Groupoid Characterization of Partial Algebras on 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.