SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} = 1 6 \pi^ {4} \int_ {\mathbb {R} ^ {d}} \omega_ {i} ^ {2} \omega_ {j} ^ {2} | \mathcal {F} [ f ] (\omega) | ^ {2} | 1 - \mathcal {F} [ g ] (v \omega) | ^ {2} d \omega \\ \leq 1 6 \pi^ {4} \pi^ {2} \int_ {t \| \omega \| \geq 1} \omega_ {i} ^ {2} \omega_ {j} ^ {2} | \mathcal {F} [ f ] (\omega) | ^ {2} d \omega \\ = 1 6 \pi^ {4} \int_ {v \| \omega \| \geq 1} \omega_ {i} ^ {2} \omega_ {j} ^ {2} (1 + \| \omega \| ^ {2}) ^ {- \beta} (1 + \| \omega \| ^ {2}) ^ {\beta} | \mathcal {F} [ f ] (\omega) | ^ {2} d \omega \\ \leq 1 6 \pi^ {4} \sup  _ {v \| \omega \| \geq 1} \frac {\omega_ {i} ^ {2} \omega_ {j} ^ {2}}{(1 + \| \omega \| ^ {2}) ^ {\beta}} \int (1 + \| \omega \| ^ {2}) ^ {\beta} | \mathcal {F} [ f ] (\omega) | ^ {2} d \omega \\ \leq 16 \pi^ {4} \sup  _ {v \| \omega \| \geq 1} \frac {\omega_ {i} ^ {2} \omega_ {j} ^ {2}}{(1 + \| \omega \| ^ {2}) ^ {\beta}} 2 ^ {2 \beta} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} ^ {2} \\ \leq 16 \pi^ {4} \sup  _ {v \| \omega \| \geq 1} \frac {1}{(1 + \| \omega \| ^ {2}) ^ {\beta - 2}} 2 ^ {2 \beta} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} ^ {2} \\ \leq \frac {2 5 6 \pi^ {4} (2 v) ^ {2 (\beta - 2)}}{(1 + v ^ {2}) ^ {\beta - 2}} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} ^ {2}. \\ \end{array}
$$
Now we have,
$$
\begin{array}{l} \left\| \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p ^ {\star} (x, t) - \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} f (x, t) \right\| _ {L ^ {1} \left(\mathbb {R} ^ {d}\right)} = \left\| 2 \sum_ {m = 1} ^ {q} \left[ \frac {\partial f _ {m}}{\partial x _ {i}} \frac {\partial f _ {m}}{\partial x _ {j}} - \frac {\partial f _ {m , v}}{\partial x _ {i}} \frac {\partial f _ {m , v}}{\partial x _ {j}} \right. \right. \\ \left. \left. + f _ {m} \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} - f _ {m, v} \frac {\partial^ {2} f _ {m , v}}{\partial x _ {i} \partial x _ {j}} \right] \right\| _ {L ^ {1} \left(\mathbb {R} ^ {d}\right)} \\ \end{array}
$$
$$
\begin{array}{l} \leq 2 \sum_ {m = 1} ^ {q} \left[ \left\| \frac {\partial f _ {m}}{\partial x _ {i}} - \frac {\partial f _ {m , v}}{\partial x _ {i}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {m}}{\partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} + \left\| \frac {\partial f _ {m}}{\partial x _ {j}} - \frac {\partial f _ {m , v}}{\partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {m , v}}{\partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \right] \\ + 2 \sum_ {m = 1} ^ {q} \left[ \left\| \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} - \frac {\partial^ {2} f _ {m , v}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \| f _ {m} \| _ {L ^ {2} (\mathbb {R} ^ {d})} + \| f _ {m} - f _ {m, v} \| _ {L ^ {2} (\mathbb {R} ^ {d})} \left\| \frac {\partial^ {2} f _ {m , v}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \right] \\ \end{array}
$$
$$
