What question did this study set out to answer?

The aim is to establish new weighted inequalities on smooth two-manifolds with explicit constants.

New weighted inequalities on two–manifolds

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New class of weighted inequalities estimated for two-manifolds, implying advancements in mathematical analysis.

The aim is to establish new weighted inequalities on smooth two-manifolds with explicit constants.
Introduce a family of weights satisfying a differential equation.
Derive weighted L^{2} Poincaré and elliptic functional inequalities on smooth two-manifolds.
Focus on weights related to geodesic distances to specific points.
Successfully establish new inequalities with explicit constants.
Weights demonstrated to be comparable to products of geodesic distances.
Provide estimates for a weighted Hodge decomposition for one-forms.

Aria Halavati (2026) studied this question.

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