Nonlocal Harnack Inequalities

• Nonlocal Harnack inequalities are comparison principles for integro-differential equations where a tail term quantifies long-range interactions beyond local neighborhoods.
• They extend classical sup-inf estimates by incorporating nonlocal tail effects, ensuring robust regularity results even without global positivity.
• Applications include elliptic, parabolic, and time-nonlocal equations, underpinning regularity theory, eigenvalue problems, and geometric analyses.

Nonlocal Harnack inequalities are comparison principles for solutions of integro-differential equations in which long-range interactions prevent purely local control by the classical supremum–infimum mechanism. In the nonlocal setting, the value of a solution inside a ball or cylinder depends on values outside it, and the correct replacement of the local Harnack principle typically includes a tail term measuring far-field influence. Beginning with elliptic minimizers and weak solutions, and later extending to parabolic equations, nonstandard growth, nonsymmetric kernels, manifold and Carnot-group geometries, and even disconnected regions, the theory has developed into a structural regularity framework in which local boundedness, weak Harnack inequalities, full Harnack inequalities, Hölder continuity, and eigenvalue theory are tightly linked (Castro et al., 2014, Strömqvist, 2018, Kassmann et al., 2023).

1. Core formulation and the tail principle

The basic distinction between local and nonlocal Harnack theory is the necessity of a nonlocal tail. For elliptic fractional $p$-type problems, a standard tail is

$$\operatorname{Tail}(u;x_0,R):=\left(R^{sp}\int_{\mathbb{R}^n\setminus B_R(x_0)}\frac{|u(y)|^{p-1}}{|y-x_0|^{n+sp}}\,dy\right)^{\frac1{p-1}},$$

while in parabolic problems one uses space-time tails such as

$$\operatorname{Tail}(v;x_0,r,t_1,t_2):=\frac{r^{2s}}{t_2-t_1}\int_{t_1}^{t_2}\int_{\mathbb{R}^n\setminus B_r(x_0)}\frac{|v(x,t)|}{|x-x_0|^{n+2s}}\,dx\,dt,$$

together with supremum-in-time variants. These quantities encode the action at a distance that is absent in local PDEs, and they enter naturally when solutions are only locally nonnegative rather than globally positive (Castro et al., 2014, Strömqvist, 2018).

A recurring structural feature is that the negative part $u_-=\max\{-u,0\}$ appears in the tail. In the elliptic and parabolic theories this allows one to compare local suprema and infima without assuming global positivity. The data indicate that such terms are not artifacts of proof but reflect genuine nonlocal phenomena: counterexamples in the literature show that they cannot in general be removed (Strömqvist, 2018).

The resulting landscape may be summarized schematically as follows.

Setting	Typical comparison	Distinctive feature
Elliptic space-nonlocal	$\sup \le C(\inf+\text{Tail})$	far-field dependence through $u_-$
Parabolic space-nonlocal	$\sup_{Q_-}\le C(\inf_{Q_+}+\text{Tail})$	time lag and parabolic tail
Time-nonlocal diffusion	dimension-dependent validity	memory effects and critical-dimension phenomena

This pattern persists across linear and nonlinear problems, De Giorgi classes, general growth functionals, and geometric settings. A plausible implication is that nonlocal Harnack inequalities are best viewed not as isolated estimates but as regularity statements whose natural right-hand side is determined by the operator’s interaction range and the admissible solution class (Chaker et al., 2022).

2. Elliptic origins and the emergence of full nonlocal Harnack inequalities

An early systematic elliptic theory appears for integral equations of the form

$$\mathcal{L}[u](x)=\int_\Omega k(x,y)u(y)\,dy-b(x)u(x)=0,$$

where the main difficulties are control of the solution outside compact sets and the lack of local uniform estimates. In that setting, a contraction result makes the $L^1$ norms of positive solutions on two compact sets $\omega_1\Subset\omega_2$ equivalent, leading to interior and boundary Harnack-type inequalities and to the construction of principal positive eigenfunctions for associated nonlocal operators (Coville, 2013).

