Poincaré inequalities, logarithmic growth and doubling

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 $L\log L(0,1)\hookrightarrow ̸X$. 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.