Reflected Dirichlet Forms and the Uniqueness of Silverstein's Extension

Reflected Dirichlet space for quasi-regular Dirichlet forms is presented in this paper. We show the closedness of the (active) reflected Dirichlet forms without using the first definition of reflected Dirichlet space by Silverstein and the characterization by Chen. As an application of the closedness, the closability of distorted forms are discussed. We also show the maximality of (active) reflected Dirichlet space in the class of Silverstein's extensions and consider the uniqueness problem. Only the techniques of the transfer method and the change of underlying measures are used.     

Carleson measures, multipliers and integration operators for spaces of Dirichlet type

For 0<p<∞ and α>−1, we let denote the space of those functions f which are analytic in the unit disc and satisfy . In this paper we characterize the positive Borel measures μ in D such that , 0<p<q<∞. We also characterize the pointwise multipliers from to (0<p<q<∞) if p−2<α<p. In particular, we prove that if the only pointwise multiplier from to (0<p<q<∞) is the trivial one. This is not longer true for and we give a number of explicit examples of functions which are multipliers from to for this range of values.