Kato space for Dirichlet forms (Q1300175)

The authors consider a wide class of regular Dirichlet forms of diffusion type incuding elliptic and subelliptic operators. For these Dirichlet forms the notion of Kato space is introduced and it is proved that the Kato space becomes a Banach space when suitably normed. The main theorem states that a positive Radon measure \(\mu\) belongs to the dual of the form domain if and only if \[ \int G(x,y) d\mu(x) d\mu(y)< \infty, \] where \(G\) is the Green function. It is then proved that a weak solution of the equation \(Lu=f\), \(L\) being the generator of the form, is continuous if \(\mu\) belongs to the Kato space.