domain $\mathcal{D}^w_{\gamma}( \Delta_{\nu,\mu})$ have the same eigenvalues $(\lambda_n)_{n \in \mathbb{N}}$. Further, if $f$ is an eigenfunction of $ \Delta_{\nu,\mu} $, then $f \circ \check{F}^{-1}_{\mu}$ is an eigenfunction of $ \Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda} $, and if $f$ is an eigenfunction of $\Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda}$, then $f \circ F_{\mu}$ is an eigenfunction of $ \Delta_{\nu,\mu}$. In particular, if $(f_n)_{n \in \mathbb{N}}$ denotes the orthonormal basis consisting of eigenfunctions of $ \Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda}$ then $(f_n \circ F_{\mu})_{n \in \mathbb{N}}$ forms an orthonormal basis consisting of $ \Delta_{\nu,\mu}$-eigenfunctions.

can be seen as a generalisation of [@KSW16] where the $\Delta_{\mu,\mu}$-Laplacian has been considered.

\[Reso\] Assume that ${\operatorname{supp}}(\nu) \subseteq {\operatorname{supp}}(\mu)$ and $\gamma \in[0,{\pi/2}]^{2}$. For $\lambda>0$, letting $R_{\nu , \mu}^{\lambda}$ denote the resolvent operator of $ \Delta_{\nu, \mu}$ with domain ${ \mathcal{D}^*_{\gamma}(\Delta_{\nu,\mu})}$, for all $f \in \S^*$ $$\begin{aligned}
R_{\nu,\mu}^{\lambda}(  f ) \circ \check{F}_{\mu}^{-1}=R_{\nu \circ F_{\mu}^{-1},\Lambda}^{\lambda}(f \circ \check{F}_{\mu}^{-1}).\end{aligned}$$ In particular, we have $$\begin{aligned}
\big \Vert R_{\nu \circ F_{\mu}^{-1},\Lambda}^{\lambda} \big\Vert_{L^2(\nu \circ F_{\mu}^{-1} ) }=\big \Vert R_{\nu,\mu}^{\lambda} \big\Vert_{L^2 (\nu) },\end{aligned}$$ and $$\begin{aligned}
\big \Vert R^{\lambda}_{\nu \circ F_{\mu}^{-1},\Lambda} \big\Vert_{C_{\nu \circ F_{\mu}^{-1},\Lambda}}
=
\big\Vert R^{\lambda}_{\nu,\mu}\big\Vert_{C_{\nu,\mu}}\end{aligned}$$

Note that the resolvent $R^{\lambda}_{\nu,\mu}$ is well defined for all $\lambda>0$ (see [@MR2017701]) and for $f \in \S^*$ we have 