A decisive formulation for nonlinear nonlocal elliptic equations was given for minimizers of possibly degenerate integro-differential operators modeled on the fractional $$\operatorname{Tail}(u;x_0,R):=\left(R^{sp}\int_{\mathbb{R}^n\setminus B_R(x_0)}\frac{|u(y)|^{p-1}}{|y-x_0|^{n+sp}}\,dy\right)^{\frac1{p-1}},$$0-Laplacian. If $$\operatorname{Tail}(u;x_0,R):=\left(R^{sp}\int_{\mathbb{R}^n\setminus B_R(x_0)}\frac{|u(y)|^{p-1}}{|y-x_0|^{n+sp}}\,dy\right)^{\frac1{p-1}},$$1 in $$\operatorname{Tail}(u;x_0,R):=\left(R^{sp}\int_{\mathbb{R}^n\setminus B_R(x_0)}\frac{|u(y)|^{p-1}}{|y-x_0|^{n+sp}}\,dy\right)^{\frac1{p-1}},$$2, then for $$\operatorname{Tail}(u;x_0,R):=\left(R^{sp}\int_{\mathbb{R}^n\setminus B_R(x_0)}\frac{|u(y)|^{p-1}}{|y-x_0|^{n+sp}}\,dy\right)^{\frac1{p-1}},$$3,

$$\operatorname{Tail}(u;x_0,R):=\left(R^{sp}\int_{\mathbb{R}^n\setminus B_R(x_0)}\frac{|u(y)|^{p-1}}{|y-x_0|^{n+sp}}\,dy\right)^{\frac1{p-1}},$$4

with $$\operatorname{Tail}(u;x_0,R):=\left(R^{sp}\int_{\mathbb{R}^n\setminus B_R(x_0)}\frac{|u(y)|^{p-1}}{|y-x_0|^{n+sp}}\,dy\right)^{\frac1{p-1}},$$5 depending only on the structural parameters. When $$\operatorname{Tail}(u;x_0,R):=\left(R^{sp}\int_{\mathbb{R}^n\setminus B_R(x_0)}\frac{|u(y)|^{p-1}}{|y-x_0|^{n+sp}}\,dy\right)^{\frac1{p-1}},$$6 in all of $$\operatorname{Tail}(u;x_0,R):=\left(R^{sp}\int_{\mathbb{R}^n\setminus B_R(x_0)}\frac{|u(y)|^{p-1}}{|y-x_0|^{n+sp}}\,dy\right)^{\frac1{p-1}},$$7, the tail vanishes and one recovers the classical-looking form $$\operatorname{Tail}(u;x_0,R):=\left(R^{sp}\int_{\mathbb{R}^n\setminus B_R(x_0)}\frac{|u(y)|^{p-1}}{|y-x_0|^{n+sp}}\,dy\right)^{\frac1{p-1}},$$8 (Castro et al., 2014).

Subsequent work broadened the elliptic theory from fixed power growth to Orlicz and nonstandard growth regimes. For generalized nonlocal $$\operatorname{Tail}(u;x_0,R):=\left(R^{sp}\int_{\mathbb{R}^n\setminus B_R(x_0)}\frac{|u(y)|^{p-1}}{|y-x_0|^{n+sp}}\,dy\right)^{\frac1{p-1}},$$9-Laplacian-type equations in fractional Orlicz-Sobolev spaces, the Harnack bound involves $$\operatorname{Tail}(v;x_0,r,t_1,t_2):=\frac{r^{2s}}{t_2-t_1}\int_{t_1}^{t_2}\int_{\mathbb{R}^n\setminus B_r(x_0)}\frac{|v(x,t)|}{|x-x_0|^{n+2s}}\,dx\,dt,$$0 and a tail built from the growth function $$\operatorname{Tail}(v;x_0,r,t_1,t_2):=\frac{r^{2s}}{t_2-t_1}\int_{t_1}^{t_2}\int_{\mathbb{R}^n\setminus B_r(x_0)}\frac{|v(x,t)|}{|x-x_0|^{n+2s}}\,dx\,dt,$$1, extending the theory to examples such as $$\operatorname{Tail}(v;x_0,r,t_1,t_2):=\frac{r^{2s}}{t_2-t_1}\int_{t_1}^{t_2}\int_{\mathbb{R}^n\setminus B_r(x_0)}\frac{|v(x,t)|}{|x-x_0|^{n+2s}}\,dx\,dt,$$2, $$\operatorname{Tail}(v;x_0,r,t_1,t_2):=\frac{r^{2s}}{t_2-t_1}\int_{t_1}^{t_2}\int_{\mathbb{R}^n\setminus B_r(x_0)}\frac{|v(x,t)|}{|x-x_0|^{n+2s}}\,dx\,dt,$$3, $$\operatorname{Tail}(v;x_0,r,t_1,t_2):=\frac{r^{2s}}{t_2-t_1}\int_{t_1}^{t_2}\int_{\mathbb{R}^n\setminus B_r(x_0)}\frac{|v(x,t)|}{|x-x_0|^{n+2s}}\,dx\,dt,$$4, and $$\operatorname{Tail}(v;x_0,r,t_1,t_2):=\frac{r^{2s}}{t_2-t_1}\int_{t_1}^{t_2}\int_{\mathbb{R}^n\setminus B_r(x_0)}\frac{|v(x,t)|}{|x-x_0|^{n+2s}}\,dx\,dt,$$5 (Fang et al., 2022). For nonlocal problems with non-standard growth governed by a convex function $$\operatorname{Tail}(v;x_0,r,t_1,t_2):=\frac{r^{2s}}{t_2-t_1}\int_{t_1}^{t_2}\int_{\mathbb{R}^n\setminus B_r(x_0)}\frac{|v(x,t)|}{|x-x_0|^{n+2s}}\,dx\,dt,$$6, a full Harnack inequality was proved for De Giorgi class functions, local minimizers, and weak solutions: $$\operatorname{Tail}(v;x_0,r,t_1,t_2):=\frac{r^{2s}}{t_2-t_1}\int_{t_1}^{t_2}\int_{\mathbb{R}^n\setminus B_r(x_0)}\frac{|v(x,t)|}{|x-x_0|^{n+2s}}\,dx\,dt,$$7 with robustness as $$\operatorname{Tail}(v;x_0,r,t_1,t_2):=\frac{r^{2s}}{t_2-t_1}\int_{t_1}^{t_2}\int_{\mathbb{R}^n\setminus B_r(x_0)}\frac{|v(x,t)|}{|x-x_0|^{n+2s}}\,dx\,dt,$$8 explicitly emphasized (Chaker et al., 2022).

