On closability of classical Dirichlet forms. (Q1427628)

Let \(H\) be a Hilbert space, \(D({\mathcal E})\) a linear subspace of \(H\) and \({\mathcal E}:D({\mathcal E})\times D({\mathcal E})\to \mathbb R\) a nonnegative symmetric bilinear form. The form is closable if \(u_n\in D({\mathcal E})\), with \(u_n\to0\) in \(H\) and \({\mathcal E}(u_n-u_m,u_n-u_m)\to0\) implies that \({\mathcal E}(u_n,u_n)\to0\). Consider the space \(L^2(\mu)\), where \(\mu\) is a probability measure on the Borel sets of \(\mathbb R^n\). It is known that if the partial Dirichlet forms \({\mathcal E}_i(f,g)=\int {\partial f\over \partial x_i} {\partial g\over \partial x_i}\,d\mu(x)\) are closable, then so is the classical gradient Dirichlet form \({\mathcal E}(f,g)=\int (\nabla f,\nabla g)\,d\mu(x)\). The converse was conjectured some years ago by M. Röckner to be, in general, false. The paper under review establishes this conjecture by constructing (via a Cantor set type procedure) a bounded open set \(F\) in \(\mathbb R^2\) such that the gradient Dirichlet form is closed on \(L^2(\lambda^2| _F)\) but the partial form \({\mathcal E}_1\) is not (here \(\lambda^2\) is Lebesgue measure in \(\mathbb R^2\)). The construction extends to a measure with support \(\mathbb R^2\).