Fonctions d'une variable $p$-adique et représentations de ${\mathrm{G}\mathrm{L}}_2({\mathbf{Q}}_p)$

We extend the dictionary between Fontaine rings and $p$-adic functionnal analysis, and we give a refinement of the $p$-adic local Langlands correspondence for principal series representations of ${\mathrm{G}\mathrm{L}}_2({\mathbf{Q}}_p)$.