no. 4
Analytic potential theory over the p-adics
Annales de l'Institut Fourier, Tome 43 (1993) no. 4, pp. 905-944.

Sur un corps non-archimédien, la valeur absolue élevée à une puissance α>0 arbitraire est une fonction définie négative et engendre (l’analogue d’un) un processus stable symétrique. Pour α∈(0,1) ce processus est transitoire et nous développons sa théorie du potentiel purement analytiquement et de manière explicite, en insistant sur la particularité résultant de la situation non-archimédienne. Par exemple, l’inégalité de Harnack devient une égalité.

Over a non-archimedean local field the absolute value, raised to any positive power α>0, is a negative definite function and generates (the analogue of) the symmetric stable process. For α∈(0,1), this process is transient with potential operator given by M. Riesz’ kernel. We develop this potential theory purely analytically and in an explicit manner, obtaining special features afforded by the non-archimedean setting ; e.g. Harnack’s inequality becomes an equality.

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     title = {Analytic potential theory over the $p$-adics},
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Haran, Shai. Analytic potential theory over the $p$-adics. Annales de l'Institut Fourier, Tome 43 (1993) no. 4, pp. 905-944. doi : 10.5802/aif.1361. http://archive.numdam.org/articles/10.5802/aif.1361/

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