  In the present paper, we study the regularity of the Andersson-Samuelsson
Koppelman integral operator on the $A_1$-singularity. Particularly, we prove
$L^p$- and $C^0$-estimates. As applications, we obtain $L^p$-homotopy formulas
for the $\bar{\partial}$-equation on the $A_1$-singularity, and we prove that
the $\mathcal{A}$-forms introduced by Andersson-Samuelsson are continuous on
the $A_1$-singularity.
