Geodesic
Analysis Based on the Dirichlet Space Theory on Some Extensions of~$\mathbb Q_p$
Informatics and Automation, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 114-124.

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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. 
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H. Kaneko. Analysis Based on the Dirichlet Space Theory on Some Extensions of~$\mathbb Q_p$. Informatics and Automation, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 114-124. http://geodesic.mathdoc.fr/item/TRSPY_2004_245_a12/

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