Alternative Proof of Keith-zhong Self-improvement and Connectivity

We find new proofs for the celebrated theorem of Keith and Zhong that a (1, p)-Poincaré inequality self-improves to a (1, p − )-Poincaré inequality. The paper consists of two proofs, the second of which identifies a novel characterization of Poincaré inequalities and 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.