\in \mathcal{P}({\mathcal{M}}),$$ where in the above and in the remainder $dx$ denotes integration with respect to ${\mathcal{M}}$’s volume form and $\mathcal{P}({\mathcal{M}})$ denotes the space of complete Borel probability measures on ${\mathcal{M}}$.

Associated to each of the densities $\rho_k$ we introduce a differential operator $\Delta_{\rho_k}$, which for smooth functions $u$ is defined by $$\Delta_{\rho_k} (u) := -\frac{1}{\rho_k}\operatorname{div}(\rho_k \nabla u) = -\Delta u -  \frac{1}{\rho_k} \nabla  \rho_k \cdot \nabla u.
\label{ContLaplacians}$$ In the above, $\nabla $ stands for the gradient operator on ${\mathcal{M}}$, $\operatorname{div}$ is the divergence on ${\mathcal{M}}$, and $\Delta$ is the Laplace-Beltrami operator on ${\mathcal{M}}$. The term $\nabla  \rho_k \cdot \nabla u $ evaluated at a point $x \in {\mathcal{M}}$, is the inner product of the vectors $\nabla  \rho_k(x), \nabla u (x) $ in the tangent space $\mathcal{T}_x{\mathcal{M}}$; seen as vectors in ${\mathbb{R}}^d$, $\nabla  \rho_k(x)\cdot \nabla u (x)$ is just their inner product in ${\mathbb{R}}^d$. Likewise, we consider the operator, $$\Delta_\rho (u) := -\frac{1}{\rho}\operatorname{div}(\rho \nabla u) = -\Delta u - \frac{1}{\rho}\nabla \rho \cdot \nabla u.
\label{ContLaplaciansRho}$$ Given a density $\varrho$ on ${\mathcal{M}}$ we define the weighted function spaces $$L^2({\mathcal{M}}, \varrho):= \left\{ u: {\mathcal{M}}\mapsto \mathbb R \Big| \langle u, u \rangle_{\varrho} < + \infty  \right\},$$ equipped with the weighted inner product 