Automorphisms of local fields of period $p$ and nilpotent class $<p$

Suppose $K$ is a finite field extension of $\mathbb{Q} _p$ containing a primitive $p$-th root of unity. Let $\Gamma _{<p}$ be the Galois group of a maximal $p$-extension of $K$ with the Galois group of period $p$ and nilpotent class $<p$. In the paper we describe the ramification filtration $\{\Gamma _{<p}^{(v)}\}_{v\geqslant 0}$ and relate it to an explicit form of the Demushkin relation for $\Gamma _{<p}$. The results are given in terms of Lie algebras attached to involved groups by the classical equivalence of the categories of $p$-groups and Lie algebras of nilpotent class $<p$.