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