Explicit constants for elliptic a priori estimates

Question

Let $V$, $W$ be vector bundles over a compact Riemannian manifold $M$ and let $F$ be a smooth elliptic operator of order $k$ from $V$ to $W$. "Standard elliptic theory" then gives us the following two estimates:

1. For any $l \geq 0$ and $\alpha \in (0,1)$ there exists a constant $C>0$ such that $$\|v\|_{C^{k+l,\alpha}} \leq C \left( \|F(v)\|_{C^{l,\alpha}}+\|v\|_{C^0} \right)$$ for all $v \in C^{k+l,\alpha}(V)$.

2. For any $l \geq 0$ and $\alpha \in (0,1)$ there exists a constant $D>0$ such that $$\|v\|_{C^{k+l,\alpha}} \leq D \|F(v)\|_{C^{l,\alpha}}$$ for all $v \in C^{k+l,\alpha}(V)$ and $v \perp \text{Ker}F$.

Similar estimates exist for the $L^p$-theory, i.e. take the $L^p_{k}$-norm instead of the $C^{k,\alpha}$-norm for $p \in (1,\infty)$. I am interested in both estimates.

Question: Are there situations in which $C$ or $D$ has been computed explicitly?

I have a personal interest in the Laplace operator acting on $1$-forms.

I know there is some work for operators acting on functions on domains in $\mathbb{R}^n$, namely the references listed in the introduction of Kouta Sekine, Kazuaki Tanaka, Shin'ichi Oishi: Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems. But that's not for compact manifolds, and only for trivial vector bundles.

Answer 1

The closest I found:

In Michael Plum: Explicit $H_2$ Estimates and Pointwise Bounds for Solutions of Second-Order Elliptic Boundary Value Problems the following explicit estimate is given (equation 3): $$||u||_{L^\infty,\Omega} \leq K ||Lu||_{L^2,\Omega}+K_B ||Bu||_{L^\infty,\partial \Omega}.$$ Here, $L$ denotes an order $2$ differential operator like the Laplacian, $\Omega$ denotes a domain in $\mathbb{R}^2$ or $\mathbb{R}^3$ and $\partial \Omega$ its boundary, and $B$ is the boundary operator of Dirichlet, Neumann, or mixed type.

Edit: Other examples are Theorem 1.1 in Batu Güneysu, Stefano Pigola: Quantitative $C^1$-estimates on manifolds, and Theorem 2.1 in Batu Güneysu, Stefano Pigola: Nonlinear Calderón–Zygmund inequalities for maps.