\wedge)$ is acyclic. Therefore the second spectral sequence associated to the above double complex degenerates at the $E_2$-term and the isomorphism $$\Omega^{p-1}_{V/T}(*D) \stackrel{dh\wedge}{\longrightarrow} {{\mathcal{K}}\!}er\left(\Omega^p_V(*D) \stackrel{dh\wedge}{\longrightarrow} \Omega^{p+1}_V(*D)\right)$$ yields the above quasi-isomorphism.

    In order to prove the formula for the action of $\partial_t$, notice that given a class $[\omega]$ defined by a relative $n$-form $\omega\in\Omega^{n-1}_{V/T}(*D)$, the class corresponding to it in ${\mathcal{H}}^0(\Phi_+{\mathcal{O}}_V(*D))$ is $[dh\wedge\omega]$. By definition, we have $\partial_t([dh\wedge\omega])=[dh\wedge\omega\otimes\partial_t]$. This class is equal in ${\mathcal{H}}^0(\Phi_+{\mathcal{O}}_V(*D))$ to $[d\omega-df\wedge\omega\otimes\partial_r]$. It follows that under the isomorphism , this equals $[\frac{d\omega}{dh}-\frac{df\wedge\omega}{dh}\otimes\partial_r]\in
    \Phi_*\Omega^{n-1}_{V/T}(*D)[\partial_r]/(d-df\wedge-\otimes\partial_r)\Phi_*\Omega^{n-2}_{V/T}(*D)$. Now notice that as $h$ is smooth outside $D$, there is a vector field $X\in\Theta_V(*D)$ which lifts $\partial_t\in\Theta_T$. Then we have that $d\omega/dh=\iota_Xd\omega$ and $(df\wedge\omega)/dh=\iota_X(df)\wedge\omega-df\wedge\iota_X\omega$. Putting this together and using once again the relation in the quotient $\Phi_*\Omega^{n-1}_{V/T}(*D)[\partial_r]/(d-df\wedge-\otimes\partial_r)\Phi_*\Omega^{n-2}_{V/T}(*D)$ one arrives at the formula $$\partial_t[\omega]=[\operatorname{\textup{Lie}}_X\omega]-[\operatorname{\textup{Lie}}_X(f)\omega]\otimes\partial_r.$$ Now the result follows as the meromorphic vector field $X$ can be taken to be $E/(n\cdot h)$, due to the homogeneity of $h$.

3.  This is obvious from the last point: Fourier-Laplace transformation and localization along $\tau=0$ transforms formula into the defining equation of $G(*D)$ and formula obviously corresponds to the second part of formula . The statements about regularity follows form the general considerations in [@DS theorem 1.11].

4.  