Sobolev and Hölder estimates for the $\bar{\mathrm{\partial }}$ equation on pseudoconvex domains of finite type in ${\mathbb{C}}^2$

We prove a homotopy formula which yields almost sharp estimates in all (positive-indexed) Sobolev and Hölder-Zygmund spaces for the $\bar{\mathrm{\partial }}$ equation on pseudoconvex domains of finite type in ${\mathbb{C}}^2$, extending the earlier results of Fefferman-Kohn (1988), Range (1990), and Chang-Nagel-Stein (1992). The main novelty of our proof is the construction of holomorphic support functions that admit precise estimates when the parameter variable lies in a thin shell outside the domain.