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 $L^p(\Omega )$ spaces defined by the action of the smooth algebra $\mathcal{K}(\Omega )$ 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.