Prompt: the kinetic Krylov–Safonov theorem

We used this Markdown research procedure together with the following prompt.

• mathematical_research_goal.md
• Plain text prompt

I am attaching `mathematical_research_goal.md`.

Read that file completely and treat it as the governing research procedure.
The authoritative mathematical problem is the complete LaTeX source between
the two TARGET\_PROBLEM markers below.

PROJECT\_WINDOW\_BUDGET: 6 hours

\<TARGET\_PROBLEM\_BEGIN>
```latex
Fix \(d\geq1\) and \(0<\lambda\leq\Lambda<\infty\), write \(z=(t,x,v)\in\mathbb{R}^{1+2d}\), and set
\[
Y=\partial_t+v\cdot\nabla_x.
\]
For \(z_0=(t_0,x_0,v_0)\) and \(r>0\), define
\[
Q_r(z_0)=\left\{(t,x,v):-r^2<t-t_0\leq0,\ |x-x_0-(t-t_0)v_0|<r^3,\ |v-v_0|<r\right\},
\]
and let \(Q_r^\circ(z_0)\) be the same set with \(t-t_0<0\). For \(z_i=(t_i,x_i,v_i)\), define the symmetric kinetic quasi-distance
\[
d_{\mathrm{kin}}(z_1,z_2)
=\min_{w\in\mathbb{R}^d}
\max\left\{
|t_1-t_2|^{1/2},
|v_1-w|,
|v_2-w|,
2^{-1/3}|x_1-x_2-(t_1-t_2)w|^{1/3}
\right\}.
\]
For \(0<\alpha<1\) and \(U\subset\mathbb{R}^{1+2d}\), set
\[
[u]_{C^\alpha_{\mathrm{kin}}(U)}
=\sup_{\substack{z,z'\in U\\z\neq z'}}
\frac{|u(z)-u(z')|}{d_{\mathrm{kin}}(z,z')^\alpha},
\qquad
\operatorname{osc}_Uu=\sup_Uu-\inf_Uu.
\]

Let \(A:Q_r^\circ(z_0)\to\operatorname{Sym}_d\) be any Borel measurable symmetric matrix field satisfying
\[
\lambda|\xi|^2\leq\xi^{\mathsf T}A(z)\xi\leq\Lambda|\xi|^2
\qquad
\text{for all }z\in Q_r^\circ(z_0)\text{ and }\xi\in\mathbb{R}^d.
\]
Let \(u\) be bounded and continuous on \(Q_r(z_0)\), assume that \(Yu\) and \(D_v^2u\) exist and are continuous in \(Q_r^\circ(z_0)\), and suppose that
\[
Yu=\operatorname{tr}(A(z)D_v^2u)
\quad\text{pointwise in }Q_r^\circ(z_0).
\]

Do there exist \(\alpha=\alpha(d,\lambda,\Lambda)\in(0,1)\) and \(C=C(d,\lambda,\Lambda)<\infty\) such that, simultaneously for every \(r>0\), every center \(z_0\), every such measurable coefficient field \(A\), and every such bounded classical solution \(u\),
\[
r^\alpha[u]_{C^\alpha_{\mathrm{kin}}(Q_{r/2}(z_0))}
\leq
C\operatorname{osc}_{Q_r(z_0)}u?
\]
```

\<TARGET\_PROBLEM\_END>

This is a bounded mathematical research project, not a request for an
immediate literature-status answer.

Begin the procedure in `mathematical_research_goal.md` now. As the first
persistence action, create and maintain `checkpoint.md`. Copy the exact marked
problem into that checkpoint before beginning heavy route work.

Do not terminate except under a stopping condition permitted by
`mathematical_research_goal.md`.