Weak Harnack Inequality

• Weak Harnack inequality is an integral-to-pointwise estimate that bounds L^p-averages of nonnegative supersolutions by their essential infimum in specified regions.
• It underpins regularity results such as Hölder continuity and maximum principles in diverse settings including symmetric stable Lévy processes, time-fractional diffusion, and mixed local–nonlocal equations.
• The proof architectures range from potential theory to Moser iteration, accommodating anisotropy, tail corrections, and nonlocal influences in various analytical frameworks.

Weak Harnack inequality is an integral-to-pointwise estimate that bounds an $L^p$-average of a nonnegative supersolution on one region by the infimum, or essential infimum, on a smaller or later region. In elliptic settings it is typically stated on nested balls; in parabolic settings it compares backward and forward cylinders; in nonlocal problems it often carries a tail term recording the influence of the exterior negative part. It is a fundamental structural estimate in regularity theory and is used to derive Hölder regularity, strong maximum principles, local boundedness, and full Harnack inequalities in settings ranging from symmetric stable Lévy processes to time-fractional diffusion and mixed local–nonlocal equations (Sertic, 2015, Zacher, 2010, Garain, 5 Oct 2025).

1. Basic form and relation to the full Harnack inequality

The defining feature of a weak Harnack inequality is that it compares an integral quantity with a pointwise lower bound. In one of the formulations used for symmetric $\alpha$-stable Lévy processes, it is written as

$$\|u\|_{L^{1}(B_{1/2})} \leq C\cdot \inf_{B_{1/4}} u,$$

for every non-negative function $u$ harmonic in $B_1$ with respect to the process (Sertic, 2015). In a time-fractional parabolic setting, the corresponding estimate is

$$\left( \frac{1}{|Q_-|}\int_{Q_-} u^p\,d\mu_{N+1} \right)^{1/p} \le C\, \operatorname*{ess\,inf}_{Q_+} u,$$

where $Q_-$ is earlier in time and $Q_+$ is later (Zacher, 2010). In a mixed local–nonlocal parabolic equation, the estimate becomes

$$\left( \iint_{V^-(r)} (u+\ell)^q\,dx\,dt \right)^{1/q} \le C\,\Bigl(\operatorname*{ess\,inf}_{V^+(r)} u + \ell\Bigr),$$

with a tail correction $\ell$ generated by the exterior negative part (Garain et al., 2021).

Setting	Representative weak Harnack form	Distinctive feature
Symmetric $\alpha$0-stable harmonic functions	$\alpha$1	Nested balls (Sertic, 2015)
Time-fractional diffusion	$\alpha$2	Earlier-to-later cylinders (Zacher, 2010)
Mixed local–nonlocal parabolic equations	$\alpha$3	Tail term from exterior negativity (Garain et al., 2021)

This estimate is weaker than the full, or strong, Harnack inequality. In the Lévy-process formulation, the strong version is the pointwise estimate

$\alpha$4

whereas the weak version only compares an average over $\alpha$5 with the infimum on $\alpha$6 (Sertic, 2015). A recurring theme in modern nonlocal theory is that weak Harnack may survive in settings where the full pointwise Harnack inequality fails.

2. Elliptic and nonlocal formulations

A prominent nonlocal setting is that of symmetric $\alpha$7-stable Lévy processes in $\alpha$8, $\alpha$9, $$\|u\|_{L^{1}(B_{1/2})} \leq C\cdot \inf_{B_{1/4}} u,$$0, with characteristic exponent

$$\|u\|_{L^{1}(B_{1/2})} \leq C\cdot \inf_{B_{1/4}} u,$$1

where $$\|u\|_{L^{1}(B_{1/2})} \leq C\cdot \inf_{B_{1/4}} u,$$2 is a spectral measure (Sertic, 2015). Under the assumptions that $$\|u\|_{L^{1}(B_{1/2})} \leq C\cdot \inf_{B_{1/4}} u,$$3 is absolutely continuous with respect to the uniform measure $$\|u\|_{L^{1}(B_{1/2})} \leq C\cdot \inf_{B_{1/4}} u,$$4 on the sphere and that its density satisfies

$$\|u\|_{L^{1}(B_{1/2})} \leq C\cdot \inf_{B_{1/4}} u,$$5

the weak Harnack inequality holds for nonnegative functions harmonic in $$\|u\|_{L^{1}(B_{1/2})} \leq C\cdot \inf_{B_{1/4}} u,$$6 (Sertic, 2015). The paper “Weak Harnack Inequality and Hölder Regularity for Symmetric Stable Lévy Processes” states, at the abstract level, the same structural framework—symmetric $$\|u\|_{L^{1}(B_{1/2})} \leq C\cdot \inf_{B_{1/4}} u,$$7-stable Lévy processes with spectral measure absolutely continuous with respect to $$\|u\|_{L^{1}(B_{1/2})} \leq C\cdot \inf_{B_{1/4}} u,$$8 and density bounds—and uses weak Harnack to prove Hölder regularity estimates (Sertic, 2015).

