Geodesic
Analysis Based on the Dirichlet Space Theory on Some Extensions of~$\mathbb Q_p$
Trudy Matematicheskogo Instituta imeni V.A. Steklova, 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. 
@article{TM_2004_245_a12,
     author = {H. Kaneko},
     title = {Analysis {Based} on the {Dirichlet} {Space} {Theory} on {Some} {Extensions} of~$\mathbb Q_p$},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {114--124},
     publisher = {mathdoc},
     volume = {245},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2004_245_a12/}
}
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H. Kaneko. Analysis Based on the Dirichlet Space Theory on Some Extensions of~$\mathbb Q_p$. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 114-124. http://geodesic.mathdoc.fr/item/TM_2004_245_a12/