The Poincaré Inequality does not improve with blow-up

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 $(1,P)$-Poincaré inequality if and only if $P>\beta$.

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