On partial actions of Hopf-Ore extensions

In this work we study how to extend a partial action of a Hopf Algebra $A$ on an algebra $R$ to a partial action of a Hopf-Ore extension of $A$ on $R$. As consequence, we characterize all partial actions of rank one Hopf algebras (in particular, generalized Taft algebras and Radford algebras), under suitable conditions.