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Wolff potential estimates and Wiener criterion for nonlocal equations with Orlicz growth

This paper establishes Wolff potential estimates for nonlocal equations with Orlicz growth and applies them to derive a necessary and sufficient Wiener criterion for the regularity of boundary points through a detailed analysis of superharmonic functions within nonlocal nonlinear potential theory.

Original authors: Minhyun Kim, Ki-Ahm Lee, Se-Chan Lee

Published 2026-08-28

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CC BY 4.0

Original authors: Minhyun Kim, Ki-Ahm Lee, Se-Chan Lee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the rules governing how things spread, settle, or smooth out are not fixed by simple, uniform laws, but instead change depending on the intensity of the situation itself. This is the realm of nonlinear potential theory, a branch of mathematics that studies how functions behave when they are governed by complex, shifting rules rather than steady, predictable ones. For over a century, mathematicians have sought to understand the "fine properties" of these functions, particularly how they behave right at the edge of a region. A central question in this field is determining whether a specific point on a boundary is "regular." In plain terms, a regular point is one where a solution to a mathematical problem behaves exactly as expected, smoothly connecting to the values imposed from the outside. If a point is irregular, the solution might jump, stutter, or fail to settle, no matter how carefully the surrounding conditions are set. The answer to whether a point is regular has long been known to depend on the geometry of the boundary and the specific nature of the mathematical rules involved.

For decades, researchers have relied on a powerful tool called the Wiener criterion to answer this question. Originally developed for simple, linear equations, this criterion was later extended to more complex, nonlinear equations that describe phenomena like fluid flow or elasticity. However, these extensions were limited to cases where the mathematical rules followed a standard, polynomial pattern. In recent years, mathematicians began exploring "nonlocal" equations, where the value of a function at one point depends on its values at distant points, not just its immediate neighbors. This nonlocal behavior mimics real-world systems where long-range interactions matter, such as in certain models of finance or biology. While progress was made in understanding these nonlocal equations with standard rules, a major gap remained for equations with more general, flexible growth patterns, known as Orlicz growth. These patterns allow the rules to change in more complex ways, making the mathematics significantly harder to tame.

In this paper, a team of researchers has successfully bridged that gap. They have established a complete set of rules for determining when a boundary point is regular for a broad new class of nonlocal equations with Orlicz growth. To do this, they first developed a precise way to measure the "strength" of a solution near a boundary using a concept called the Wolff potential. Think of this potential as a mathematical gauge that accumulates information about how a solution behaves across different scales, from the immediate vicinity of a point out to the wider region. The authors proved that this gauge provides both an upper and a lower limit for the value of a superharmonic function—a type of function that tends to be larger than its average neighbors. By proving these limits, they showed that the Wolff potential is not just a theoretical curiosity but a fundamental tool that accurately captures the behavior of these complex solutions.

With these estimates in hand, the researchers then derived the Wiener criterion for this new, general framework. They proved that a boundary point is regular if and only if a specific integral involving the capacity of the boundary's complement diverges to infinity. In simpler terms, this means that a point is regular if the "missing" part of the space outside the boundary is sufficiently "thick" or substantial in a very specific mathematical sense. If the boundary is too thin or sparse near that point, the solution will fail to behave regularly. This result is significant because it unifies previous findings and extends them to a much wider range of mathematical behaviors. The authors also corrected a subtle error in earlier work regarding how solutions behave when the growth of the equation is very fast, ensuring that the theory is now robust across all possible scenarios.

The findings are not merely a technical refinement; they provide a necessary and sufficient condition for regularity, meaning the test is both required and enough to guarantee the answer. The researchers demonstrated that this regularity depends entirely on the local geometry of the boundary and the specific growth rules of the equation, but surprisingly, it does not depend on the specific constants that describe the equation's ellipticity. This independence suggests a deep universality in how these nonlocal systems behave at their edges. By introducing new tools like the concept of quasicontinuity, which allows mathematicians to ignore sets of points that are negligible in a specific capacity sense, the team was able to handle cases where a single point might have a non-zero "weight" in the system. This careful handling of edge cases ensures that the theory holds up even in the most extreme mathematical landscapes.

Ultimately, this work completes a long-standing chapter in the study of nonlocal equations. It moves the field from a state where results were fragmented and limited to specific types of growth, to a comprehensive framework that covers a vast array of possibilities. The authors have shown that despite the complexity of nonlocal interactions and flexible growth rules, the fundamental condition for a boundary point to be regular remains a clear, geometric property that can be calculated and understood. This clarity allows mathematicians to predict the behavior of solutions in complex systems with greater confidence, paving the way for future applications in fields where long-range interactions and variable growth rates are the norm. The paper stands as a definitive proof, leaving no ambiguity about the conditions required for regularity in this sophisticated mathematical setting.

Technical Summary of "Wolff Potential Estimates and Wiener Criterion for Nonlocal Equations with Orlicz Growth"

Problem Statement
The paper addresses the development of nonlinear potential theory for a class of nonlocal operators with Orlicz growth. While the theory for standard $p$ -growth (polynomial nonlinearity) and fractional $p$ -Laplace equations is well-established—particularly regarding pointwise estimates via Wolff potentials and the Wiener criterion for boundary regularity—these results have not been fully extended to the more general Orlicz setting for nonlocal equations. Specifically, the authors aim to establish pointwise upper and lower bounds for superharmonic functions associated with nonlocal operators in terms of Wolff potentials and to derive a necessary and sufficient condition (the Wiener criterion) for the regularity of boundary points in this generalized framework.

