$$\int_{\voldomaxis} f\, \rho \, \dd\rho \dd z \dd \theta~.$$ This leads naturally to the use of weighted Hilbert spaces, which are defined as follows: $$\begin{alignedat}{2}
&\Lsp_\rho^2(\srfdomflat) &&= \parG{u: \int_\srfdomflat
    u^2 \,\rho\,\dd\rho \dd z < \infty}~, \\ 
&\Lsp_\rho^2(\srfdomflat;\, \R^3) &&= \parG{\uv:
    \int_\srfdomflat\norm{\uv}_e^2\, \rho\,\dd\rho \dd z <
    \infty}~, 
\end{alignedat}$$ where $\norm{\uv}_e = \sqrt{\uv \cdot \uv}$ is the Euclidean norm and $\uv$ is a vectorial function defined on $\srfdomflat$ which takes values in $\R^3$. The associated norms are the standard ones induced by the inner products $$\idot{u}{v}_{\Lsp_\rho^2(\srfdomflat)} = \int_\srfdomflat uv
\,\rho\,\dd\rho \dd z~, \quad
\idot{\uv}{\vv}_{\Lsp_\rho^2(\srfdomflat;\, \R^3)} =
\int_\srfdomflat \uv \cdot \vv \,\rho\,\dd\rho \dd z~,$$ and will be often simply denoted using the subscript $\rho$, *i.e.* $\norm{u}_\rho = \sqrt{\idot{u}{u}_\rho}$. We also introduce the following spaces: $$\label{eq:Zm_notation}
\begin{alignedat}{3}
&\CySpT^{m, \, 0} && = \Hsp_\rho(\gradm;\,\srfdomflat)&& =
\parG{u \in \Lsp_\rho^2(\srfdomflat): \gradm u \in
    \Lsp_\rho^2(\srfdomflat; \,\R^3)},\\ 
&\CySpT^{m, \, 1} && = \Hsp_\rho(\curlm;\,\srfdomflat) &&=
\parG{\uv \in \Lsp_\rho^2(\srfdomflat; \,\R^3): \curlm \uv \in
    \Lsp_\rho^2(\srfdomflat; \,\R^3)},\\ 
&\CySpT^{m, \, 2} && = \Hsp_\rho(\divergm;\,\srfdomflat)&& =
\parG{\uv \in \Lsp_\rho^2(\srfdomflat; \,\R^3): \divergm \uv \in
    \Lsp_\rho^2(S)}~,\\ 
&\CySpT^{m, \, 3} && = \Lsp_\rho^2(\srfdomflat)~,&&
\end{alignedat}$$ where the differential operators have to be interpreted in a weak sense and the corresponding norms are $$\begin{aligned}
{2}
&\norm{u}_{\Hsp_\rho(\gradm;\,\srfdomflat)}^2 &&= \norm{u}_\rho^2 +
\norm{\gradm 