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

TL;DR

The paper develops a framework in which a smooth algebra $\mathscr{K}(\Omega)$ acts on $L^p(\Omega)$ through convolution nets $\{\phi_\varepsilon\}$ to define a partial dynamical system on function spaces. It then shows that Sobolev spaces $W^{k,p}(\Omega)$ are invariant under this action and emerge as slices or quotients within $L^p(\Omega)$, with weak derivatives acting equivariantly as part of the dynamics. The key technical contributions include the commutativity $D_i(f_\varepsilon) = (D_i f)_\varepsilon$ ensuring $\mathscr{K}(\Omega)$-invariance, and the slice-theoretic interpretation $W^{k,p}(\Omega) = D^{-k}(\mathscr{N}(\Omega))$, providing a distributional viewpoint on regularity. The results yield a unified viewpoint linking partial dynamical systems, smoothing operators, and Sobolev regularity, with potential implications for PDE solvability and generalized flow analysis in $L^p$ 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.

Theorems & Definitions (31)

• Definition 1.1
• Remark 1.2
• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Proposition 2.6
• proof