Koppelman formulas on affine cones over smooth projective complete intersections

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 $\bar{\mathrm{\partial }}$-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 $\alpha <1$.