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

Abstract

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