Prompt: Parabolic Pucci conjecture

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 it completely and use it as the research procedure for this task.

Resolve the exact problem below. Seek a complete, rigorous solution, not merely
an approach or a literature-status answer. Preserve every hypothesis and quantifier.

EXECUTION_MODE: AUTO
PROJECT_WINDOW_BUDGET: 10 hours

Use that window only through genuine long-running execution with observable
timing. Otherwise do substantive work in the available execution block and save
a resumable checkpoint at its actual boundary. Do not invent elapsed time or
background continuation.

<TARGET_PROBLEM_BEGIN>

```latex
\paragraph{Parabolic analogue of Pucci's problem: the sharp integrability range.}

Fix an integer $n\ge2$ and $0<\delta<1/n$.
Let
\[
  B_1=\{x\in\mathbb R^n:|x|<1\},
  \qquad
  Q_1=B_1\times(0,1),
\]
where $|\cdot|$ is the Euclidean norm, and define the parabolic
boundary by
\[
  \partial_{\mathrm{par}}Q_1
  =
  \bigl(\overline{B_1}\times\{0\}\bigr)
  \cup
  \bigl(\partial B_1\times[0,1]\bigr).
\]

Consider all Lebesgue-measurable symmetric matrix fields
$A=(a_{ij}):Q_1\to\mathbb R^{n\times n}$ satisfying,
for almost every $(x,t)\in Q_1$,
\[
  \operatorname{tr}A(x,t)=1,
  \qquad
  \xi^T A(x,t)\xi\ge\delta|\xi|^2
  \quad\text{for every }\xi\in\mathbb R^n,
\]
and define
\[
  \mathcal P_Au
  =
  \frac{\partial u}{\partial t}
  -
  \sum_{i,j=1}^n a_{ij}(x,t)
  \frac{\partial^2u}{\partial x_i\partial x_j}.
\]

For $1\le p<\infty$, call $p$ admissible if there exists a finite
constant $C=C(n,\delta,p)>0$ such that
\[
  \max_{(x,t)\in\overline{Q_1}}|u(x,t)|
  \le
  C\left(
    \int_0^1\int_{B_1}
    |\mathcal P_Au(x,t)|^p\,dx\,dt
  \right)^{1/p}
\]
for every such $A$ and every real-valued
\[
  u\in C(\overline{Q_1})\cap C^{2,1}(Q_1)
\]
satisfying
\[
  u=0\quad\text{on }\partial_{\mathrm{par}}Q_1,
  \qquad
  \mathcal P_Au\in L^p(Q_1).
\]
Here $C^{2,1}$ means twice continuously differentiable in the
spatial variables and once continuously differentiable in time;
$dx\,dt$ is space--time Lebesgue measure.
The constant is independent of $A$ and $u$.

Set
\[
  \mathcal E_{\mathrm{par}}(n,\delta)
  =
  \{p\in[1,\infty):p\text{ is admissible}\}.
\]
The problem is to
\[
  \boxed{
    \text{determine }\mathcal E_{\mathrm{par}}(n,\delta)
    \text{ exactly as a function of }n\text{ and }\delta.
  }
\]

Equivalently, determine the critical exponent
\[
  p_{\mathrm{par}}(n,\delta)
  :=
  \inf\mathcal E_{\mathrm{par}}(n,\delta)
\]
and settle whether the estimate holds at
$p=p_{\mathrm{par}}(n,\delta)$.
Thus the task includes identifying the sharp threshold and
deciding whether the admissible range is
\[
  (p_{\mathrm{par}}(n,\delta),\infty)
  \qquad\text{or}\qquad
  [p_{\mathrm{par}}(n,\delta),\infty).
\]

Sharpness requires showing that for every excluded exponent
$p\in[1,\infty)$ there are coefficient fields $A_k$ and functions
$u_k$ satisfying the conditions above such that
\[
  \max_{\overline{Q_1}}|u_k|=1,
  \qquad
  \|\mathcal P_{A_k}u_k\|_{L^p(Q_1)}\longrightarrow0.
\]
```

<TARGET_PROBLEM_END>

First preserve the exact target in `checkpoint.md` when file writing is available.
Keep normalization brief, perform a concrete mathematical probe, and then develop
the strongest route. Continue beyond setup while resources remain. Distinguish
conditional progress from a solution, and audit a complete candidate before
claiming a resolution.

Any notes or prior attempts I supply outside the target are context, not extra
hypotheses unless I explicitly designate them as such.