Nonlocal Pucci Operators

• Nonlocal Pucci operators are fully nonlinear integro-differential extremal operators defined via supremal and infimal envelopes over linear nonlocal operators with prescribed ellipticity, generalizing the classical Pucci operators.
• They underpin regularity theory by establishing maximum principles, Harnack inequalities, and Hölder continuity for solutions in both Euclidean and Riemannian settings.
• Their role in non-divergence form equations offers quantitative control on boundary regularity and provides a framework for understanding free boundary and singularity phenomena in nonlocal PDEs.

Nonlocal Pucci operators are fully nonlinear integro-differential extremal operators that generalize the classical (second-order) Pucci operators to the nonlocal, fractional-order setting. These operators are defined through supremal and infimal envelopes over linear nonlocal operators with kernels subject to prescribed ellipticity bounds, serving as canonical representatives of the class of uniformly elliptic equations of non-divergence form. Nonlocal Pucci operators are central to regularity theory, comparison principles, and qualitative analysis for fully nonlinear equations involving fractional Laplacians and related integro-differential operators, both in the Euclidean setting and on Riemannian manifolds with nonnegative curvature (Kim et al., 2021, Guillen et al., 2010, Cabeza et al., 13 Feb 2026).

1. Definition and Structural Properties

Let $s\in(0,1)$ (order parameter, $\sigma=2s$), and $0<\lambda\leq \Lambda$ (ellipticity constants). Consider the family $\mathcal{L}_*$ of linear, translation-invariant integro-differential operators $L$ of the form

$$L[u](x)=\int_{\mathbb{R}^n} \left(u(x+y)-u(x)-\nabla u(x)\cdot y \,\chi_{B_1}(y)\right)\,K(y)\,dy,$$

where $K(y)$ is symmetric, $K(ty)=t^{-n-2s}K(y)$ (homogeneity), and $$\lambda|y|^{-n-2s} \leq K(y) \leq \Lambda |y|^{-n-2s}$$ for all $y\neq 0$. The nonlocal Pucci extremal operators are defined by

$\sigma=2s$0

In the context of Riemannian manifolds $\sigma=2s$1 with nonnegative sectional curvature, the definition is adapted using geodesic balls, the exponential map, and measure $\sigma=2s$2, with symmetry imposed by reflection along geodesics and volume comparison replacing translation invariance (Kim et al., 2021).

Key structural features include:

• Uniform ellipticity in the sense of the kernel class,
• Homogeneity and symmetry to ensure well-defined extremality,
• Recovering second-order Pucci operators as $\sigma=2s$3,
• Scaling naturality for fractional orders,
• Local isometry invariance on manifolds, rather than translation invariance,
• Adaptability to both Euclidean and geometric contexts.

2. Operator Classes and Admissible Kernels

The kernel class $\sigma=2s$4 consists of all measurable, symmetric, homogeneous of order $\sigma=2s$5 kernels $\sigma=2s$6 satisfying

$\sigma=2s$7

almost everywhere. In the Euclidean case, this specializes to

$\sigma=2s$8

In the geometric setting, a kernel $\sigma=2s$9 is defined for $0<\lambda\leq \Lambda$0 by

$0<\lambda\leq \Lambda$1

with the symmetry condition $0<\lambda\leq \Lambda$2 for local reflection $0<\lambda\leq \Lambda$3.

These classes allow sup/inf over a uniform family of "maximal" and "minimal" nonlocal uniformly elliptic equations, analogous to their local second-order counterparts.

3. Principal Analytic Properties

Nonlocal Pucci operators retain fundamental qualitative and regularity features of the local (second-order) Pucci operators, including maximum principles, Harnack inequalities, and Hölder regularity. Notable results include:

• Krylov–Safonov Harnack inequality: Nonnegative viscosity solutions $0<\lambda\leq \Lambda$4 of $0<\lambda\leq \Lambda$5, $0<\lambda\leq \Lambda$6 in a ball (in either $0<\lambda\leq \Lambda$7 or Riemannian context) satisfy

$0<\lambda\leq \Lambda$8

with $0<\lambda\leq \Lambda$9 depending only on ellipticity, dimension, and parameters of the geometric space (Kim et al., 2021, Guillen et al., 2010).

• Interior Hölder regularity: Under the same assumptions, $\mathcal{L}_*$0 with quantitative estimates (Kim et al., 2021).
• Aleksandrov–Bakelman–Pucci (ABP) estimate: For bounded, lower-semicontinuous $\mathcal{L}_*$1 solving $\mathcal{L}_*$2, the maximum of $\mathcal{L}_*$3 is controlled by $\mathcal{L}_*$4 and $\mathcal{L}_*$5 norms of $\mathcal{L}_*$6 on the contact set, i.e.,

$\mathcal{L}_*$7

(Guillen et al., 2010). This generalizes the classical second-order ABP to nonlocal operators.

• Barrier and Hopf-type lemma: The existence of explicit, annulus-supported barriers shows non-trivial vanishing at boundaries, and a Hopf lemma for nonlocal Pucci operators ensures strict positivity of solutions near boundary points under natural geometric conditions (Cabeza et al., 13 Feb 2026).
• Strong maximum principle: Nontrivial nonnegative viscosity solutions in a domain are strictly positive unless they are identically zero (Cabeza et al., 13 Feb 2026).

