  In the present paper, we study regularity of the Andersson-Samuelsson
Koppelman integral operator on affine cones over smooth projective complete
intersections. Particularly, we prove $L^p$- and $C^\alpha$-estimates, and
compactness of the operator, when the degree is sufficiently small. As
applications, we obtain homotopy formulas for different
$\overline{\partial}$-operators acting on $L^p$-spaces of forms, including the
case $p=2$ if the varieties have canonical singularities. We also prove that
the $\mathcal{A}$-forms introduced by Andersson-Samuelsson are $C^\alpha$ for
$\alpha < 1$.
