Sobolev and Hölder estimates for homotopy operators of the $\bar{\mathrm{\partial }}$-equation on convex domains of finite multitype

We construct homotopy formulas for the $\bar{\mathrm{\partial }}$-equation on convex domains of finite type that have optimal Sobolev and Hölder estimates. For a bounded smooth finite type convex domain $\mathrm{\Omega }\subset {\mathbb{C}}^n$ that has $q$-type $m_q$ for $1\le q\le n$, our $\bar{\mathrm{\partial }}$ solution operator ${\mathcal{H}}_q$ on $(0,q)$-forms has (fractional) Sobolev boundedness ${\mathcal{H}}_q:H^{s,p}\to H^{s+1\mathrm{/}{m}_q,p}$ and Hölder-Zygmund boundedness ${\mathcal{H}}_q:{\mathcal{C}}^s\to {\mathcal{C}}^{s+1\mathrm{/}{m}_q}$ for all $s\in \mathbb{R}$ and $1<p<\mathrm{\infty }$. We also show the $L^p$-boundedness ${\mathcal{H}}_q:H^{s,p}\to H^{s,p{r}_q\mathrm{/}(r_q-p)}$ for all $s\in \mathbb{R}$ and $1<p<r_q$, where $r_q:=(n-q+1)m_q+2q$.