Groupoid Characterization of Partial Algebras on Sobolev Spaces

TL;DR

This paper characterizes Sobolev spaces within a Lie groupoid framework, revealing their structure as partial algebras and dynamical systems, and demonstrates how this approach simplifies understanding their properties.

Contribution

It introduces a novel Lie groupoid characterization of Sobolev spaces and partial algebras, linking partial actions, differential operators, and groupoid structures.

Findings

Sobolev spaces are invariant under partial actions of smooth algebras.

Partial algebras define partial dynamical systems on $L^p$-spaces.

Unitary representations of Lie groupoids simplify the analysis of Sobolev spaces.

Abstract

The $L^p$-spaces, with $p \not = \infty$, form a partial algebra $(L^p(\Omega), \Gamma, \cdot)$ with pointwise multiplication of functions. The Sobolev spaces $W^{k,p}(\Omega)$, delineated by weak derivatives as subspaces of $L^p$-spaces is shown to contain the partial algebra $(L^p(\Omega), \Gamma, \cdot)$ generalized by the partial action of the smooth algebra $\mathscr{K}(\Omega)$ by convolution on the Banach spaces $L^p(\Omega)$. We characterised the Sobolev space $W^{k,p}(\Omega)$, invariant under $\mathscr{K}(\Omega)$ partial action, using Lie groupoid framework, and study the partial algebra as defining the partial dynamical systems on the $L^p$-space associated with the weak differential operators. The locally convex partial $^*$-algebra $(L^p(\Omega), \Gamma, \cdot,^*)$ defines the stable local flows coinciding with local bisections of the Lie groupoid. The unitary representation of resulting Lie groupoid $\mathscr{W} \rightrightarrows W^{k,p}(\Omega)$ on the associated Hilbert bundle demonstrates the simplification achieved by the characterisation.