Kinetic Schauder estimate for time-irregular Landau coefficients

Let $Q_1$ and $Q_{1/2}$ denote the kinetic cylinders appearing in the equation, let $a$, $c$, $f$, $g$, and $\bar a$, $\bar c$ be the corresponding coefficients and functions, and let $D_v^2f$ denote the Hessian of $f$ in the velocity variable. For $\alpha\in(0,1)$, write $[\cdot]_{C_v^\alpha(Q)}$ for the velocity Hölder seminorm and assume that the ellipticity condition holds. Kinetic Schauder conjecture. Fix any $\alpha\in(0,1)$. If $f$, $D_v^2f$, $\bar a$, $\bar c$, and $g$ belong to $C_v^\alpha(Q_1)$, then

$$[D_v^2 f]_{C_v^\alpha(Q_{1/2})} \lesssim \left(1+[c]_{C_v^\alpha(Q_1)}+[a]_{C_v^\alpha(Q_1)}^{1+\frac{2}{\alpha}}\right)\|f\|_{L^\infty(Q_1)}+\left(1+[a]_{C_v^\alpha(Q_1)}\right)[g]_{C_v^\alpha(Q_1)}.$$

The implied constant depends only on $d$, $\alpha$, and $\Lambda$. This would extend the parabolic Schauder analogy to the kinetic setting without requiring regularity in the spatial variable; the paper presents it as a conjectural strengthening of its estimates, and its status is not otherwise resolved in the supplied text.