Partial Dynamical Systems of Lp-Spaces and their Stability Spaces

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