New weighted inequalities on two-manifolds

TL;DR

This paper introduces new weighted elliptic estimates on two-manifolds, enabling advanced analysis of differential forms with applications to Hodge theory.

Contribution

It develops a novel class of $L^2$-weighted inequalities on two-manifolds, including weights related to geodesic distances, with explicit constants.

Findings

Established new weighted elliptic estimates for two-manifolds.

Included weights comparable to powers of geodesic distances.

Facilitated weighted Hodge decomposition for one-forms.

Abstract

We establish a new class of $L^2$-weighted elliptic estimates on smooth two-manifolds for a family of weights satisfying an equation with explicit constants. This family includes weights that are comparable to the product of positive powers of the geodesic distance to a given collection of points. Our primary motivation is to derive estimates related to a weighted Hodge decomposition for one-forms.