I would be interested to see how the formula looks for a more complicated equation. For example, Eq 3.12 or 3.31 in the link https://arxiv.org/pdf/2203.09346.pdf
$
cal(C)(M) = cal(E)_T^t (theta, cal(S)_t)^2 + C_t M_t^(-2/d)
+ cal(E)_T^"PDE" (theta_t, cal(S)_"int")^2 + C_"PDE" M_"int"^(-2/(d+1)) \
+ C_1T^(1/2) [cal(E)_T^"div" (theta, cal(S)_"int") + C_"div" M_"int"^(-1/(d+1))
+ (1+nu)(cal(E)_T^s (theta, cal(S)_s) + C_s M_s^(-1/d))]
$
Just to now push it a little. I wonder if I can make the `1/(d+1)` in the superscript be not a real fraction but inline fraction.
\begin{align}
\begin{split}
\mathcal{C}(M) = &\: \Et^t(\theta,\S_t)^2
+ C_t M_t^{-\frac{2}{d}}
+ \Et^\pde(\theta,\S_\inte)^2 +C_\pde
M_\mathrm{int}^{-\frac{2}{d+1}}\\&+ C_1{T}^{\frac{1}{2}}
\bigg[\Et^\divv(\theta,\S_\inte) +C_\divv
M_\mathrm{int}^{-\frac{1}{d+1}}+(1+\nu)
big( \Et^s(\theta,\S_s) + C_s M_s^{-\frac{1}{d}}\big)\bigg],\\
\end{split}
\end{align}
\begin{align}
\begin{split}
\norm{(u_0)_j-\hu_j(t=0)}_{L^2(\mathbb{T}^d)} &\leq
\norm{u_j-\hu_j}_{L^2(\partial \Omega)} \\&\leq
\sqrt{\frac{2\max\left\{2h_\Omega,d+1\right\}}
{\rho_\Omega}}\norm{u_j-\hu_j}_{H^1(\Omega)}\\
&\leq \sqrt{\frac{2\max\left\{2h_\Omega,d+1\right\}}
{\rho_\Omega}}C_{1,k,d+1, u_1} \lambda_{1}(N)N^{-k+1},
\end{split}
\end{align}
$
norm((u_0)_j-hat(u)_j (t=0))_(L^2 TT^d) &<= norm(u_j - hat(u)_j)_(L^2 delta Omega) \
&<= sqrt(2 max {2h_Omega,d+1} / rho_Omega) norm(u_j - hat(u)_j)_(H^1 (Omega)) \
&<= sqrt(2 max {2h_Omega,d+1} / rho_Omega) C_(1,k,d+1,u_1) lambda_1 (N) N^(-k+1)
$