On the closability of classical Dirichlet forms in the plane. (English. Russian original) Zbl 1085.46023

Dokl. Math. 64, No. 2, 197-200 (2001); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 380, No. 3, 315-318 (2001).
The author exhibits a measure \(\mu\) on the plane such that the Dirichlet form \(E(f,g)=\int(\nabla f,\nabla g)\,d\mu\) is closable, whereas the form \(E_x(f,g)=\int \partial_xf\partial_xg\,d\mu\) is not. This gives a positive answer to a question of S. Albeverio and M. R√∂ckner [J. Funct. Anal. 88, No. 2, 395–436 (1990; Zbl 0737.46036)]. The measure \(\mu\) is restriction of Lebesgue measure to an open subset of the unit square and the construction is based on a Cantor set of positive measure.


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
31C25 Dirichlet forms
46N20 Applications of functional analysis to differential and integral equations
47A07 Forms (bilinear, sesquilinear, multilinear)
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)


Zbl 0737.46036