Alternative proof of Keith-Zhong self-improvement and connectivity
arXiv:1610.02129
Abstract
We find a new proof for the celebrated theorem of Keith and Zhong that a -Poincaré inequality self-improves to a -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 , directly proportional to , 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.
arXiv commentSome proofs and typos corrected. More details for some constructions. Accepted to Ann. Acad. Sci. Fenn