On supnorm estimates for $\bar{\mathrm{\partial }}$ on infinite type convex domains in ${\mathbb{C}}^2$

In this paper, we study the $\bar{\mathrm{\partial }}$-equation on some convex domains of infinite type in ${\mathbb{C}}^2$. In detail, we prove that supnorm estimates hold for infinite exponential type domains provided the exponent is less than 1.