up near domain boundaries ($s\to 1$, $r\to 1$), matching fundamental solution behavior for the model equations and domains. Application domains span potential theory, nonlinear analysis, geometric flows, homogenization, and stochastic control, with quasi-Harnack and geometric proof techniques applicable broadly in regularity theory and fine properties of PDE solutions.

Table: Classical vs. Quantitative Unit-Scale Harnack Constants

Equation Class	Domain/Scale	Harnack Constant $C(s)$
Harmonic ($\mathbb{R}^n$)	$B(x, sr)$	$((1+s)/(1-s))^n$
Singular parabolic ($1	Time-space	$\bar{\sigma} (\bar{\rho}/\rho)^{p/(2-p)}$
Subelliptic	CC ball	$C(\Lambda/\lambda)$
n-Liouville, Quasilinear	$B_1$	$C_2(n, a, b, \Lambda, C_1)$ for $C_1>n-1$
Riemannian metric-measure	$B_1$	$C_2(n,N,\sqrt{K})$, cf. curvature

• Euclidean and subelliptic classical: (Kargar, 2023, Montanari, 2014, Athreya et al., 2015).
• Singular parabolic and metric: (Düzgün et al., 2015, Marola et al., 2013).
• Quasi-Harnack and comparison-based: (Silva et al., 2018).
• Ricci lower bound and Riemannian measure: (Wang et al., 2011, Garofalo et al., 2013).
• Nonlinear exponential: (Esposito et al., 2021).
• Discrete JKO scheme: (Lee, 2015).
• Stronger harmonic disk inequalities: (Svetlik, 16 Jan 2025).

The quantitative unit-scale Harnack inequality thus serves as a precise bridge from local structural conditions of PDEs and associated geometric settings to global regularity and oscillation control, with explicit constants illuminating the influence of analytic, geometric, and probabilistic data.