Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $V$ be a p-adic representation of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/K)$. Write $K_\infty=K(\mu_{p^{\infty}})$ for the cyclotomic extension of $K$. A construction of Sen attaches to $V$ a $K_\infty$-vector space $\mathrm{D}_\mathrm{Sen}(V)$ of dimension $\mathrm{dim}_{\mathbb{Q}_p}(V)$; it is defined as the union of all finite dimensional $\mathrm{Gal}(K_\infty/K)$-stable subspaces of $(\mathbb{C}_p\otimes V)^{\mathrm{Gal}(\overline{K}/K_\infty)}$.
Later, Fontaine similarly defined a $K_{\infty}((t))$-vector space $\mathrm{D}_\mathrm{dif}(V)$, by letting $\mathrm{D}^+_\mathrm{dif}(V)$ be the union of all finite rank $\mathrm{Gal}(K_\infty/K)$-stable $K_\infty[[t]]$-submodules of $(\mathrm{B}_{\mathrm{dR}}^+\otimes V)^{\mathrm{Gal}(\overline{K}/K_\infty)}$ and $\mathrm{D}^+_\mathrm{dif}(V)=\mathrm{D}^+_\mathrm{dif}(V)\otimes_{K_\infty[[t]]}K_\infty((t))$.
In his paper "Sur un resultat de S. Sen", Colmez constructs a period ring $\mathrm{B}_{\mathrm{Sen}}$ such that $(\mathrm{B}_{\mathrm{Sen}}\otimes_{\mathbb{Q}_p} V)^{\mathrm{Gal}(\overline{K}/K_\infty)}$ is isomorphic to $\mathrm{D}_\mathrm{Sen}(V)$. In fact $\mathrm{B}_{\mathrm{Sen}}$ is the ring $\mathbb{C}_p\{\{{\log{t}\}\}}$ of power series with nonzero radius of convergence in the variable $\log{t}$, where $t$ is Fontaine's period.
Does there exist a similar ring $\mathrm{B}_{\mathrm{dif}}$ which would be a period ring for $\mathrm{D}_\mathrm{dif}$? If so, what is it?
Thanks!
