On a Frequency Localized Bernstein Inequality and Some Generalized Poincaré-type Inequalities

We consider a frequency localized Bernstein inequality for the fractional Laplacian operator, which has wide applications in fluid dynamics such as dissipative surface quasi-geostrophic equations. We use a heat flow reformulation and prove the inequality for the full range of parameters and in all dimensions. A crucial observation is that after frequency projection the zeroth frequency part of the Lévy semigroup does not participate in the inequality and therefore can be freely adjusted. Our proof is based on this idea and a careful perturbation of the Lévy semigroup near the zero frequency, which preserves the positivity and improves the time decay. As an application we also give new proofs of some generalized Poincaré-type inequalities.