Riesz型不等式在粘合下的稳定性
Stability of the Riesz-type inequalities under gluing
- Sun Yat-sen (Zhongshan) University（中山大学）
- Macquarie University（麦考瑞大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明在紧致粘合下逆平方根不等式和Riesz变换的稳定性，通过模型流形粘合与Fredholm构造，控制误差并推广至外部域边值问题。
AI中文摘要：
我们建立了在紧致粘合下逆平方根不等式和Riesz变换的稳定性原理。在共同的Sobolev维数以下，逆估计从有限多个模型流形传递到它们的连通和。一个边界相容的Poisson参数化还证明了对于外部Lipschitz域上的实对称一致椭圆散度型算子，对每个$1<p<\infty$成立Neumann逆不等式。对于正向变换，我们将一个有界核心模型和一个独立的整体空间端模型粘合在一起，这两个模型在重叠区域与系数一致。然后，一个齐次Fredholm构造解决了形式值误差。所得范围由两个模型指数控制，对于Dirichlet数据还有额外的限制$p<n$。更广泛地说，我们的方法阐明了有界内部和外部问题之间的相互作用，并为研究其他边值问题提供了一个灵活的框架。
英文摘要：
We establish stability principles under compact gluing for the reverse square-root inequality and the Riesz transform. Below a common Sobolev dimension, the reverse estimate passes from finitely many model manifolds to their connected sum. A boundary-compatible Poisson parametrix also proves the Neumann reverse inequality for every $1<p<\infty$ for real symmetric uniformly elliptic divergence-form operators on exterior Lipschitz domains. For the forward transform, we glue a bounded-core model and an independent whole-space end model that agree with the coefficients on overlapping regions. A homogeneous Fredholm construction then resolves the form-valued error. The resulting range is controlled by the two model exponents, with the additional restriction $p<n$ for Dirichlet data. More broadly, our method illuminates the interplay between the bounded interior and exterior problems, and provides a flexible framework for studying other boundary value problems.
