An explicit d-bar-integration formula for weighted homogeneous varieties II: forms of higher degree

TL;DR

This paper develops explicit integral formulas to solve the d-bar-equation on weighted homogeneous varieties, providing L^p-bounded solutions even with singularities, advancing complex analysis techniques for such varieties.

Contribution

It introduces a new class of explicit d-bar-integration formulas for weighted homogeneous varieties, including cases with singularities, and establishes L^p-bounded solution operators.

Findings

Explicit integral formulas for d-bar-equation solutions

L^p-bounded solution operators for weighted homogeneous varieties

Applicable to varieties with arbitrary singular loci

Abstract

Let Y be a weighted homogeneous (singular) subvariety of C^n. The main objective of this paper is to present a class of explicit integral formulae for solving the d-bar-equation $\omega=\dbar\lambda$ on the regular part of Y, where $\omega$ is a d-bar-closed (0,q)-form with compact support and degree q>=1. Particular cases of these formulae yield L^p-bounded solution operators for $1<=p<=\infty$ if Y is a homogeneous and pure dimensional subvariety with an arbitrary singular locus.