**** 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 ) 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 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.

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