Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 44, 2019, 407-425

# ALTERNATIVE PROOF OF KEITH–ZHONG
SELF-IMPROVEMENT AND CONNECTIVITY

## Sylvester Eriksson-Bique

UCLA, Department of Mathematics

520 Portola Plaza, Los Angeles CA 90095, U.S.A.; syerikss 'at' math.ucla.edu

**Abstract.**
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.

**2010 Mathematics Subject Classification:**
Primary 30L99, 42B25, 39B72.

**Key words:**
Poincaré inequality, self-improvement, metric spaces, PI-spaces, analysis on metric spaces,
connectivity, Muckenhoupt-weights.

**Reference to this article:** S. Eriksson-Bique:
Alternative proof of Keith–Zhong self-improvement and connectivity.
Ann. Acad. Sci. Fenn. Math. 44 (2019), 407-425.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4424

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