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.