The elliptic theory also reaches fully nonlinear operators. For general nonlocal elliptic equations with zero order terms, viscosity solutions satisfy

$$\operatorname{Tail}(v;x_0,r,t_1,t_2):=\frac{r^{2s}}{t_2-t_1}\int_{t_1}^{t_2}\int_{\mathbb{R}^n\setminus B_r(x_0)}\frac{|v(x,t)|}{|x-x_0|^{n+2s}}\,dx\,dt,$$9

and this Harnack inequality is then used to construct principal eigenvalues in general, possibly unbounded, domains and to study self-similar solutions of related parabolic problems (Dávila et al., 2019).

3. Parabolic equations with spatial nonlocality

For parabolic equations with spatially nonlocal diffusion, the model form is

$u_-=\max\{-u,0\}$0

with symmetric kernels satisfying two-sided bounds of fractional type. The fractional heat equation is the prototypical example, obtained when $u_-=\max\{-u,0\}$1 (Strömqvist, 2018).

A central advance was the parabolic Harnack inequality for locally nonnegative solutions without any global positivity assumption. In one formulation, if $u_-=\max\{-u,0\}$2 in a local cylinder, then

$u_-=\max\{-u,0\}$3

where the estimate has the usual parabolic time lag and $u_-=\max\{-u,0\}$4 depends only on $u_-=\max\{-u,0\}$5. The same work also establishes a weak Harnack inequality for supersolutions and a local boundedness estimate in terms of a local $u_-=\max\{-u,0\}$6-norm and an outside tail. These results are stated to be new even for the fractional heat equation and to apply to time-inhomogeneous symmetric kernels, not only to the fractional Laplacian (Strömqvist, 2018).

A related parabolic theory for weak solutions of

$u_-=\max\{-u,0\}$7

was obtained via the De Giorgi method, with a weak Harnack inequality containing a parabolic tail, a tail-free corollary for globally nonnegative solutions, and an $u_-=\max\{-u,0\}$8-weak Harnack estimate. The proof develops nonlocal Caccioppoli estimates, a parabolic fractional Poincaré inequality, De Giorgi iteration, a nonlocal Krylov–Safonov covering lemma, and viscosity/weak solution equivalence (Kim, 2018).

A later analytic treatment in the variational framework completed the local regularity program for linear parabolic nonlocal equations with bounded measurable coefficients. In that setting, for globally nonnegative weak solutions,

$u_-=\max\{-u,0\}$9

and for merely nonnegative solutions a full Harnack estimate with tails is obtained. A key technical development is the replacement of the supremum-in-time tail by an $\sup \le C(\inf+\text{Tail})$0-in-time tail, which is finite for natural energy solutions and sharp for Hölder regularity (Kassmann et al., 2023).

