The authors give conditions for the existence of partitions of unity on certain infinite dimensional manifolds. Separable Banach spaces and many Banach manifolds have partitions of unity consisting of functions that are locally Lipschitz (and hence are weakly differentiable in a certain sense), but the authors seek conditions giving global integrability of the derivatives.
Let \(E\) be a topological space, \(W\) a linear space of real valued functions on \(E\). If \(\{U_{\alpha}\}\) is an open covering of \(E, \{\Psi_i\}\) is a partition of unity of class \(W\) subordinated to \(\{U_{\alpha}\}\) if (i) \(\Psi_i \in W\) for each \(i\). (ii) There exists a locally finite open covering \(\{V_i\}\) subordinated to the \(\{U_{\alpha}\}\) such that supp \(\Psi_i := \overline{\{ \Psi_i \neq 0 \}}\) is contained in \(\{V_i \}\) for each \(i\). (iii) \(\Psi_i(x) \geq 0\), and \(\sum_i \Psi_i(x) =1 \) for each \(x \in E\).
They start with two results that are essentially known. They prove the existence of partitions of unity for \(E\) an abstract Wiener space with \(W\) the continuous functions in \(D^{\infty}\), the Malliavin test function space. They prove the existence of partitions of unity for a regular Dirichlet form \((\mathcal E, \text{D}(\mathcal E))\) on \(L^2(E,m)\) (\(E\) is a Hausdorff topological space and \(m\) is a \(\sigma\)-finite Borel measure, \((\mathcal E, \text{D}(\mathcal E))\) is called a Dirichlet space) with \(W\) the space of continuous functions which are measurable versions of elements in \({\text{D}}(\mathcal E)\); they say the result is known but they are unable to find a suitable reference.
They then consider the case of more general Dirichlet forms on Polish state spaces. If \((\mathcal E, {\text{D}}(\mathcal E))\) is quasi-regular, but not regular, there may be no non-vanishing continuous function in the domain \({\text{D}}(\mathcal E)\). They give sufficient conditions which guarantee the existence of sufficiently many continuous functions of class \(W\) (where \(W\) is defined as for regular Dirichlet forms). Their construction gives partitions of unity on Dirichlet spaces on pinned and free loop spaces, on Dirichlet spaces of gradient type on Banach spaces, and on Dirichlet spaces on spaces of measures.