4. Viscosity Solutions and Comparison Principles

Viscosity theory for nonlocal Pucci operators relies on the structure of the extremal operators and barriers constructed from their kernel class. Viscosity sub- and super-solutions are defined as follows: $\mathcal{L}_*$8 is a (sub/super)solution to $\mathcal{L}_*$9 if, whenever a $L$0 test function $L$1 touches $L$2 from above/below at $L$3, one has $L$4. This setup facilitates the application of comparison and maximum principles, as well as regularity theory.

A central property is the comparison principle: if $L$5 is fully nonlinear elliptic with respect to $L$6 and $L$7 are bounded viscosity sub- and supersolutions, then their difference $L$8 satisfies

$L$9

which, combined with the ABP estimate, yields comparison and uniqueness results (Guillen et al., 2010).

5. Geometric and Euclidean Contexts

In $$L[u](x)=\int_{\mathbb{R}^n} \left(u(x+y)-u(x)-\nabla u(x)\cdot y \,\chi_{B_1}(y)\right)\,K(y)\,dy,$$0, the prototypical nonlocal Pucci operators are built from isotropic kernels $$L[u](x)=\int_{\mathbb{R}^n} \left(u(x+y)-u(x)-\nabla u(x)\cdot y \,\chi_{B_1}(y)\right)\,K(y)\,dy,$$1 and correspond to extremal fractions of the fractional Laplacian. On Riemannian manifolds $$L[u](x)=\int_{\mathbb{R}^n} \left(u(x+y)-u(x)-\nabla u(x)\cdot y \,\chi_{B_1}(y)\right)\,K(y)\,dy,$$2 with nonnegative sectional curvature, the theory adapts using local geodesics, the exponential map, volume comparability, and ellipticity conditions on $$L[u](x)=\int_{\mathbb{R}^n} \left(u(x+y)-u(x)-\nabla u(x)\cdot y \,\chi_{B_1}(y)\right)\,K(y)\,dy,$$3. Classical results for the fractional Laplacian and second-order Pucci operators are recovered in the limit $$L[u](x)=\int_{\mathbb{R}^n} \left(u(x+y)-u(x)-\nabla u(x)\cdot y \,\chi_{B_1}(y)\right)\,K(y)\,dy,$$4 (Kim et al., 2021).

Examples illustrating this include:

Setting	Kernel Class	Reduction as $$L[u](x)=\int_{\mathbb{R}^n} \left(u(x+y)-u(x)-\nabla u(x)\cdot y \,\chi_{B_1}(y)\right)\,K(y)\,dy,$$5
$$L[u](x)=\int_{\mathbb{R}^n} \left(u(x+y)-u(x)-\nabla u(x)\cdot y \,\chi_{B_1}(y)\right)\,K(y)\,dy,$$6	$$L[u](x)=\int_{\mathbb{R}^n} \left(u(x+y)-u(x)-\nabla u(x)\cdot y \,\chi_{B_1}(y)\right)\,K(y)\,dy,$$7, $$L[u](x)=\int_{\mathbb{R}^n} \left(u(x+y)-u(x)-\nabla u(x)\cdot y \,\chi_{B_1}(y)\right)\,K(y)\,dy,$$8	Local Pucci, $$L[u](x)=\int_{\mathbb{R}^n} \left(u(x+y)-u(x)-\nabla u(x)\cdot y \,\chi_{B_1}(y)\right)\,K(y)\,dy,$$9
Sphere, Product Manifold	$K(y)$0 with geodesic symmetry and volume comparison	Local Pucci on tangent space

6. Applications and Open Problems

Nonlocal Pucci operators underpin regularity theory for integro-differential equations, nonlocal obstacle problems, and geometric PDEs in both Euclidean and non-Euclidean geometries. Recent developments focus on:

• Maximum principles and Hopf lemmas in the nonlocal regime, including implications for dead-core formation and positivity (Cabeza et al., 13 Feb 2026).
• Harnack inequalities and Hölder regularity of viscosity solutions under minimal assumptions (Kim et al., 2021, Guillen et al., 2010).
• Liouville-type theorems and classification of blow-up profiles, particularly for sublinear equations (Cabeza et al., 13 Feb 2026).
• Parabolic extensions and time-dependent regularity, where the nonlocal Pucci operators govern models with memory or long-range interactions.

A plausible implication is that control of nonlocal tail behavior (i.e., $K(y)$1 outside the domain) is critical in the nonlocal setting, a significant distinction from the local Pucci case, influencing regularity and maximum principles (Cabeza et al., 13 Feb 2026).

Open questions include fine classification of singularities, boundary regularity phenomena unique to nonlocal equations, and quantitative versions of comparison principles for more general nonlocal structures.

7. Relation to Classical Theory

As $K(y)$2, for functions $K(y)$3 sufficiently regular, nonlocal Pucci extremal operators converge to the second-order local Pucci operators:

$K(y)$4

where $K(y)$5 are the eigenvalues of $K(y)$6. All standard second-order theory (strong maximum principle, Hopf lemma, nonlinear eigenvalues, etc.) is recovered in this limit (Kim et al., 2021, Cabeza et al., 13 Feb 2026). However, the nonlocal dependence on $K(y)$7 outside the domain (the “tail effect”) is unique to the nonlocal framework and requires specific estimates and control, particularly for dead-core and boundary regularity phenomena.

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Key references: Kim–Kim–Lee (Kim et al., 2021), Guillen–Schwab (Guillen et al., 2010), Cabeza–Nornberg–dos Prazeres (Cabeza et al., 13 Feb 2026).