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We extend the dictionary between Fontaine rings and $p$-adic functionnalanalysis, and we give a refinement of the $p$-adic local Langlandscorrespondence for principal series representations of ${\rmGL}_2(\mathbf{Q}_p)$.

Fonctions d'une variable $p$-adique et repr\'esentations de ${\rm GL}_2(\mathbf{Q}_p)$