Methodology
The authors build upon the foundational work of Kilpeläinen–Malý and Lukkari–Maeda–Marola for local equations and extend the recent nonlocal results of Kim–Lee–Lee (which focused on $p$ -growth) to the Orlicz setting. The methodology involves several key technical components:

Operator and Growth Framework: The study considers the nonlocal operator $L$ defined by a kernel $k(x,y)$ and a nonlinearity governed by a Young function $G$ (with derivative $g$ ). The growth of $G$ is controlled by parameters $1 < p \le q$ , satisfying specific structural conditions (e.g., $pG(t) \le tg(t) \le qG(t)$ ).
Superharmonic Functions and Riesz Measures: The authors define superharmonic functions in the nonlocal Orlicz context. A crucial step is proving that for any superharmonic function $u$ , the equation $Lu = \mu$ holds in the sense of distributions, where $\mu$ is a unique nonnegative Borel measure (the Riesz measure). This is established without requiring the function to be in a standard Sobolev space, utilizing approximation techniques and weak convergence.
Wolff Potentials: The paper introduces the Wolff potential $W^\mu_{s,G}$ adapted to Orlicz growth, defined via an integral involving the inverse of the derivative of the Young function and the measure of balls.
Iterative Estimates:
- Upper Bound: The proof of the pointwise upper bound involves a De Giorgi-type iteration scheme. It utilizes a reverse Hölder inequality with measure data, the weak Harnack inequality for supersolutions, and careful handling of the nonlocal "tail" terms.
- Lower Bound: The lower bound is derived by iterating a Caccioppoli-type estimate and a nonlocal analogue of the Kilpeläinen–Malý lemma, relating the measure of the Riesz measure on balls to the infimum of the superharmonic function.
Capacity and Quasicontinuity: The authors develop the theory of $(s, G)$ -capacity and $L$ -potentials. They establish the existence of $(s, G)$ -quasicontinuous representatives for functions in the relevant Orlicz-Sobolev spaces. This is essential for handling cases where the capacity of a single point may be zero or positive.

Key Contributions and Results

Pointwise Wolff Potential Estimates (Theorems 1.2 and 1.3):
- The paper proves that for a nonnegative superharmonic function $u$ in a ball $B_R(x_0)$ , the value $u(x_0)$ is bounded above by the sum of the infimum of $u$ in a smaller ball, the Wolff potential of its Riesz measure $\mu$ , and a nonlocal tail term.
- Conversely, the Wolff potential of the Riesz measure is bounded above by the value of the function at the center and the tail term.
- These results hold for the full range of growth parameters $p > 1$ and establish the existence of the Riesz measure $\mu = Lu$ for superharmonic functions, a result that was previously unknown for the full range in the nonlocal setting.
The Wiener Criterion (Theorem 1.4):
- The authors establish a necessary and sufficient condition for a boundary point $x_0 \in \partial \Omega$ to be regular. A point is regular if and only if the Wiener integral diverges:
  $\int_0^1 \rho^s g^{-1}\left( \frac{\text{cap}_{s,G}(B_\rho(x_0) \setminus \Omega, B_{2\rho}(x_0))}{\rho^{n-s}} \right) \frac{d\rho}{\rho} = \infty$
- This criterion characterizes regularity purely in terms of the $(s, G)$ -capacity of the complement of the domain near the boundary point.
Refinement of Previous Work:
- The paper corrects an error in a previous work (Kim–Lee–Lee [18]) regarding the case $p > n/s$ , where the continuity of harmonic functions up to the boundary was incorrectly deduced solely from Sobolev embedding without assuming boundary regularity. The current work resolves this by distinguishing cases based on whether a single point has positive $(s, G)$ -capacity.
- It unifies the treatment of different growth regimes by introducing $(s, G)$ -quasicontinuity, avoiding the need to split cases based on the dimension and growth exponent as was done in the $p$ -growth setting.

Significance and Claims
The paper claims to provide a comprehensive extension of nonlinear potential theory to nonlocal equations with Orlicz growth. Its primary significance lies in:

Generalization: Moving beyond standard polynomial growth ( $p$ -Laplace type) to general Orlicz growth, which allows for a broader class of nonlocal operators.
Completeness: Providing both upper and lower Wolff potential estimates for the full range of $p > 1$ and establishing the existence of the associated Riesz measure for superharmonic functions.
Boundary Regularity: Deriving the Wiener criterion in this general setting, showing that the regularity of a boundary point is determined by the local geometry of the domain's complement and the specific growth function $G$ , independent of the ellipticity constant of the kernel.
Theoretical Foundation: The development of $(s, G)$ -capacity and the associated potential theory tools (like $L$ -potentials and quasicontinuity) is presented as a contribution of independent interest for future studies in nonlocal nonlinear analysis.

The authors emphasize that their approach relies on a fine analysis of superharmonic functions within the framework of nonlocal nonlinear potential theory, bridging the gap between local and nonlocal theories in the context of Orlicz spaces.

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