  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
zero frequency part of the L\'{e}vy 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 L\'{e}vy semigroup near the zero
frequency which preserves the positivity and improves the time decay. Several
alternative proofs (with weaker results) are also included. As an application
we also give new proofs of some generalized Poincare type inequalities.
