Several counterexamples to kinetic Schauder estimates

In this paper, we construct counterexamples to the Schauder estimate for kinetic Fokker--Planck equations, disproving \cite[Conjecture~1.3]{HW}.

The conjectured estimate would control second velocity derivatives using only velocity Hölder regularity of the coefficients and forcing, with no spatial regularity.

We also give a zero-forcing variant, where the same {conditions are imposed on} a bounded zeroth-order coefficient.

Finally, in dimensions $d\geq 2$, we give endpoint examples showing that bounded velocity-independent forcing may produce bounded distributional stationary solutions whose second velocity derivatives are not locally bounded.