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.