Semi-local behaviour of non-local hypoelliptic equations: divergence form

Technical Summary: Semi-Local Behaviour of Non-Local Hypoelliptic Equations

Problem Formulation
The paper investigates the regularity and quantitative bounds for a class of kinetic integro-differential equations in divergence form:
$$\partial_t f + v \cdot \nabla_x f = Lf + h,$$
where $L$ is a non-local operator defined by a principal value integral with a non-negative kernel $K$. The operator models fluctuations in velocity for local time and space, distinct from local diffusion. The primary challenge addressed is the derivation of the Strong Harnack inequality for these equations. While the Weak Harnack inequality is known to hold even in bounded velocity domains, the Strong Harnack inequality (a linear bound of the local supremum by the local infimum) was previously known to fail if the equation is satisfied only in a bounded velocity domain. The paper aims to establish this inequality under the condition that the equation holds globally in velocity ($v \in \mathbb{R}^d$).

Methodology
The proof strategy departs from standard local parabolic techniques by addressing the non-local "tail" terms that typically prevent purely local estimates. The methodology proceeds in three main stages:

1. Tail Bound on Upper Level Sets:
   The authors derive a bound on the non-local tail term (the contribution of the solution outside the local domain) in terms of local quantities. Unlike previous approaches that required tail terms in the final estimate, this work exploits the coercivity of cross terms arising from the non-local operator when testing against level set functions. Specifically, by testing the equation with a power of the level set function $(f-l)_+^{-(1-\zeta)}$, the authors show that the tail on upper level sets can be bounded by local Lebesgue norms. This step crucially relies on the global validity of the equation in velocity to utilize the polynomial decay of the solution at infinity.

2. De Giorgi Iteration in $L^1$:
   Standard De Giorgi arguments for non-local equations often yield a linear $L^2$-to-$L^\infty$ bound only with tail terms. Here, the authors perform the iteration in $L^1$. By incorporating the previously derived local tail bound into the energy estimate, they establish a fully local linear $L^1$-to-$L^\infty$ bound (Proposition 5). This result is significant because it is the first instance of a First De Giorgi Lemma for hypoelliptic integro-differential equations where the supremum is bounded linearly by the $L^1$ norm without any tail error terms. A key distinction from the local case is that this result holds for solutions, whereas local results often extend to sub-solutions.

3. Fundamental Solution Bounds via Aronson's Method:
   Leveraging the Strong Harnack inequality and the linear $L^1$-$L^\infty$ bound, the authors adapt Aronson's method to the non-local kinetic setting. They construct a specific decay function $H$ satisfying a differential inequality involving the transport operator and the non-local part. This allows them to derive off-diagonal bounds by testing the equation with the product of the solution and $H$.

Key Results

• Strong Harnack Inequality (Theorem 1): The paper proves that for non-negative solutions satisfying the equation globally in velocity, the local supremum in a past cylinder is bounded linearly by the local infimum in a future cylinder, plus a source term. This holds without non-local tail terms, provided the equation is satisfied for all $v \in \mathbb{R}^d$.
• Polynomial Upper Bounds (Theorem 2): Using the derived Harnack inequality and Aronson's method, the authors establish polynomial upper bounds on the fundamental solution $J$. The decay rate is $O((\sigma-\tau_0)^{-2d(1+s)/2s})$ in time, with polynomial decay in space and velocity. The authors note these bounds are sub-optimal in space and velocity compared to the constant-coefficient case, primarily due to the treatment of the non-singular part of the kernel without assuming pointwise upper bounds.
• Exponential Lower Bounds (Theorem 3): For the range $s \in (1/2, 1)$, the paper derives exponential lower bounds on the fundamental solution. This is achieved by applying the Strong Harnack inequality repeatedly along a path connecting two points, yielding a pointwise Harnack inequality that translates into an exponential decay rate.
• Conditional Optimal Upper Bounds (Theorem 4): The authors show that if the non-singular part of the fundamental solution can be separated (a condition analogous to Meyer's decomposition in the parabolic case), the upper bounds become optimal in velocity.

Significance and Claims
The paper claims to provide the first Strong Harnack inequality for hypoelliptic integro-differential equations without additional tail error terms. This result improves upon the "Not-so-Strong Harnack inequality" previously established in [35], which involved a non-linear relation between the supremum and infimum or required non-local tail terms.

The significance lies in:

1. Local Regularity: Capturing the local regularity of solutions to a class of hypoelliptic integro-differential equations purely in terms of local quantities, provided the global velocity assumption holds.
2. Fundamental Solution Estimates: Providing the first two-sided estimates (polynomial upper and exponential lower) for the fundamental solution of a kinetic integro-differential equation with rough, non-local kernels.
3. Methodological Advance: Introducing a De Giorgi iteration in $L^1$ that successfully eliminates tail terms for solutions, a feat not previously achieved for non-local hypoelliptic equations.

The authors remain modest regarding the optimality of the derived bounds. They explicitly state that the polynomial upper bounds are sub-optimal in space and velocity compared to the fractional Kolmogorov equation (constant coefficients) and that the exponential lower bound relies on the assumption $s > 1/2$. They identify the lack of a pointwise lower bound on the kernel and the inability to separate non-singular contributions without pointwise upper bounds as the primary reasons for the sub-optimality. The results are framed as a step toward understanding the non-cutoff Boltzmann equation, though the full application to that specific physical model is reserved for future work.