The Poincar\'e Inequality does not improve with blow-up

TL;DR

The paper constructs a family of metric measure spaces demonstrating that the Poincaré inequality's validity does not necessarily improve under blow-up operations, challenging assumptions about regularity enhancement.

Contribution

It provides explicit examples of spaces where Poincaré inequalities do not improve with blow-ups, showing limitations of regularity transfer in metric measure spaces.

Findings

Poincaré inequality does not necessarily improve under blow-up.

Constructs a family of spaces closed under weak-tangents.

Shows the threshold for Poincaré inequality depends on parameter eta.

Abstract

For each $\beta>1$ we construct a family $F_\beta$ of metric measure spaces which is closed under the operation of taking weak-tangents (i.e.~blow-ups), and such that each element of $F_\beta$ admits a $(1,P)$-Poincar\'e inequality if and only if $P>\beta$.