Keith-Zhong自我改进和连通性的替代证明
Alternative proof of Keith-Zhong self-improvement and connectivity
摘要：
对Keith和Zhong的著名定理$(1,p)$ -Poincaré不等式自改进为$(1,p-ε)$ -Poincaré不等式给出了一个新的证明.本文对Poincaré不等式进行了新的刻画，并利用它给出了一个与Muckenhoupt-权密切相关的全新证明.这个新的特征，和替代的证明，证明了Muckenhoupt权重和庞加莱不等式之间的形式相似性。我们给出的证明是简短的，而且更直接一些。有了它们，我们可以给出第一个完全透明的自我改进的数量和所涉及的常数的界限。我们观察到，对于大的$p$，自我改进的量与$p$成正比，与倍增常数和庞加莱不等式中常数的幂成反比。证明可以局部化，从而得到局部Poincaré不等式自改进的更透明的证明。
英文摘要：
We find a new proof for the celebrated theorem of Keith and Zhong that a $(1,p)$-Poincaré inequality self-improves to a $(1,p-ε)$-Poincaré inequality. The paper consists of a novel characterization of Poincaré inequalities and then uses it to give an entirely new proof which is closely related to Muckenhoupt-weights. This new characterization, and the alternative proof, demonstrate a formal similarity between Muckenhoupt-weights and Poincaré inequalities. The proofs we give are short and somewhat more direct. With them we can give the first completely transparent bounds for the quantity of self-improvement and the constants involved. We observe that the quantity of self-improvement is, for large $p$, directly proportional to $p$, and inversely proportional to a power of the doubling constant and the constant in the Poincaré inequality. The proofs can be localized and thus we obtain more transparent proofs of the self-improvement of local Poincaré inequalities.
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