The kinetic Schauder estimate for velocity Hessians

Let $Q_r=(-r^2,0]\times B_{r^3}\times B_r\subset\mathbb{R}\times\mathbb{R}^d\times\mathbb{R}^d$. For a bounded function $h$ on $Q$, define

$$[h]_{C_v^\alpha(Q)}:=\sup_{\substack{(t,x,v),(t,x,v')\in Q\\0<|v-v'|<1/2}}\frac{|h(t,x,v)-h(t,x,v')|}{|v-v'|^\alpha}$$

and analogously define $[h]_{C_x^\beta(Q)}$ using spatial differences. For matrix-valued functions, let $|M|$ denote the Frobenius norm. Let $\alpha\in(0,1)$ and $\Lambda>1$. There is a constant $C=C(d,\alpha,\Lambda)$ such that every smooth solution $f$ of the kinetic Fokker–Planck equation in $Q_1$ satisfying

$$\Lambda^{-1}\operatorname{Id}\le a\le\Lambda\operatorname{Id},\qquad |c|\le\Lambda,\qquad |g|\le\Lambda,$$

and with $f,D_v^2f,a,c,g\in C_v^\alpha(Q_1)$ obeys

$$[D_v^2f]_{C_v^\alpha(Q_{1/2})}\le C\Bigl(1+[c]_{C_v^\alpha(Q_1)}+[a]_{C_v^\alpha(Q_1)}^{1+2/\alpha}\Bigr)\|f\|_{L^\infty(Q_1)}+C\Bigl(1+[a]_{C_v^\alpha(Q_1)}\Bigr)[g]_{C_v^\alpha(Q_1)}.$$

The kinetic Schauder estimate. The displayed estimate asserts a uniform interior Hölder bound for the velocity Hessian of smooth solutions in terms of the solution, the coefficient Hölder seminorms, and the source-term Hölder seminorm. The paper states that this conjecture is false, so the proposed estimate is refuted.