\begin{array}{l} \left\| \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} \left[ \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} \left(D _ {i j} p ^ {\star} (x, t)\right) - \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} \left(D _ {i j} f (x, t)\right) \right] \right\| _ {L _ {1} \left(\mathbb {R} ^ {d}\right)} \\ \leq \underbrace {\sup  _ {i , j} D _ {i j}} _ {:= R _ {d}} \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} \left\| \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p ^ {\star} (x, t) - \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} f (x, t) \right\| _ {L _ {1} \left(\mathbb {R} ^ {d}\right)}. \tag {29} \\ \end{array}
$$
$$
\begin{array}{l} \left\| \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p ^ {\star} (x, t) - \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} f (x, t) \right\| _ {L ^ {1} \left(\mathbb {R} ^ {d}\right)} \\ \leq 2 \sum_ {m = 1} ^ {q} \left[ \left\| \frac {\partial f _ {m}}{\partial x _ {i}} - \frac {\partial f _ {m , v}}{\partial x _ {i}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {m}}{\partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} + \left\| \frac {\partial f _ {m}}{\partial x _ {j}} - \frac {\partial f _ {m , v}}{\partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {m , v}}{\partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \right] \\ \end{array}
$$
$$
+ 2 \sum_ {m = 1} ^ {q} \left[ \left\| \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} - \frac {\partial^ {2} f _ {m , v}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \| f _ {m} \| _ {L ^ {2} (\mathbb {R} ^ {d})} + \| f _ {m} - f _ {m, v} \| _ {L ^ {2} (\mathbb {R} ^ {d})} \left\| \frac {\partial^ {2} f _ {m , v}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \right]
$$
Let's consider all the terms separately.
$$
2 \sum_ {m = 1} ^ {q} \left\| \frac {\partial f _ {m}}{\partial x _ {i}} - \frac {\partial f _ {m , v}}{\partial x _ {i}} \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d}\right)} \left\| \frac {\partial f _ {m}}{\partial x _ {j}} \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d}\right)} \leq \frac {1 6 \pi (2 v) ^ {(\beta - 1)}}{(1 + v ^ {2}) ^ {\frac {\beta - 1}{2}}} \sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {m}}{\partial x _ {j}} \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d}\right)}
$$
$$
2 \sum_ {m = 1} ^ {q} \left\| \frac {\partial f _ {m}}{\partial x _ {j}} - \frac {\partial f _ {m , v}}{\partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {m , v}}{\partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \leq \frac {1 6 \pi (2 v) ^ {(\beta - 1)}}{(1 + v ^ {2}) ^ {\frac {\beta - 1}{2}}} \sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {m}}{\partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \| g \| _ {L ^ {1} (\mathbb {R} ^ {d})}
$$
$$
2 \sum_ {m = 1} ^ {q} \left\| \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} - \frac {\partial^ {2} f _ {m , v}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \| f _ {m} \| _ {L ^ {2} (\mathbb {R} ^ {d})} \leq \frac {3 2 \pi^ {2} (2 v) ^ {(\beta - 2)}}{(1 + v ^ {2}) ^ {\frac {\beta - 2}{2}}} \sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \| f _ {m} \| _ {L ^ {2} (\mathbb {R} ^ {d})}
$$
$$