4. Time-nonlocal diffusion and critical-dimension effects

A distinct branch of the subject concerns nonlocality in time. For equations of the form

$\sup \le C(\inf+\text{Tail})$1

the memory term changes the Harnack picture qualitatively. A counterexample shows that the classical local parabolic Harnack inequality fails for globally positive solutions whenever $\sup \le C(\inf+\text{Tail})$2. The mechanism is that the fundamental solution is singular at the origin for all $\sup \le C(\inf+\text{Tail})$3 in dimensions $\sup \le C(\inf+\text{Tail})$4, so concentration in the initial data persists and destroys any universal local Harnack constant (Dier et al., 2018).

The same analysis identifies a critical-dimension phenomenon. In the purely space-fractional case $\sup \le C(\inf+\text{Tail})$5, the classical local Harnack inequality holds for all $\sup \le C(\inf+\text{Tail})$6. With a fractional time derivative, the situation splits: the local Harnack inequality holds if $\sup \le C(\inf+\text{Tail})$7, but fails if $\sup \le C(\inf+\text{Tail})$8. The paper also proves a non-local Harnack inequality involving the potential of the initial data, showing that if the initial datum already satisfies an elliptic Harnack inequality on balls, then a Harnack estimate propagates to the solution after a suitable time lag, with constants depending on the initial datum (Dier et al., 2018).

Recent work completes the low-dimensional picture for general nonlocal-in-time subdiffusion equations

$\sup \le C(\inf+\text{Tail})$9

In one space dimension, for globally nonnegative local weak solutions, a full Harnack inequality holds in parabolic boxes whose time-length is governed by a scale function $u_-$0 adapted to the kernel $u_-$1. The same source states that in dimension $u_-$2 the proof relies crucially on the embedding $u_-$3, whereas the classical Harnack inequality fails for these equations in dimensions $u_-$4 (Ryszewska et al., 20 Oct 2025).

This contrast between spatial and temporal nonlocality is one of the most striking structural features of the area. A plausible implication is that “nonlocal Harnack inequality” is not a single phenomenon but a family of comparison principles whose validity depends sharply on whether nonlocality acts in space, in time, or in both variables.

5. Nonlinear, nonsymmetric, and De Giorgi-class extensions

The modern theory extends well beyond linear symmetric kernels. For doubly nonlinear mixed local and nonlocal parabolic equations,

$u_-$5

a Harnack inequality was established for nonnegative weak solutions by combining a comparison principle, local boundedness, an integral Harnack-type inequality, and expansion of positivity. The resulting estimate is intrinsic: both the temporal scale and the spatial scale depend on the local value $u_-$6 (Radulescu et al., 2024).

For nonlocal double phase equations with operator

$u_-$7

nonlocal Harnack, weak Harnack, and explicit local boundedness estimates are proved under the assumptions $u_-$8, $u_-$9, and without any Hölder condition on $\sup_{Q_-}\le C(\inf_{Q_+}+\text{Tail})$0. The full Harnack inequality has the familiar form

$\sup_{Q_-}\le C(\inf_{Q_+}+\text{Tail})$1

for locally nonnegative weak solutions (Kim, 9 Sep 2025).

Nonlocal Harnack theory has also been developed for nonsymmetric forms. For operators

$\sup_{Q_-}\le C(\inf_{Q_+}+\text{Tail})$2

with decomposition $\sup_{Q_-}\le C(\inf_{Q_+}+\text{Tail})$3, local boundedness and Harnack inequalities are derived for parabolic and elliptic equations under structural conditions such as (K2), (Cutoff), (Sob), (Poinc), and either (K1loc) or (K1glob), with two independent proofs: one based on De Giorgi iteration and the other on Moser iteration (Kassmann et al., 2022).

A further abstraction replaces the equation by an energy class. For the nonlocal parabolic $\sup_{Q_-}\le C(\inf_{Q_+}+\text{Tail})$4-homogeneous De Giorgi class $\sup_{Q_-}\le C(\inf_{Q_+}+\text{Tail})$5, local boundedness, several weak Harnack inequalities, propagation lemmas, a full Harnack inequality, Hölder continuity, and a Liouville-type rigidity result are proved under minimal tail assumptions in $\sup_{Q_-}\le C(\inf_{Q_+}+\text{Tail})$6 in time. The results are stated to be new even for the linear case and to show that recent parabolic achievements are structural properties of the class rather than of a particular equation (Ciani et al., 22 Aug 2025).

