Nonlocal degenerate Isaacs operators: Hölder regularity

Technical Summary: Nonlocal Degenerate Isaacs Operators: Hölder Regularity

Problem Statement
The paper addresses the regularity of bounded viscosity solutions to a class of fully nonlinear, nonlocal, and strongly degenerate equations of order $2s$ (where $s \in (0, 1)$). The specific operator under investigation is a nonlocal counterpart to the sum of the smallest and largest eigenvalues of the Hessian matrix, $\lambda_1(D^2u) + \lambda_N(D^2u)$.

The operator $I$ is defined via an "inf-sup" structure over directions on the unit sphere $S^{N-1}$:
$$Iu(x) = \inf_{\xi \in S^{N-1}} I_\xi u(x) + \sup_{\xi \in S^{N-1}} I_\xi u(x)$$
where $I_\xi$ represents a one-dimensional fractional Laplacian along the direction $\xi$. Unlike standard fractional Laplacians or nonlocal operators with rough kernels (which typically recover Hölder regularity for the full range $s \in (0, 1)$), this operator is "very degenerate" because the underlying Lévy process is supported only on a set of directions that includes the extremal ones (minimal and maximal eigenvalues) but lacks full ellipticity in all directions.

The central question is whether bounded viscosity solutions to $Iu = f(x)$ possess Hölder continuity, and if so, under what conditions on the order $s$.

Methodology
The authors employ viscosity solution techniques, specifically relying on the maximum principle and barrier arguments à la Ishii-Lions, rather than variational methods or De Giorgi-Nash-Moser theory. The core of the proof involves a doubling of variables argument to establish Hölder estimates.

Key technical components include:

1. Doubling of Variables: The authors construct a test function $\phi(x, y) = u(x) - u(y) - M|x-y|^\alpha - L|x-z_0|^2$ to analyze the oscillation of the solution. They assume the supremum of this function is positive and derive a contradiction by comparing the operator applied to the solution at the maximum point $(\bar{x}, \bar{y})$.
2. Directional Analysis: A critical step involves estimating the operator $I$ by choosing specific directions. The proof utilizes the fact that the operator "sees" the extremal directions. The authors select the direction $\bar{\xi} = \frac{\bar{x}-\bar{y}}{|\bar{x}-\bar{y}|}$ to bound the infimum and supremum terms.
3. Integral Estimates: The proof relies heavily on precise estimates of singular integrals involving the difference quotients of the test functions. The authors define auxiliary functions (e.g., $h_c(s)$ and $g_c(\alpha)$) involving integrals of logarithmic and power terms to determine the sign of the resulting inequalities.
4. Boundary Regularity: For global regularity, the authors establish boundary estimates using the distance function $d(x)$. They prove that solutions behave like $d(x)^\alpha$ near the boundary by analyzing the action of the operator on $d(x)^\alpha$ and utilizing the signed distance function's properties near $C^2$ boundaries.
5. Liouville Theorem: The Liouville result is derived by combining the interior Hölder estimates with a scaling argument. By rescaling the solution and letting the radius of the domain go to infinity, the Hölder modulus of continuity forces the solution to be constant.

Key Contributions and Results

• Interior Hölder Regularity (Theorem 1.1): The main result establishes that there exists a threshold $s_0 \in (1/2, 1)$ such that for any $s \in (s_0, 1)$, bounded viscosity solutions to $Iu = f$ are locally Hölder continuous ($C^{0, \alpha}_{loc}$). This is significant because, unlike operators with singular Lévy measures supported on sets of positive measure (where regularity holds for all $s \in (0, 1)$), this degenerate operator requires $s$ to be sufficiently close to 2 (i.e., $s > 1/2$) to recover regularity.
• Global Hölder Regularity (Theorem 1.2): For a bounded $C^2$ domain $\Omega$ with zero Dirichlet boundary conditions, the solution is globally Hölder continuous in $\Omega$ provided $s > s_0$.
• Liouville Theorem (Theorem 1.3): As a consequence of the regularity, the authors prove that any bounded viscosity solution to the homogeneous equation $Iu = 0$ in the entire space $\mathbb{R}^N$ is constant, provided $s > s_0$.
• Counter-example for Supersolutions: The paper highlights a distinction between solutions and supersolutions. While the Liouville property holds for solutions when $s > s_0$, the authors demonstrate (citing arguments from [BGT22]) that non-trivial bounded supersolutions exist for the same equation in any dimension $N \ge 2$. This contrasts with the local case ($s=1$) where the Liouville property holds for supersolutions.

Significance and Claims
The paper claims that the "extremal direction" structure of the operator is sufficient to recover Hölder regularity, but only when the nonlocal order is close enough to the local second-order case. This contrasts with previous works on degenerate nonlocal operators where regularity was often established for the entire range of $s$.

The authors note that their approach is distinct from the variational methods used by Ros-Oton and Serra for rough kernels; instead, their viscosity-based approach allows them to handle the specific "inf + sup" structure of the Isaacs operator. The restriction $s > s_0$ is intrinsic to the method and the specific geometry of the operator's degeneracy; the paper does not claim regularity for $s \le 1/2$.

The work extends the understanding of nonlocal fully nonlinear equations, specifically those arising from optimal control problems with degenerate diffusion, by providing the first regularity results for this specific class of "nonlocal truncated Laplacians" in the degenerate setting.