[Paper Review] Alternative proof of Keith-Zhong self-improvement and connectivity

Sylvester Eriksson‐Bique|arXiv (Cornell University)|Oct 7, 2016

Geometric Analysis and Curvature Flows8 references3 citations

TL;DR

This paper presents a new, direct proof of the Keith-Zhong self-improvement theorem for (1,p)-Poincaré inequalities in doubling metric measure spaces, using a novel characterization of Poincaré inequalities via Muckenhoupt-type conditions along curves. The proof establishes explicit quantitative bounds showing that self-improvement occurs for q < p − ε, with ε proportional to p and inversely proportional to powers of the doubling constant and Poincaré constant, offering the first transparent estimates for the improvement quantity.

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.

Motivation & Objective

• To provide a new, more transparent proof of the Keith-Zhong self-improvement theorem for (1,p)-Poincaré inequalities in doubling metric measure spaces.
• To establish explicit quantitative bounds for the self-improvement exponent ε in terms of p, the doubling constant D, and the Poincaré constant C_PI.
• To reveal and formalize a deep structural similarity between Muckenhoupt weights and Poincaré inequalities through a novel characterization involving curve fragments and iteration.
• To extend the proof to local Poincaré inequalities by localizing the arguments and tracking scale dependencies.

Proposed method

• Introduce a new characterization of (1,p)-Poincaré inequalities using a Muckenhoupt-type condition along curves, defined via infima of integral averages of upper gradients over curves.
• Use iterative curve fragment decomposition to control the oscillation of functions and relate it to the upper gradient L^p norm.
• Define the quantity α^p(L,τ) as the supremum over all pairs x,y of the infimum of (1/d(x,y))∫_γ g ds over curves γ connecting x and y, with g in a suitable class of functions.
• Establish a recursive inequality: α^q(L,τ) ≤ C M^k τ + δ max_{i=1,…,k} M^{-i} α^q(L, M^i τ), which enables iteration and leads to the self-improvement estimate.
• Use the A_p-connectivity condition as a bridge: show that A_p-connectivity implies the (1,p)-Poincaré inequality, and that A_p-connectivity implies A_q-connectivity for q < p − ε.
• Localize the proof by restricting to scales r_0 and tracking the required doubling and Poincaré inequality scales, ensuring the argument holds at finite scales.

Experimental results

Research questions

• RQ1 Can the self-improvement of (1,p)-Poincaré inequalities be proven with explicit, transparent quantitative bounds?
• RQ2 What is the precise dependence of the self-improvement exponent ε on the doubling constant D and the Poincaré constant C_PI?
• RQ3 Is there a formal structural analogy between Muckenhoupt weights and Poincaré inequalities at the level of proofs and definitions?
• RQ4 Can the self-improvement result be localized to finite scales, and what are the required scale conditions?

Key findings

• The self-improvement exponent ε satisfies ε ≤ p / (2^{13p+3} C_PI^p D^{3p+4})^{1/(p−1)} for p > 1, showing explicit dependence on p, C_PI, and D.
• For large p, the self-improvement quantity ε behaves asymptotically as p / (2^7 C_PI), indicating direct proportionality to p and inverse proportionality to C_PI.
• The proof establishes that A_p-connectivity implies A_q-connectivity for all q < p − ε(D,p,C_A), with ε depending quantitatively on D and C_A.
• The method yields the first completely transparent and direct bounds for the self-improvement phenomenon, resolving long-standing opacity in the original proof.
• The localized version of the result holds under the condition r_0 ≤ min{r_PI/4, r_D/(20C)}, ensuring the doubling and Poincaré properties are valid at required scales.
• The connection between Muckenhoupt weights and Poincaré inequalities is formalized via a curve-based characterization, revealing a deep structural analogy at the proof level.