$F_p$-espaces vectoriels de formes diff\'erentielles logarithmiques sur la droite projective

Let k be an algebraically closed field of characteristic p >0. Let $m \in \N$,(m,p)=1.Westudy$\fp$-vector spaces of logarithmic differential forms on the projective line such that each non zero form has a unique zero at $\mathrm{\infty }$ of given order m-1. We discuss the existence of such vectors spaces according to the value of m. We give applications to the lifting to characteristic 0 of $(\mathbb{Z}\mathrm{/}p\mathbb{Z}{)}^n$ actions as k-automorphisms of $k[[t]]$.