The kinetic Krylov–Safonov conjecture

Let $f:[0,1]\times B_1\times B_1\to\mathbb R$ be a classical solution of

$$f_t+v\cdot\nabla_x f=a_{ij}(t,x,v)\partial_{v_i v_j}f.$$

Assume that the coefficient matrix is uniformly elliptic: there are constants $\Lambda\geq\lambda>0$ such that, for $(t,x,v)\in[-1,0]\times B_1\times B_1$,

$$\lambda\mathrm I\leq\{a_{ij}(t,x,v)\}\leq\Lambda\mathrm I.$$

Kinetic Krylov–Safonov conjecture. There exist constants $\alpha>0$ and $C$, depending only on the dimension and $\lambda,\Lambda$, such that

$$\|f\|_{C^\alpha((1/2,1)\times B_{1/2}\times B_{1/2})}\leq C\|f\|_{L^\infty([0,1]\times B_1\times B_1)}.$$

This is the kinetic analogue of the Krylov–Safonov estimate for uniformly parabolic equations in non-divergence form. The source explicitly identifies it as an open problem.