Partial Dynamical Systems of $L^p$-Spaces and their Stability Spaces

Using the convolution product and weak derivatives, we consider the partial dynamical systems of the locally convex $L^p(\Omega)$ spaces defined by the action of the smooth algebra $\mathscr{K}(\Omega)$ through its nets. Slice analysis is then employed to show that the Sobolev spaces $W^{k,p}(\Omega)$ are the stable states or space of these partial dynamical systems as limit spaces of the convolution actions of the smooth algebra $K(\Omega)$ on the Banach spaces $L^p(\Omega)$. Thus, the Sobolev spaces $W^{k,p}(\Omega)$ are closed subspaces of the $Lp(\Omega)$-spaces under convolution product and weak derivatives, with the weak derivative operators acting as equivariant maps of the slice spaces.