Kato Class Of Smooth Measures And Its Properties

Let (ε, D(ε)) be a quasi-regular semi-Dirichlet form on L2(E;m) which is associated with a m-tight special standard process M=(Xt, Px).A positive measure μ on (E) is said to be smooth (μ∈S in notation)if Cap(N)=0 implies μ(N)=0 for all N∈B(E) and there exists an ε-net of compact sets {Fk}k=1∞ such that μ(Fk)<∞ for all k∈N.In the framework of semi-Dirichlet form,we define Kato class of smooth measure with respect to M,give the equivalent class of Kato class of smooth measure under some conditions. We mainly get some properties of Kato class of smooth measure.In Chapter 1, first we describe the basic definitions and the notations associated with Dirichlet form and processes,then we will give some results of Kato class measures that have be talked about.In Chapter l,in the framework of semi-Dirichlet form,in section 2,we define Kato class of smooth measure with respect to M, give the equivalent class of Kato class of smooth measure under heat kernel estimates according to the detailed proof;in section 3,we get some properties of Kato class of smooth measure,for example,theorem 2.3.1 which is if μ∈S,then there exists an ε- net of compact sets.Fn,then for (?)n IFn∈Sk tells us the relationship between smooth measure and Kato class of smooth measure.It is a generalization of [3]theorem 2.4 in the framework of symmetric Dirichlet form.These properties talked about in section 3 are very important in research of perturbation of semi-Dirichlet form, strong continuity of generalized Feynman-Kac semigroups and large deviations.InIn Chapter 2,excepting considering the properties of Kato class of smooth measure in the framework of semi-Dirichlet form.In section 4,we also talk about the problem about the bounded linear mappings on Dirichlet space.