This nonlocal elliptic theory is not a mere reformulation of the classical local case. For a specially constructed family of symmetric $$\|u\|_{L^{1}(B_{1/2})} \leq C\cdot \inf_{B_{1/4}} u,$$9-stable Lévy processes in $u$0 with $u$1, the usual pointwise Harnack inequality fails; the counterexample is built from explicit harmonic functions

$u$2

whose values at two interior points become arbitrarily separated (Sertic, 2015). The same paper shows that weak Harnack still holds for a broader bounded-density spectral class, thereby isolating the integral estimate as the more robust phenomenon.

A distinct boundary-sensitive nonlocal version appears for antisymmetric $u$3-harmonic functions. If $u$4 with respect to the plane $u$5, then the natural global quantity is

$u$6

and the weak Harnack inequalities take two forms. In the interior one has

$u$7

while near the symmetry plane the correct boundary version is

$u$8

reflecting the fact that $u$9, rather than $B_1$0, is the natural boundary-scale quantity (Dipierro et al., 2022).

3. Parabolic, kinetic, time-fractional, and stochastic variants

For time-fractional diffusion in divergence form,

$B_1$1

with bounded measurable uniformly elliptic coefficients, the weak Harnack inequality compares an $B_1$2-average on an earlier cylinder with the essential infimum on a later cylinder (Zacher, 2010). The admissible exponents satisfy

$B_1$3

and the paper proves that $B_1$4 is optimal (Zacher, 2010). A defining feature of this theory is that positivity cannot be localized only near $B_1$5: because the Riemann–Liouville derivative is nonlocal in time, the whole previous interval matters.

In kinetic Fokker–Planck theory, the model equation

$B_1$6

leads to a weak Harnack inequality of the form

$B_1$7

where the cylinders are adapted to kinetic scaling,

$B_1$8

The anisotropy $B_1$9 is essential: diffusion acts only in $$\left( \frac{1}{|Q_-|}\int_{Q_-} u^p\,d\mu_{N+1} \right)^{1/p} \le C\, \operatorname*{ess\,inf}_{Q_+} u,$$0, while transport propagates information through $$\left( \frac{1}{|Q_-|}\int_{Q_-} u^p\,d\mu_{N+1} \right)^{1/p} \le C\, \operatorname*{ess\,inf}_{Q_+} u,$$1 according to Galilean geometry (Guerand et al., 2021).

For linear nonlocal parabolic equations

$$\left( \frac{1}{|Q_-|}\int_{Q_-} u^p\,d\mu_{N+1} \right)^{1/p} \le C\, \operatorname*{ess\,inf}_{Q_+} u,$$2

with kernels comparable to $$\left( \frac{1}{|Q_-|}\int_{Q_-} u^p\,d\mu_{N+1} \right)^{1/p} \le C\, \operatorname*{ess\,inf}_{Q_+} u,$$3, a weak Harnack inequality is proved without assuming global nonnegativity. Instead,

$$\left( \frac{1}{|Q_-|}\int_{Q_-} u^p\,d\mu_{N+1} \right)^{1/p} \le C\, \operatorname*{ess\,inf}_{Q_+} u,$$4

is bounded by a later infimum plus a correction involving

$$\left( \frac{1}{|Q_-|}\int_{Q_-} u^p\,d\mu_{N+1} \right)^{1/p} \le C\, \operatorname*{ess\,inf}_{Q_+} u,$$5

so the exterior negative part enters explicitly (Strömqvist, 2018). An analogous phenomenon appears in the mixed local–nonlocal parabolic equation

$$\left( \frac{1}{|Q_-|}\int_{Q_-} u^p\,d\mu_{N+1} \right)^{1/p} \le C\, \operatorname*{ess\,inf}_{Q_+} u,$$6

where the weak Harnack estimate also contains a tail term and is valid for sign-changing supersolutions that are only locally nonnegative in the comparison cylinder (Garain et al., 2021).

