De Giorgi-Nash-Moser theory for kinetic equations with nonlocal diffusions

Technical Summary: De Giorgi-Nash-Moser Theory for Kinetic Equations with Nonlocal Diffusions

Problem Statement
The paper addresses the regularity theory for a class of nonlocal hypoelliptic equations arising in kinetic theory, specifically integro-differential Kolmogorov equations of the form:
$$(\partial_t + v \cdot \nabla_x)f = L_v f + h$$
where $L_v$ is a nonlocal diffusion operator acting only on the velocity variable $v$, typically involving fractional derivatives (e.g., a fractional Laplacian $(-\Delta_v)^s$ with rough coefficients). The kernel $K$ satisfies standard ellipticity/coercivity conditions but may be degenerate.

A central difficulty in this setting is the failure of classical De Giorgi-Nash-Moser (DGNM) estimates. Recent counterexamples by Kaßmann and Weidner [45] demonstrate that for such nonlocal kinetic equations, the classical Harnack inequality and local $L^2-L^\infty$ estimates generally fail, even for stationary solutions. This failure stems from the interplay between the nonlocality of the diffusion and the anisotropic transport term ($v \cdot \nabla_x$), which prevents the standard compensation of nonlocal effects via supremum tails used in purely elliptic or parabolic nonlocal settings.

Methodology
The authors develop a refined DGNM theory tailored to the specific geometry of kinetic equations, which is governed by a homogeneous Lie group structure (Galilean transformations) and non-Euclidean scaling. The methodology involves:

1. Geometric Framework: The analysis is conducted on "slanted cylinders" $Q_r(z_0)$ adapted to the kinetic scaling $\delta_r(t, x, v) = (r^{2s}t, r^{1+2s}x, rv)$. This respects the intrinsic scaling of the transport operator.
2. Tail Control: Instead of assuming global boundedness or using supremum tails, the authors introduce a condition based on the $L^p$-summability of the "nonlocal tail" of the solution along the drift variables. The tail is defined as:
   $$\text{Tail}(f; B_r(v_0)) := r^{2s} \int_{\mathbb{R}^n \setminus B_r(v_0)} \frac{|f(v)|}{|v_0 - v|^{n+2s}} \, dv$$
   The key assumption is that this tail quantity belongs to $L^p_{loc}$ with respect to the transport variables $(t, x)$, where $p > N_s/2s$ and $N_s$ is the homogeneous dimension.
3. Integrability Gain (Caccioppoli Estimates): The proof relies on a new kinetic Caccioppoli estimate that incorporates the tail term. By utilizing the hypoelliptic nature of the equation and the fundamental solution of the fractional Kolmogorov equation (via a parametrix approach), the authors establish a local gain of integrability for weak subsolutions. This allows them to bypass the lack of a direct Sobolev inequality in the time-velocity space.
4. Iterative Scheme: A De Giorgi-type iterative argument is employed, modified to handle the nonlocal tail. The iteration balances local contributions with the nonlocal tail contribution, controlled by a parameter $\delta$.

Key Contributions and Results

• Theorem 1.1 ($\delta$-interpolative $L^2-L^\infty$ Estimate): The authors prove a local boundedness estimate for weak subsolutions. The $L^\infty$ norm of the solution on a smaller cylinder is bounded by its $L^2$ norm on a larger cylinder, the $L^p$ norm of the source term $h$, and crucially, the $L^p$ norm of the nonlocal tail. The estimate includes a parameter $\delta$ that allows for a quantitative balance between local and nonlocal effects.
  $$\text{ess sup}_{Q_{r/2}} f \leq c(\delta) \langle v_0 \rangle^{\dots} \|f\|_{L^2} + \dots + \delta \|\text{Tail}(f^+)\|_{L^p}$$
  This result resolves the issue raised by the counterexample in [45] by showing that the failure of the classical estimate is due to the lack of control on the tail's integrability.

• Theorem 1.4 (Local Gain of Integrability): The paper establishes that weak subsolutions belong to $L^q_{loc}$ for a specific range of $q$, provided the tail and source terms satisfy the $L^p$ condition. This is a foundational step for the $L^2-L^\infty$ estimate.

• Theorem 1.5 (Strong Harnack Inequality): A new formulation of the strong Harnack inequality is proven for non-negative weak solutions. Unlike classical versions, this inequality includes a nonlocal remainder term involving the $L^p$ norm of the tail.
  $$\sup_{Q^-_{r_0}} f \leq c \inf_{Q^+_{r_0}} f + c \|\text{Tail}(f)\|_{L^p}$$
  This formulation is shown to be consistent with the counterexample in [45], as the tail term prevents the ratio of values from blowing up when the tail is controlled.

• Geometric Characterization (Theorem 4.7): Extending the work of Aronson and Serrin [7] to the nonlocal kinetic setting, the authors provide a geometric characterization of the Harnack inequality. They define "nonlocal Harnack chains" connecting points via admissible curves in the underlying Lie group, showing that the maximum of the solution on a compact set is controlled by the value at a starting point plus the accumulated tail contributions along the chain.

Significance and Claims
The paper claims to complete the nonlocal DGNM picture for Kolmogorov equations within the considered class. Its primary significance lies in:

1. Resolving the Counterexample: It identifies the precise nonlocal contribution (the $L^p$-summable tail) required to restore regularity estimates that fail in the classical form.
2. Removing A Priori Assumptions: The results allow for the proof of Hölder continuity and Harnack inequalities without assuming a priori global boundedness of solutions, a common assumption in previous kinetic literature.
3. Unification: The work aligns the kinetic theory with the estimates available in fractional elliptic and parabolic settings, providing integro-differential counterparts to recent results for local kinetic equations with rough coefficients.
4. Sharpness: The authors argue that the $L^p$ condition on the tail is sharp (or nearly sharp) in the context of the counterexamples provided by Kaßmann and Weidner, as lower integrability leads to the failure of the estimates.

The paper concludes by noting that these techniques could serve as a basis for further developments, such as nonlinear $p$-Laplacian type operators, viscosity approaches, and applications to mean field games, though these are presented as potential future directions rather than established results within the text.