Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New weighted inequalities on two-manifolds

Aria Halavati

TL;DR

The paper develops a general framework for L^2-weighted elliptic estimates on smooth two-manifolds by introducing weights $\omega$ solving $\omega^2 \Delta \log(\omega) = -\kappa(x) \omega^2$ and proving a robust weighted Hodge decomposition. The main contributions include a 2D generalization of Caffarelli–Kohn–Nirenberg inequalities, a homogeneous elliptic estimate and a refined $\varepsilon$-elliptic inequality with explicit constants, and a precise weighted Hodge decomposition for $A = *d\xi_1 + d\xi_2$ and $\omega A = *\omega d\phi_1 + \omega^{-1} d\phi_2$, together with a quantitative bound on $d(\xi_1-\phi_1)$. The results apply to weights that vanish at multiple points, notably $\omega(x) \sim \prod_i d_M(x,x_i)^{\alpha_i}$ and exponentials of Green's functions, extending CK N-type theory in 2D and providing tools for stability analyses in gauge theories. These methods rely on elementary variational and integration-by-parts techniques and yield constants explicit in terms of the geometry and weight parameters, making the results applicable beyond the immediate Yang–Mills–Higgs context.

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.

New weighted inequalities on two-manifolds

TL;DR

The paper develops a general framework for L^2-weighted elliptic estimates on smooth two-manifolds by introducing weights solving and proving a robust weighted Hodge decomposition. The main contributions include a 2D generalization of Caffarelli–Kohn–Nirenberg inequalities, a homogeneous elliptic estimate and a refined -elliptic inequality with explicit constants, and a precise weighted Hodge decomposition for and , together with a quantitative bound on . The results apply to weights that vanish at multiple points, notably and exponentials of Green's functions, extending CK N-type theory in 2D and providing tools for stability analyses in gauge theories. These methods rely on elementary variational and integration-by-parts techniques and yield constants explicit in terms of the geometry and weight parameters, making the results applicable beyond the immediate Yang–Mills–Higgs context.

Abstract

We establish a new class of -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.
Paper Structure (5 sections, 5 theorems, 61 equations)

This paper contains 5 sections, 5 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. General formulation and examples of weights
  4. Main results
  5. The Proof

Key Result

Lemma 1.1

Let $(\mathcal{M}^2,g)$ be a Riemannian two--manifold and let $\Omega \in \mathcal{M}^2$ be a smooth open domain and $\omega$ is a weight as in weak-weight-equation with $\kappa = 0$. Any smooth one-form $A\in C_c^{\infty}(\bigwedge^1\Omega)$ has a Hodge decomposition and a weighted Hodge decomposit for $4$ compactly supported functions $\xi_1,\xi_2,\phi_1,\phi_2$. Moreover for any $0 \leq \vareps

Theorems & Definitions (14)

  • Lemma 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark
  • Remark
  • Definition 2.1
  • proof : Proof of \ref{['Generalized-CKN-thm']}
  • proof : Proof of \ref{['elliptic-inequality-thm']}
  • Lemma 2.2
  • ...and 4 more