\begin{array}{l} 2 \sum_ {m = 1} ^ {q} \| f _ {m} - f _ {m, v} \| _ {L ^ {2} (\mathbb {R} ^ {d})} \left\| \frac {\partial^ {2} f _ {m , v}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \leq 2 (2 v) ^ {\beta} \sum_ {m = 1} ^ {q} \| f _ {m} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial^ {2} f _ {m , v}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \\ = 2 (2 v) ^ {\beta} \sum_ {m = 1} ^ {q} \| f _ {m} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial^ {2} f _ {m} \star g _ {v}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \\ = 2 (2 v) ^ {\beta} \sum_ {m = 1} ^ {q} \| f _ {m} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \| g \| _ {L ^ {1} (\mathbb {R} ^ {d})}. \\ \end{array}
$$
$$
\begin{array}{l} \left\| \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p ^ {\star} (x, t) - \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} f (x, t) \right\| _ {L ^ {1} (\mathbb {R} ^ {d})} \leq \frac {1 6 \pi (2 v) ^ {(\beta - 1)}}{(1 + v ^ {2}) ^ {\frac {\beta - 1}{2}}} (1 + \| g \| _ {L ^ {1} (\mathbb {R} ^ {d})}) \sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {m}}{\partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \\ + \frac {3 2 \pi^ {2} (2 v) ^ {(\beta - 2)}}{(1 + v ^ {2}) ^ {\frac {\beta - 2}{2}}} \sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \| f _ {m} \| _ {L ^ {2} (\mathbb {R} ^ {d})} + 2 (2 v) ^ {\beta} \sum_ {m = 1} ^ {q} \| f _ {m} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \| g \| _ {L ^ {1} (\mathbb {R} ^ {d})}. \\ \end{array}
$$
Hence, if we choose,
$$
v = \min \left(\left(\frac {\varepsilon}{4 8 \pi 2 ^ {\beta - 1} C _ {1}}\right) ^ {\frac {1}{\beta - 1}}, \left(\frac {\varepsilon}{9 6 \pi^ {2} 2 ^ {\beta - 2} C _ {2}}\right) ^ {\frac {1}{\beta - 2}}, \left(\frac {1}{2 ^ {\beta + 2} C _ {3}}\right) ^ {\frac {1}{\beta}}\right)
$$
where  $C_1 = \left(1 + \| g\|_{L^1 (\mathbb{R}^d)}\right)\sum_{m = 1}^{q}\| f\|_{W_2^\beta (\mathbb{R}^d)}\left\| \frac{\partial f_m}{\partial x_j}\right\|_{L^2 (\mathbb{R}^d)},$ $C_2 =$
$$
\sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \| f _ {m} \| _ {L ^ {2} (\mathbb {R} ^ {d})} \quad \text {a n d} \quad C _ {3} \quad = \quad \sum_ {m = 1} ^ {q} \| f _ {m} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \| g \| _ {L ^ {1} (\mathbb {R} ^ {d})},
$$
$$
\left\| \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p ^ {\star} (x, t) - \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} f (x, t) \right\| _ {L ^ {1} (\mathbb {R} ^ {d})} \leq \varepsilon .
$$
$$
\left\| \sum_ {i, j = 1} ^ {d} D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p ^ {\star} (x, t) - D _ {i j} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} \hat {p} (x, t) \right\| _ {L ^ {1} \left(\mathbb {R} ^ {d}\right)} \leq d ^ {2} R _ {d} \varepsilon .
$$
In similar way, we can also obtain bound for  $L^2$  metric. Now we have,
$$
\left\| \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p ^ {\star} (x, t) - \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} f (x, t) \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d}\right)} = \left\| 2 \sum_ {m = 1} ^ {q} \left[ \frac {\partial f _ {m}}{\partial x _ {i}} \frac {\partial f _ {m}}{\partial x _ {j}} - \frac {\partial f _ {m , v}}{\partial x _ {i}} \frac {\partial f _ {m , v}}{\partial x _ {j}} \right. \right.