At the level of rough transport, Harnack estimates have now been obtained for

$\sup_{Q_-}\le C(\inf_{Q_+}+\text{Tail})$7

with divergence-free drift $\sup_{Q_-}\le C(\inf_{Q_+}+\text{Tail})$8 in critical or supercritical regimes and $\sup_{Q_-}\le C(\inf_{Q_+}+\text{Tail})$9 in negative-index fractional Sobolev spaces. The weak and full Harnack inequalities again contain nonlocal tail terms and support applications to critical stochastic quasi-geostrophic equations, two-dimensional fractional Navier–Stokes equations with measure-valued initial vorticity, and generalized martingale problems (Chen et al., 17 Nov 2025).

6. Geometric settings, boundary-type phenomena, and applications

Nonlocal Harnack inequalities persist in settings where the underlying geometry is not Euclidean. On complete Riemannian manifolds with nonnegative sectional curvature, nonlocal Pucci operators are defined by kernels controlled by

$$\mathcal{L}[u](x)=\int_\Omega k(x,y)u(y)\,dy-b(x)u(x)=0,$$0

and fully nonlinear Harnack and Hölder estimates are proved for operators elliptic with respect to the corresponding class $$\mathcal{L}[u](x)=\int_\Omega k(x,y)u(y)\,dy-b(x)u(x)=0,$$1. The constants are uniform as $$\mathcal{L}[u](x)=\int_\Omega k(x,y)u(y)\,dy-b(x)u(x)=0,$$2, recovering the classical local theory in the limit (Kim et al., 2021).

In the Heisenberg group $$\mathcal{L}[u](x)=\int_\Omega k(x,y)u(y)\,dy-b(x)u(x)=0,$$3, a general Harnack inequality is proved for weak solutions of nonlinear integro-differential problems whose prototype is the Dirichlet problem for the $$\mathcal{L}[u](x)=\int_\Omega k(x,y)u(y)\,dy-b(x)u(x)=0,$$4-fractional subLaplacian. The estimate includes the Heisenberg-tail term

$$\mathcal{L}[u](x)=\int_\Omega k(x,y)u(y)\,dy-b(x)u(x)=0,$$5

and for $$\mathcal{L}[u](x)=\int_\Omega k(x,y)u(y)\,dy-b(x)u(x)=0,$$6 the asymptotic behavior as $$\mathcal{L}[u](x)=\int_\Omega k(x,y)u(y)\,dy-b(x)u(x)=0,$$7 shows robustness toward the local subelliptic limit (Palatucci et al., 2022).

Two recent directions highlight genuinely nonlocal geometric phenomena. First, a Harnack inequality in a disconnected region compares values on one connected component with values on another: $$\mathcal{L}[u](x)=\int_\Omega k(x,y)u(y)\,dy-b(x)u(x)=0,$$8 for weak solutions in $$\mathcal{L}[u](x)=\int_\Omega k(x,y)u(y)\,dy-b(x)u(x)=0,$$9. This has no local analogue and is proved both by a localized maximum principle and by Poisson kernel estimates (Lee, 22 Aug 2025). Second, for antisymmetric functions one has a boundary-type inequality

$L^1$0

derived by reducing the problem via Bochner’s relation from an antisymmetric equation in $L^1$1 to an interior Harnack inequality in $L^1$2 (Dipierro et al., 2024).

There are also sharp, explicitly geometric results for the fractional heat equation with $L^1$3. Positive classical solutions satisfy a sharp double-sided Harnack inequality expressed through the Poisson kernel

$L^1$4

where the upper and lower bounds are determined by intersection points of a circle or sphere through the two space-time points with the initial time plane. The bounds are attained by the kernel itself, linking Harnack ratios to circular geometry in higher dimensions (Dembny et al., 9 Jun 2025).

Applications run throughout the literature. Harnack inequalities support Hölder continuity, local boundedness, Liouville-type theorems, principal eigenvalue theory in bounded and unbounded domains, decay estimates for self-similar profiles, and probabilistic interpretations involving Lévy processes and jump processes (Kassmann et al., 2023, Dávila et al., 2019). The data also mention applications in signal and image processing for parabolic nonlocal equations and the construction of principal positive eigenfunctions for integral operators (Strömqvist, 2018, Coville, 2013). Taken together, these developments suggest that nonlocal Harnack inequalities now serve as one of the main organizing principles of regularity theory for nonlocal PDE, with the tail term functioning as the analytic signature of long-range interaction.