Analysis Based on the Dirichlet Space Theory on Some Extensions of $\mathbb Q_p$

The space $\mathcal F_{r,p}$, which was designed so as to play a role similar to the ordinary Sobolev space $W_{r,p}$, is introduced as a cornerstone for analyzing nonlinear potential theoretic features of the state space with a measure-symmetric semigroup. The aim of this article is to reveal a sufficient condition for the coincidence of the counterparts of the Sobolev space and to derive the equivalence of the norms associated with those counterparts.