Let $\Omega$ be a an open bounded domain in $\mathbb{R}^N$. When studying elliptic boundary problems involving the Laplacian (e.g. $-\Delta u =\lambda u, \lambda\in\mathbb{R}$ with Dirichlet boundary condition) often the right space in which place the problem is $H^1_0(\Omega)$ and usually one refers to direct sum decomposition $$H^1_0(\Omega)= V\oplus W$$ where $dim W<+\infty$, $W=\oplus_{\lambda_n\le \lambda} M_{\lambda_n}$ and $V=\overline{\oplus_{\lambda_n>\lambda}M_{\lambda_n}}$, where $0<\lambda_1<\dots<\lambda_n<\dots$ denotes the sequences of the eigenvalues of the Laplacian and $M_{\lambda_n}$ denotes the eigenspace relative to the eigenfunction $\lambda_n$.
My question is: if $1<p<+\infty$, it is possible to make a similar direct sum decomposition for the Banach space $W_0^{1, p}(\Omega)$? And what about the space $L^p(\Omega)$?
Could someone please help me and/or give some reference?
Thank you in advance!
${\bf EDIT:}$ In particular, I would be intersted in a decomposition of the type $$W_0^{1, p}(\Omega)= V\oplus W,$$ where $W= M_{\lambda_1}$ denotes the eigenspace referred to the first eigenvalue of the $p$-Laplacian $\lambda_1$, $dim(W)<+\infty$ and $V=\oplus_{\lambda_n>\lambda_1}M_{\lambda_n}$. Is that possible?