  The purpose of this paper is to characterize the zero sets of holomorphic
functions in the Nevanlinna class on a class of convex domains of infinite type
in $\mathbb{C}^2$. Moreover, we also obtain $L^p$ estimates, $1 \leq p \leq
\infty$, for a particular solution of the tangential Cauchy-Riemann equation on
the boundaries of these domains.
