Abstract. Let $f\in \Psi(\mathbb R)$ and $I^nf\in N_\Psi(\mathbb R)$,
for all $n = 1, 2, ...$. Then
\[\mathop {\lim }\limits_{n \to
\infty } I^n
f_{N_\Psi ()}^{1/n} = \sigma ^{  1},\]
where $\sigma := \inf\{\xi: \xi \in \supp
f\}, ._N_\Psi(\mathbb R)$ is the norm in the
Lorentz space $N_\Psi (\mathbb R)$, and for $g\in
S'(\mathbb R)$, the tempered generalized function
$Ig$ is a primitive of $g$ if $D(Ig) = g$, that is
\[<Ig,\varphi^'>\<g,\varphi>, \forall \varphi \in S(\mathbb R)\]
and
$S(\mathbb R)$ is the Schwartz space of rapidly
decreasing functions.
In other words, in this paper
we characterize behavior of the sequence of $N_\Psi(\mathbb R)  norms of primitives of a function by its
spectrum (the support of its Fourier transform).
