C^k estimates for convex domains of finite type in ${ℂ}^n$ are known from Alexandre (C. R. Acad. Paris, Ser. I 335 (2002) 23–26). We now want to show the same result for annuli. Precisely, we show that for all convex domains D and D′ relatively compact of ${ℂ}^n$, of finite type m and m′ such that $\overline{D}\subset \mathrm{D}\text{'}$, for all q=1,…,n−2, there exists a linear operator ${\tilde{T}}_q^{*}$ from $C_{0,\mathrm{q}}\left(\overline{\mathrm{D}\text{'}⧹\mathrm{D}}\right)$ to $C_{0,\mathrm{q}-1}\left(\overline{\mathrm{D}\text{'}⧹\mathrm{D}}\right)$ such that for all $\mathrm{k}\in \mathbb{N}$ and all (0,q)-form f, $\overline{\partial }$-closed of regularity C^k up to the boundary, ${\tilde{T}}_q^{*}f$ is of regularity C^{k+1/max(m,m′)} up to the boundary and $\overline{\partial }{\tilde{T}}_q^{*}\mathrm{f}=\mathrm{f}$. We fit the method of Diederich, Fisher and Fornaess to the annuli by switching z and ζ. However, the integration kernel will not have the same behavior on the frontier as in the Diederich–Fischer–Fornaess case and we have to alter the Diederich–Fornaess support function which will not be holomorphic anymore. Also, we take care of the so generated residual term in the homotopy formula and show that it is extremely regular so that solve the $\overline{\partial }$ problem for it will not be difficult.