$$
$$
\begin{array}{l} \left. + f _ {m} \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} - f _ {m, v} \frac {\partial^ {2} f _ {m , v}}{\partial x _ {i} \partial x _ {j}} \right] \Bigg \| _ {L ^ {2} \left(\mathbb {R} ^ {d}\right)} \\ \leq 2 \sum_ {m = 1} ^ {q} \left[ \left\| \frac {\partial f _ {m}}{\partial x _ {i}} - \frac {\partial f _ {m , v}}{\partial x _ {i}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {m}}{\partial x _ {j}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} + \left\| \frac {\partial f _ {m}}{\partial x _ {j}} - \frac {\partial f _ {m , v}}{\partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {m , v}}{\partial x _ {j}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \right] \\ + 2 \sum_ {m = 1} ^ {q} \left[ \left\| \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} - \frac {\partial^ {2} f _ {m , v}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \| f _ {m} \| _ {L ^ {\infty} (\mathbb {R} ^ {d})} + \| f _ {m} - f _ {m, v} \| _ {L ^ {2} (\mathbb {R} ^ {d})} \left\| \frac {\partial^ {2} f _ {m , v}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \right] \\ \end{array}
$$
Let us now compute the following,
$$
\left\| \frac {\partial f _ {m , v}}{\partial x _ {j}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \leq \left\| \frac {\partial f _ {m}}{\partial x _ {j}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \| g \| _ {L ^ {1} (\mathbb {R} ^ {d})}
$$
$$
\left\| \frac {\partial^ {2} f _ {m , v}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \leq \left\| \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \| g \| _ {L ^ {1} (\mathbb {R} ^ {d})}
$$
$$
\begin{array}{l} \left\| \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} \left[ \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} \left(D _ {i j} p ^ {\star} (x, t)\right) - \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} \left(D _ {i j} f (x, t)\right) \right] \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d}\right)} \\ \leq \underbrace {\sup  _ {i , j} D _ {i j}} _ {:= R _ {d}} \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} \left\| \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p ^ {\star} (x, t) - \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} f (x, t) \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d}\right)}. \tag {30} \\ \end{array}
$$
Combining everything together, we have
$$
\begin{array}{l} \left\| \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p ^ {\star} (x, t) - \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} f (x, t) \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d}\right)} \leq \frac {1 6 \pi (2 v) ^ {(\beta - 1)}}{(1 + v ^ {2}) ^ {\frac {\beta - 1}{2}}} (1 + \| g \| _ {L ^ {1} \left(\mathbb {R} ^ {d}\right)}) \sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} \left(\mathbb {R} ^ {d}\right)} \left\| \frac {\partial f _ {m}}{\partial x _ {j}} \right\| _ {L ^ {\infty} \left(\mathbb {R} ^ {d}\right)} \\ + \frac {3 2 \pi^ {2} (2 v) ^ {(\beta - 2)}}{(1 + v ^ {2}) ^ {\frac {\beta - 2}{2}}} \sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \| f _ {m} \| _ {L ^ {\infty} (\mathbb {R} ^ {d})} + 2 (2 v) ^ {\beta} \sum_ {m = 1} ^ {q} \| f _ {m} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \| g \| _ {L ^ {1} (\mathbb {R} ^ {d})}. \\ \end{array}
$$
Hence, we get the final result by replacing  $f(x,t)$  with  $\hat{p}(x,t)$ .
A.2 Approximation Result by a PSD Model (Proof of Theorem 1)
Proof. Now, we just need to collect results from Lemmas 2 and 3 to get the final bound. Let us assume that there exist an optimal density function  $p^{\star}(x,t)$  which is the solution of fokkerplanck equation. Hence, We would like to show that there exists a psd operator  $M_{\varepsilon}$  such that if  $\hat{p} (x,t) = f(x,t) = \phi_{X,T}(x,t)^{\top}M_{\varepsilon}\phi_{X,T}(x,t)$  such that we can obtain bound for following three quantities for  $r\in \{1,2\}$ :
1.  $\left\| \frac{\partial p^{\star}(x,t)}{\partial t} - \frac{\partial \hat{p}(x,t)}{\partial t} \right\|_{L^{r}(\mathbb{R}^{d + 1})} = O(\varepsilon)$ .
