Koppelman formulas on the A_1-singularity

TL;DR

This paper investigates the regularity properties of Koppelman integral operators on the $A_1$-singularity, establishing $L^p$ and continuity estimates, and applying these to solve the $ar{ ext{d}}$-equation and analyze form continuity.

Contribution

It provides new $L^p$- and $C^0$-estimates for Koppelman operators on the $A_1$-singularity, advancing understanding of their regularity and applications.

Findings

Proved $L^p$-estimates for the Koppelman operator.

Established $C^0$-regularity on the $A_1$-singularity.

Derived $L^p$-homotopy formulas for the $ar{ ext{d}}$-equation.

Abstract

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.