  We characterize the set of functions $u\_0\in L^2(R^n)$ such that the
solution of the problem $u\_t=\mathcal{L}u$ in $R^n\times(0,\infty)$ starting
from $u\_0$ satisfy upper and lower bounds of the form $c(1+t)^{-\gamma}\le
\|u(t)\|\_2\le c'(1+t)^{-\gamma}$.Here $\mathcal{L}$ is in a large class of
linear pseudo-differential operator with homogeneous symbol (including the
Laplacian, the fractional Laplacian, etc.). Applications to nonlinear PDEs will
be discussed: in particular our characterization provides necessary and
sufficient conditions on $u\_0$ for a solution of the Navier--Stokes system to
satisfy sharp upper-lower decay estimates as above.In doing so, we will revisit
and improve the theory of \emph{decay characters} by C. Bjorland, C. Niche, and
M.E. Schonbek, by getting advantage of the insight provided by the
Littlewood--Paley analysis and the use of Besov spaces.