$$
\begin{array}{l} \left\| \sum_ {i = 1} ^ {d} \frac {\partial}{\partial x _ {i}} \left(\mu_ {i} (x, t) p ^ {\star} (x, t) - \mu_ {i} (x, t) \hat {p} (x, t)\right) \right\| _ {L ^ {r} \left(\mathbb {R} ^ {d + 1}\right)} = \left\| \sum_ {i = 1} ^ {d} \left[ \mu_ {i} (x, t) \left[ \frac {\partial p ^ {\star} (x , t)}{\partial x _ {i}} - \frac {\partial \hat {p} (x , t)}{\partial x _ {i}} \right] \right. \right. \\ \left. + \frac {\partial \mu_ {i} (x , t)}{\partial x _ {i}} \left(p ^ {\star} (x, t) - \hat {p} (x, t)\right) \right] \Bigg \| _ {L ^ {1} \left(\mathbb {R} ^ {d + 1}\right)} = O (\varepsilon), \\ \end{array}
$$
3.  $\left\| \sum_{i = 1}^{d}\sum_{j = 1}^{d}\left[\frac{\partial^2}{\partial x_i\partial x_j} (D_{ij}p^\star (x,t)) - \frac{\partial^2}{\partial x_i\partial x_j} (D_{ij}\hat{p} (x,t))\right]\right\|_{L^1 (\mathbb{R}^{d + 1})} = O(\varepsilon).$
If the above three condition holds, then, we can show that for  $r \in \{1, 2\}$ ,
$$
\left\| \frac {\partial \hat {p} (x , t)}{\partial t} + \sum_ {i = 1} ^ {d} \frac {\partial}{\partial x _ {i}} (\mu_ {i} (x, t) \hat {p} (x, t)) - \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} (D _ {i j} \hat {p} (x, t)) \right\| _ {L ^ {r} (\mathbb {R} ^ {d + 1})} = O (\varepsilon).
$$
We have from Lemma 2 and equation (26), if we choose
$$
v = \min \left(\left(\frac {\varepsilon}{2 ^ {\beta - 1} 1 6 \pi C _ {1}}\right) ^ {\frac {1}{\beta - 1}}, \left(\frac {\varepsilon}{2 ^ {\beta + 1} C _ {2}}\right) ^ {\frac {1}{\beta}}\right)
$$
where  $C_1 = \sum_{i=1}^q \|f_i\|_{W_2^\beta(\mathbb{R}^{d+1})} \|f_i\|_{L^\infty(\mathbb{R}^{d+1})}$  and  $C_2 =$
$$
\sum_ {i = 1} ^ {q} \left\| f _ {i} \right\| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} \left\| \frac {\partial f _ {i}}{\partial t} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})} \left\| g \right\| _ {L _ {1} (\mathbb {R} ^ {d + 1})}. H e n c e,
$$
$$
\left\| \frac {\partial p ^ {\star} (x , t)}{\partial t} - \frac {\partial f (x , t)}{\partial t} \right\| _ {L ^ {2} (\mathbb {R} ^ {d + 1})} \leq \varepsilon .
$$
From [RC21, Theorem D.4 (Eq. D.25)], we have,
$$
\| p ^ {\star} (x, t) - \hat {p} (x, t) \| _ {L ^ {2} (\mathbb {R} ^ {d + 1})} \leq (2 v) ^ {\beta} (1 + \| g \| _ {L ^ {1} (\mathbb {R} ^ {d + 1})}) \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} \| f _ {i} \| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})}.
$$
This proves the first part. Going further,
$$
\begin{array}{l} \left\| \sum_ {i = 1} ^ {d} \frac {\partial}{\partial x _ {i}} \left(\mu_ {i} (x, t) p ^ {\star} (x, t) - \mu_ {i} (x, t) \hat {p} (x, t)\right) \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} = \left\| \sum_ {i = 1} ^ {d} \left[ \mu_ {i} (x, t) \left[ \frac {\partial p ^ {\star} (x , t)}{\partial x _ {i}} - \frac {\partial \hat {p} (x , t)}{\partial x _ {i}} \right] \right. \right. \\ \left. + \left. \frac {\partial \mu_ {i} (x , t)}{\partial x _ {i}} (p ^ {\star} (x, t) - \hat {p} (x, t)) \right] \right\rVert_ {L ^ {2} (\mathbb {R} ^ {d + 1})} \\ \end{array}
$$
$$
\leq \sum_ {i = 1} ^ {d} \left\| \mu_ {i} (x, t) \left[ \frac {\partial p ^ {\star} (x , t)}{\partial x _ {i}} - \frac {\partial \hat {p} (x , t)}{\partial x _ {i}} \right] \right\| _ {L ^ {2} (\mathbb {R} ^ {d + 1})} + \sum_ {i = 1} ^ {d} \left\| \frac {\partial \mu_ {i} (x , t)}{\partial x _ {i}} (p ^ {\star} (x, t) - \hat {p} (x, t)) \right\| _ {L ^ {2} (\mathbb {R} ^ {d + 1})}
$$
