A note on relativistic kinetic Schauder estimates

Technical Summary: A Note on Relativistic Kinetic Schauder Estimates

Problem Statement
This note addresses the regularity theory for linear kinetic equations with relativistic transport, specifically investigating whether Schauder estimates can be formulated using only momentum Hölder norms, thereby omitting spatial regularity requirements. The equation under study is:
$$\partial_t f + \frac{p}{\langle p \rangle} \cdot \nabla_x f = \text{tr}(A(p)D^2_p f) + B(p) \cdot \nabla_p f + s,$$
where $\langle p \rangle = (1 + |p|^2)^{1/2}$.

Recent work by Henderson, Snelson, Tarfulea, and Tasković [4] established conditional regularity and decay for the relativistic Landau equation, relying on a Schauder estimate formulated in Lorentzian kinetic Hölder spaces. This estimate assumes an intrinsic Hölder modulus in all nontrivial variables, including a spatial modulus. In the Galilean (non-relativistic) setting, the endpoint question—whether spatial regularity can be entirely removed so that the momentum Hessian is controlled solely by velocity Hölder norms of coefficients and forcing—was recently answered negatively by Dong and the author in [2] for the Kolmogorov equation. The purpose of this note is to settle the same endpoint question for relativistic transport.

Methodology
The construction of counterexamples relies on an exact conjugation between the relativistic transport operator and the constant-coefficient Galilean Kolmogorov operator.

1. Conjugation Map: The author utilizes the diffeomorphism $\Phi(p) = p/\langle p \rangle$, which maps momentum space $\mathbb{R}^3$ onto the unit velocity ball $B_1$. Its inverse is $\Psi(v) = v/\sqrt{1-|v|^2}$.
2. Coefficient Selection: Specific smooth coefficients $A(p)$ and $B(p)$ are chosen (defined in Eq. 2.1) such that under the change of variables $v = \Phi(p)$, the relativistic equation transforms exactly into the constant-coefficient Kolmogorov equation:
   $$\partial_t u + v \cdot \nabla_x u = \Delta_v u + s(t, x).$$
   Crucially, the induced first-order terms cancel, and the diffusion becomes the standard Laplacian.
3. Counterexample Construction: Leveraging the "one-frequency" construction from [2], the author defines a family of smooth stationary solutions $f_N$ driven by a forcing $s_N = \cos(Nx_1)$. The solution is constructed via an explicit integral representation involving the function $I_N(\eta)$, which solves the transformed Kolmogorov equation.
4. Scaling Analysis: The analysis focuses on the behavior of the momentum Hessian $D^2_p f_N$ on a unit kinetic cylinder. The construction exploits the fact that the forcing $s_N$ is independent of momentum (vanishing momentum Hölder seminorm) but possesses high spatial frequency $N$.

Key Results
The paper establishes the following main results:

• Theorem 3.1 (Relativistic Momentum-Only Counterexample): For the fixed smooth, uniformly elliptic coefficients $A$ and $B$, there exists a family of smooth stationary solutions $f_N$ and forcings $s_N$ such that:

  • The solution norm is small: $\|f_N\|_{L^\infty} \leq C N^{-2/3}$.
  • The forcing has zero momentum Hölder seminorm: $[s_N]_{C^\alpha_p} = 0$.
  • The momentum Hessian exhibits large growth: $[D^2_p f_N]_{C^\alpha_p} \geq c N^{\alpha/3}$.

  Consequently, no local Schauder estimate of the form $[D^2_p f]_{C^\alpha_p} \leq C(\|f\|_{L^\infty} + [s]_{C^\alpha_p})$ can hold. The failure is uniform for a single fixed operator, rather than requiring a frequency-dependent family of coefficients.

• Corollary 3.4: The failure persists even if the estimate is allowed to depend on the momentum Hölder norms of the coefficients and forcing of any order, provided no spatial modulus is included.

• Proposition 3.5 (Zero-Source Variant): The obstruction remains even when the external forcing is zero. The author constructs solutions where the "hidden" spatial oscillation is encoded in a bounded zeroth-order coefficient $c_N(x, p)$. For this variant, the momentum Hölder seminorm of the coefficient $[c_N]_{C^\alpha_p}$ tends to zero as $N \to \infty$, yet the momentum Hessian of the solution still grows as $N^{\alpha/3}$.

Significance and Claims
The paper claims to demonstrate that the full Lorentzian Hölder control used in [4] cannot, in general, be replaced by momentum-only Hölder control.

• Identification of the Missing Quantity: The construction precisely identifies the spatial modulus as the necessary component. The forcing $s_N = \cos(Nx_1)$ is invisible to all momentum seminorms, but its spatial Hölder seminorm at the kinetic exponent scales as $N^{\alpha/3}$, exactly matching the growth of the Hessian.
• Consistency with Existing Theory: The results are consistent with the Schauder estimate in [4, Theorem 6.5], which retains a Lorentzian spatial modulus. The counterexample shows that removing this spatial regularity leads to a breakdown of the estimate.
• Structural Insight: The failure is realized through a structural compatibility between the nonlinear momentum-to-velocity map and a single fixed pair of coefficients, rather than through perturbative arguments or frequency-dependent families.

In summary, the note settles the endpoint question for relativistic transport by proving that spatial regularity is indispensable for controlling the momentum Hessian in linear kinetic equations, even when coefficients and forcing are smooth and uniformly elliptic.