The kinetic integro-differential Krylov–Safonov conjecture

Let $f:[0,1]\times B_1\times\mathbb R^d\to\mathbb R$ be a classical solution of the specified kinetic integro-differential equation on $(0,1]\times B_1\times B_1$, with bounded right-hand side $h$. Assume the kernel is symmetric,

$$K(t,x,v,v+w)=K(t,x,v,v-w),$$

and satisfies, for some $\Lambda\geq\lambda>0$,

$$\lambda|v'-v|^{-d-2s}\leq K(t,x,v,v')\leq\Lambda|v'-v|^{-d-2s}.$$

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

$$\|f\|_{C^\alpha((1/2,1)\times B_{1/2}\times B_{1/2})}\leq C\left(\|f\|_{L^\infty([0,1]\times B_1\times\mathbb R^d)}+\|h\|_{L^\infty([0,1]\times B_1\times B_1)}\right).$$

This is a proposed fractional integro-differential analogue of the kinetic Krylov–Safonov estimate. The source presents it immediately as a conjectural extension, and no resolution is given.