$$
\begin{array}{l} \leq \underbrace{\sup_{x,t\in(\mathbb{X},\mathcal{T})}\sup_{i}\mu_{i}(x,t)}_{\leq R_{\mu}}\sum_{i = 1}^{d}\left\| \frac{\partial p^{\star}(x,t)}{\partial x_{i}} -\frac{\partial\hat{p} (x,t)}{\partial x_{i}}\right\|_{L^{2}(\mathbb{R}^{d + 1})} \\ + d \underbrace {\sup  _ {x , t \in (\mathfrak {X} , \mathfrak {T})} \sup  _ {i} \frac {\partial \mu_ {i} (x , t)}{\partial x _ {i}}} _ {\leq R _ {\mu_ {p}}} \| p ^ {\star} (x, t) - \hat {p} (x, t) \| _ {L ^ {2} (\mathbb {R} ^ {d + 1})} \\ \end{array}
$$
$$
\leq R _ {\mu} \sum_ {i = 1} ^ {d} \left\| \frac {\partial p ^ {\star} (x , t)}{\partial x _ {i}} - \frac {\partial \hat {p} (x , t)}{\partial x _ {i}} \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} + d R _ {\mu_ {p}} \| p ^ {\star} (x, t) - \hat {p} (x, t) \| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \tag {31}
$$
$$
\begin{array}{l} \left\| \sum_ {i = 1} ^ {d} \frac {\partial}{\partial x _ {i}} \left(\mu_ {i} (x, t) p ^ {\star} (x, t) - \mu_ {i} (x, t) \hat {p} (x, t)\right) \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \\ \leq R _ {\mu} \sum_ {i = 1} ^ {d} \left\| \frac {\partial p ^ {\star} (x , t)}{\partial x _ {i}} - \frac {\partial \hat {p} (x , t)}{\partial x _ {i}} \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} + d R _ {\mu_ {p}} \| p ^ {\star} (x, t) - \hat {p} (x, t) \| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \\ \end{array}
$$
$$
\| p ^ {\star} (x, t) - \hat {p} (x, t) \| _ {L ^ {1} (\mathbb {R} ^ {d + 1})} \leq (2 v) ^ {\beta} (1 + \| g \| _ {L ^ {1} (\mathbb {R} ^ {d + 1})}) \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} \| f _ {i} \| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})}.
$$
And previously, we showed in lemma 2
$$
\left\| \frac {\partial p ^ {\star} (x , t)}{\partial x _ {k}} - \frac {\partial \hat {p} (x , t)}{\partial x _ {k}} \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \leq 8 \pi (2 v) ^ {(\beta - 1)} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} \left(\mathbb {R} ^ {d + 1}\right)} \| f _ {i} \| _ {L ^ {\infty} \left(\mathbb {R} ^ {d + 1}\right)} \tag {32}
$$
$$
+ 2 (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} \left\| \frac {\partial f _ {i}}{\partial x _ {k}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})} \| g \| _ {L _ {1} (\mathbb {R} ^ {d + 1})} \tag {33}
$$
If we choose,
$$
v = \min \left(\left(\frac {\varepsilon}{2 ^ {\beta - 1} 1 6 \pi C _ {1}}\right) ^ {\frac {1}{\beta - 1}}, \left(\frac {\varepsilon}{2 ^ {\beta + 1} C _ {2}}\right) ^ {\frac {1}{\beta}}, \left(\frac {\varepsilon}{2 ^ {\beta + 1} C _ {3}}\right) ^ {\frac {1}{\beta}}\right)
$$
where  $C_1 = \sum_{i=1}^{q} \|f_i\|_{W_2^\beta(\mathbb{R}^{d+1})} \|f_i\|_{L^\infty(\mathbb{R}^{d+1})},$ $C_2 =$
$$
\sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} \left\| \frac {\partial f _ {i}}{\partial t} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})} \| g \| _ {L _ {1} (\mathbb {R} ^ {d + 1})} \quad \text {a n d} \quad C _ {3} \quad = \quad (1 \quad +
$$
$\| g\|_{L^1 (\mathbb{R}^{d + 1})}\big)\sum_{i = 1}^q\| f_i\|_{W_2^\beta (\mathbb{R}^{d + 1})}\| f_i\|_{L_2(\mathbb{R}^{d + 1})}$  , then, we have
$$
\left\| \sum_ {i = 1} ^ {d} \frac {\partial}{\partial x _ {i}} \left(\mu_ {i} (x, t) p ^ {\star} (x, t) - \mu_ {i} (x, t) \hat {p} (x, t)\right) \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \leq d \left(R _ {\mu} + R _ {\mu_ {p}}\right) \varepsilon .
