Norm estimates for the $\bar\partial$-equation on a non-reduced space

TL;DR

This paper establishes $L^p$-estimates for solving the $ar ext{d}$-equation on non-reduced Cohen-Macaulay analytic spaces with smooth underlying reduced spaces, advancing understanding of complex analysis in singular spaces.

Contribution

It provides the first $L^p$-estimates for the $ar ext{d}$-equation on a class of non-reduced spaces, extending classical results to more singular settings.

Findings

Existence of $L^p$-solutions for the $ar ext{d}$-equation on non-reduced Cohen-Macaulay spaces

Extension of $ar ext{d}$-problem solutions to spaces with singularities

Improved understanding of complex analysis on non-reduced analytic spaces

Abstract

We study norm-estimates for the $\bar\partial$-equation on non-reduced analytic spaces. Our main result is that on a non-reduced analytic space, which is Cohen-Macaulay and whose underlying reduced space is smooth, the $\bar\partial$-equation for $(0,1)$-forms can be solved with $L^p$-estimates.