In this article, we follow the arguments in a paper of Y-T. Siu to study the effective termination of Kohn's algorithm for special domains in $\mathbb{ C}^{ 3}$. We make explicit the effective constants and generic conditions that appear there, and we obtain an explicit expression for the regularity of the Dolbeault laplacian for the $\overline{ \partial}$-Neumann problem. Specifically, on a local peudoconvex domain of the special shape \[ \Omega:= \bigg\{ (z_{ 1},z_{ 2},z_{ 3})\in\mathbb{ C}^{ 3}:\ 2\text{ Re}\ z_{ 3}+ \sum_{ i=1}^{ N}|F_{ i}(z_{ 1},z_{ 2})|^{ 2}<0 \bigg\} \] with holomorphic function germs $F_{ 1},\dots,F_{ N}\in\mathcal{ O}_{ \mathbb{ C}^{ 2},0}$ of finite intersection multiplicity \[ s:=\dim_{ \mathbb{ C}}\ \mathcal{ O}_{ \mathbb{ C}^{ 2},0} \big/ \langle F_{ 1},\dots, F_{ N} \rangle < \infty, \] we show that an $\varepsilon$-subelliptic regularity for $(0,1)$-forms holds whenever, just in terms of $s$, \[ \varepsilon \geqslant \frac{ 1}{ 2^{ (4s^{ 2}-1)s+3} s^{ 2}(4s^{ 2}-1)^{ 4} \binom{ 8s+1}{ 8s-1}}. \]
Categories: math.CV