New weighted inequalities on two-manifolds

We establish a new class of weighted $L^2$ Poincaré and elliptic functional inequalities on smooth two-manifolds with explicit constants, for a family of weights satisfying a differential equation.

This family includes, in particular, weights comparable to products of positive powers of the geodesic distance to finitely many points.

Our primary motivation is the derivation of estimates associated with a weighted Hodge decomposition for one-forms.