  We prove that the Harnack inequality fails for nonlocal kinetic equations.
Such equations arise as linearized models for the Boltzmann equation without
cutoff and are of hypoelliptic type. We provide a counterexample for the
simplest equation in this theory, the fractional Kolmogorov equation. Our
result reflects a purely nonlocal phenomenon since the Harnack inequality holds
true for local kinetic equations like the Kolmogorov equation.
