Poincaré inequalities, logarithmic growth and doubling

Abstract

We investigate when a Poincaré inequality forces the underlying measure to be doubling, with particular attention to the space \(X\) controlling the oscillation. We identify \(L\log L\) as the sharp threshold: under a natural local growth condition on the measure, doubling is forced precisely when \(L\log L(0,1)\not\hookrightarrow X\). We also obtain a doubling criterion under a weaker geometric assumption when the oscillation norm is sufficiently stronger than \(L\log L\). We show that this threshold is sharp in every fixed Euclidean dimension, using a dimension-free local \(L\log L\)-Poincaré inequality for log-concave measures and non-doubling Gaussian examples. We then turn to infinite-dimensional spaces, where doubling is impossible for the measures under consideration. Nevertheless, the \(L\log L\)-Poincaré inequality continues to hold, and we establish sharp characterizations for infinite product models, Gaussian measures on Hilbert spaces under power-type small-ball estimates, and Wiener measure with the uniform norm.