On the closability of some positive definite symmetric differential forms on \(C_ 0^{\infty}(\Omega)\) (Q1062327)

Let D be an open subset of \({\mathbb{R}}^ n\), \(n\geq 2\) and \({\mathcal E}^ a \)Dirichlet form in \(L^ 2(D,m)\), where m is the Radon measure on D. For each integer k with \(1\leq k<n\), let \({\mathcal E}_ k\) be a Dirichlet form on some k-dimensional submanifold \(D_ k\) of D. The paper under review is devoted to the study of the closability problem for the forms \({\mathcal E}_{\alpha}\) and their sums in the real Hilbert space \(L^ 2(D;dx)\), where dx is the Lebesgue measure on D. The main results are proved using a simplified version of the imbedding theorem for the weighted Sobolev spaces \(W^ m_ p(D;(\eta (X))^{\alpha})\).