$$
We showed in lemma 3 that
$$
\begin{array}{l} \left\| \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p ^ {\star} (x, t) - \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} \hat {p} (x, t) \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \\ \leq \frac {1 6 \pi (2 v) ^ {(\beta - 1)}}{\left(1 + v ^ {2}\right) ^ {\frac {\beta - 1}{2}}} \left(1 + \| g \| _ {L ^ {1} (\mathbb {R} ^ {d + 1})}\right) \sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} \left\| \frac {\partial f _ {m}}{\partial x _ {j}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})} \\ + \frac {3 2 \pi^ {2} (2 v) ^ {(\beta - 2)}}{(1 + v ^ {2}) ^ {\frac {\beta - 2}{2}}} \sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} \| f _ {m} \| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})} + 2 (2 v) ^ {\beta} \sum_ {m = 1} ^ {q} \| f _ {m} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} \left\| \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})} \| g \| _ {L ^ {1} (\mathbb {R} ^ {d + 1})}. \\ \end{array}
$$
where  $C_1 = (1 + \| g\|_{L^1 (\mathbb{R}^{d + 1})})\sum_{m = 1}^q\| f\|_{W_2^\beta (\mathbb{R}^{d + 1}})\left\| \frac{\partial f_m}{\partial x_j}\right\|_{L^\infty (\mathbb{R}^{d + 1})},$ $C_2 =$
$$
\sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} \| f _ {m} \| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})} \mathrm {a n d} C _ {3} = \sum_ {m = 1} ^ {q} \| f _ {m} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} \left\| \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d + 1})} \| g \| _ {L ^ {1} (\mathbb {R} ^ {d + 1})},
$$
$$
\left\| \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p ^ {\star} (x, t) - \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} \hat {p} (x, t) \right\| _ {L ^ {2} (\mathbb {R} ^ {d + 1})} \leq \varepsilon .
$$
$$
\left\| \frac {\partial \hat {p} (x , t)}{\partial t} + \sum_ {i = 1} ^ {d} \frac {\partial}{\partial x _ {i}} (\mu_ {i} (x, t) \hat {p} (x, t)) - \sum_ {i = 1} ^ {d} \sum_ {j = 1} ^ {d} \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} (D _ {i j} \hat {p} (x, t)) \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d + 1}\right)} \leq \varepsilon (1 + d (R _ {\mu} + R _ {\mu_ {p}}) + d ^ {2} R _ {d}).
$$
We can bound the trace of the matrix as follows. We have,
From [RC21, Equation D.21], we have
$$
T r (M _ {\varepsilon}) = \sum_ {i = 1} ^ {q} \| f _ {i, v} \| _ {\mathcal {H}} = c _ {\eta} 2 ^ {2 \beta} (1 + (\frac {v}{3}) ^ {2 \beta} e ^ {\frac {8 9}{\eta_ {0} v ^ {2}}}) \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d + 1})} ^ {2}
$$
Since, we choose,
$$
v = \left(\frac {\varepsilon}{9 6 \pi^ {2} 2 ^ {\beta - 2} C}\right) ^ {\frac {1}{\beta - 2}},
$$