A stochastic analogue arises for divergence-form SPDEs with multiplicative first-order noise,

$$\left( \frac{1}{|Q_-|}\int_{Q_-} u^p\,d\mu_{N+1} \right)^{1/p} \le C\, \operatorname*{ess\,inf}_{Q_+} u,$$7

Here the weak Harnack principle is probabilistic: if the initial state is positive on a set of positive measure, then for every $$\left( \frac{1}{|Q_-|}\int_{Q_-} u^p\,d\mu_{N+1} \right)^{1/p} \le C\, \operatorname*{ess\,inf}_{Q_+} u,$$8 there exists an event $$\left( \frac{1}{|Q_-|}\int_{Q_-} u^p\,d\mu_{N+1} \right)^{1/p} \le C\, \operatorname*{ess\,inf}_{Q_+} u,$$9 with

$Q_-$0

such that on the good event

$Q_-$1

The conclusion is therefore pathwise only outside an explicitly quantified exceptional set (Dareiotis et al., 2015).

4. Nonlinear, mixed, and non-standard growth generalizations

A major nonlinear extension concerns mixed local and nonlocal $Q_-$2-Laplace equations with source term,

$Q_-$3

and, more generally,

$Q_-$4

For weak supersolutions that satisfy $Q_-$5 in $Q_-$6, the exact weak Harnack inequality proved in that setting is

$Q_-$7

for every $Q_-$8, where $Q_-$9 (Garain, 5 Oct 2025). This is a genuinely mixed estimate: it records both the nonlocal exterior influence and the forcing from $Q_+$0.

In equations with generalized Orlicz growth,

$Q_+$1

weak Harnack for unbounded supersolutions is no longer function-independent. The theory requires an a priori Lebesgue or Sobolev assumption and a matching continuity condition on the growth function $Q_+$2. Under those assumptions, the estimate takes the form

$Q_+$3

and the optimal exponent range is

$Q_+$4

The paper also proves sharpness of the central continuity assumptions (Benyaiche et al., 2020).

For nonlocal problems with non-standard growth, the natural weak Harnack quantity is not $Q_+$5 but

$Q_+$6

and the tail is encoded through $Q_+$7 and $Q_+$8. The abstract De Giorgi-class theorem controls

$Q_+$9

by the $$\left( \iint_{V^-(r)} (u+\ell)^q\,dx\,dt \right)^{1/q} \le C\,\Bigl(\operatorname*{ess\,inf}_{V^+(r)} u + \ell\Bigr),$$0-value of the local infimum plus the $$\left( \iint_{V^-(r)} (u+\ell)^q\,dx\,dt \right)^{1/q} \le C\,\Bigl(\operatorname*{ess\,inf}_{V^+(r)} u + \ell\Bigr),$$1-value of $$\left( \iint_{V^-(r)} (u+\ell)^q\,dx\,dt \right)^{1/q} \le C\,\Bigl(\operatorname*{ess\,inf}_{V^+(r)} u + \ell\Bigr),$$2, and this becomes the main input for the full Harnack inequality for local minimizers and weak solutions (Chaker et al., 2022).

At the endpoint $$\left( \iint_{V^-(r)} (u+\ell)^q\,dx\,dt \right)^{1/q} \le C\,\Bigl(\operatorname*{ess\,inf}_{V^+(r)} u + \ell\Bigr),$$3, a tail-free weak Harnack inequality has been established for nonlocal $$\left( \iint_{V^-(r)} (u+\ell)^q\,dx\,dt \right)^{1/q} \le C\,\Bigl(\operatorname*{ess\,inf}_{V^+(r)} u + \ell\Bigr),$$4-subminimizers on complete, connected, doubling metric measure spaces. In that case,

$$\left( \iint_{V^-(r)} (u+\ell)^q\,dx\,dt \right)^{1/q} \le C\,\Bigl(\operatorname*{ess\,inf}_{V^+(r)} u + \ell\Bigr),$$5

and no nonlocal tail term appears in the conclusion (Lahti et al., 22 Mar 2026). This places weak Harnack theory in a borderline metric-space setting analogous to the local $$\left( \iint_{V^-(r)} (u+\ell)^q\,dx\,dt \right)^{1/q} \le C\,\Bigl(\operatorname*{ess\,inf}_{V^+(r)} u + \ell\Bigr),$$6 theory.

Further mixed-order developments include weighted homogeneous equations

$$\left( \iint_{V^-(r)} (u+\ell)^q\,dx\,dt \right)^{1/q} \le C\,\Bigl(\operatorname*{ess\,inf}_{V^+(r)} u + \ell\Bigr),$$7

with scaling-subcritical $$\left( \iint_{V^-(r)} (u+\ell)^q\,dx\,dt \right)^{1/q} \le C\,\Bigl(\operatorname*{ess\,inf}_{V^+(r)} u + \ell\Bigr),$$8, where weak Harnack holds for nonnegative weak supersolutions with a negative-part tail (Biswas et al., 16 Apr 2026), and superposition operators

$$\left( \iint_{V^-(r)} (u+\ell)^q\,dx\,dt \right)^{1/q} \le C\,\Bigl(\operatorname*{ess\,inf}_{V^+(r)} u + \ell\Bigr),$$9

where the decisive new object is a nonlocal superposition tail with $\ell$0-scaling, used to prove weak Harnack, Harnack, and Hölder continuity for mixed fractional orders (Bhowmick et al., 31 May 2026).

