Locally analytic vectors and overconvergent $(\varphi, \tau)$-modules

Let $p$ be a prime, let $K$ be a complete discrete valuation field of characteristic $0$ with a perfect residue field of characteristic $p$, and let $G_K$ be the Galois group. Let $\pi$ be a fixed uniformizer of $K$, let $K_\infty$ be the extension by adjoining to $K$ a system of compatible $p^n$-th roots of $\pi$ for all $n$, and let $L$ be the Galois closure of $K_\infty$. Using these field extensions, Caruso constructs the $(\varphi, \tau)$-modules, which classify $p$-adic Galois representations of $G_K$. In this paper, we study locally analytic vectors in some period rings with respect to the $p$-adic Lie group $\mathrm{Gal}(L/K)$, in the spirit of the work by Berger and Colmez. Using these locally analytic vectors, and using the classical overconvergent $(\varphi, \Gamma)$-modules, we can establish the overconvergence property of the $(\varphi, \tau)$-modules.