# New paper: Weak Poincaré inequalities in the absence of spectral gaps

*** Update: this paper is now published in Ann. Henri Poincaré ***

Jonathan Ben-Artzi recently uploaded a new paper entitled Weak Poincaré inequalities in absence of spectral gaps, co-authored with Amit Einav.

For Markov semigroups it is well-known that the following are equivalent:

• The generator has a spectral gap,
• The generator satisfies a Poincaré inequality,
• Solutions decay exponentially

In this paper, they study semigroups which lack a spectral gap (such as the heat semigroup in $\mathbb{R}^d$) and try to see how much of the above theorem remains true. They prove that an estimate on the density of the spectrum near 0 leads to a weak Poincaré inequality, which in turn leads to an algebraic decay rate.

This is applied to the heat semigroup, where the optimal decay rate $t^{-d/2}$ is recovered. In this case, the weak Poincaré inequality is no more than the Nash inequality. This is done for the fractional Laplacian as well, with similar results.