5. Proof architectures

Classical proofs of weak Harnack inequalities rely on decay-of-measure estimates, localization, and covering arguments; this is stated explicitly for fully nonlinear parabolic equations in non-divergence form (Sen, 3 Jun 2026). Modern work shows that there is no single canonical proof strategy. Instead, the analytic architecture depends strongly on the operator class and on whether the problem is local, nonlocal, kinetic, or fractional in time.

For symmetric stable Lévy processes, the proof can be potential-theoretic rather than iterative. One represents harmonic functions by exit distributions, rewrites the Poisson kernel through the Lévy density and the killed Green function, and compares an averaged Green kernel on $\ell$1 with a pointwise Green kernel from $\ell$2 (Sertic, 2015). In that framework weak Harnack emerges from Green-function comparison and a maximum-principle argument, not from De Giorgi iteration.

For mixed local–nonlocal $\ell$3-Laplace equations with nonhomogeneity, two analytic proofs are available. Both begin with energy estimates, a logarithmic estimate for $\ell$4, and a reverse Hölder inequality for supersolutions; one then uses the John–Nirenberg lemma, while the other uses the Bombieri–Giusti lemma. Both approaches avoid the Krylov–Safonov covering lemma and the expansion of positivity argument (Garain, 5 Oct 2025).

Time-fractional diffusion requires yet another mechanism. Because Steklov averages do not commute with the Riemann–Liouville convolution kernel, the proof uses Yosida approximation of the fractional derivative, a fundamental identity for $\ell$5, Moser iteration, logarithmic estimates, and an abstract Bombieri–Giusti lemma (Zacher, 2010). In kinetic Fokker–Planck theory the central tools are a logarithmic transform $\ell$6, a weak Poincaré inequality adapted to kinetic geometry, and a kinetic ink-spots covering theorem (Guerand et al., 2021).

A recent global proof for uniformly parabolic equations in non-divergence form replaces localization and covering by spacetime paraboloid envelopes $\ell$7. The contact sets $\ell$8 are tracked as the opening increases, and the weak Harnack estimate is recovered from a global measure-theoretic analysis of these envelopes (Sen, 3 Jun 2026). This shows that even in a classical setting the theorem admits genuinely different proofs.

6. Consequences, scope, and limitations

Weak Harnack inequalities serve as a structural bridge from energy estimates to regularity and qualitative analysis. In the symmetric $\ell$9-stable Lévy setting, weak Harnack is used to obtain Hölder regularity (Sertic, 2015). For time-fractional diffusion with measurable coefficients, it yields a strong maximum principle, continuity of weak solutions at $\alpha$00, and a Liouville-type theorem (Zacher, 2010). For fully nonlinear uniformly parabolic equations with unbounded coefficients and inhomogeneous terms, the weak Harnack inequality for $\alpha$01-viscosity supersolutions leads to Hölder continuity, a local maximum principle, and a Harnack inequality for solutions (Koike et al., 2018). In mixed local–nonlocal parabolic problems, it combines with local boundedness and tail estimates to produce the full Harnack inequality (Garain et al., 2021).

The theorem is also a point of separation between what is robust and what is not. Weak Harnack does not automatically upgrade to a full pointwise Harnack principle under arbitrary anisotropy. For symmetric stable Lévy processes, the full Harnack inequality can fail for specific anisotropic examples even though a weak Harnack inequality holds under bounded-density assumptions on the spectral measure (Sertic, 2015). In time-fractional equations, positivity assumptions cannot generally be localized because the memory term depends on the full past interval (Zacher, 2010). In nonlocal elliptic and parabolic equations, tail terms are not technical artifacts but reflect genuine dependence on the exterior negative part; they disappear only when global nonnegativity is available (Strömqvist, 2018).

Across these formulations, weak Harnack inequality retains the same conceptual role: it is the estimate that turns distributed positivity into pointwise lower control. What changes from one setting to another is the geometry of the comparison sets, the appearance or absence of tail terms, the admissible exponent range, and the proof technology needed to accommodate nonlocality, rough coefficients, memory, transport structure, or stochasticity.