Ramification filtrations of certain abelian Lie extensions of local fields

Let G⊂xFq〚x〛G⊂xFq〚x〛 (q is a power of the prime p) be a subset of formal power series over a finite field such that it forms a compact abelian p  -adic Lie group of dimension d≥1d≥1. We establish a necessary and sufficient condition for the APF extension of local field corresponding to (Fq⸨x⸩,G)(Fq⸨x⸩,G) under the field of norms functor to be an extension of p-adic fields. We then apply this result to study invertible power series over a ring of p-adic integers which commute with a fixed noninvertible power series under composition.