  This paper systematically investigates Paley-Wiener-type theorems in the
context of hypercomplex variables. To this end, we introduce and study the
so-called generalized Bernstein spaces endowed by the fractional Dirac operator
$D_{\alpha}^{\theta}$ - a space-fractional operator of order $\alpha$ and
skewness $\theta$, encompassing the Dirac operator $D$. We will show that such
family of function spaces seamlessly characterizes the interplay between
Clifford-valued $L^p-$functions satisfying the support condition
$\mathrm{supp}\ \widehat{f}\subseteq B(0,R)$, and the solutions of the Cauchy
problems endowed by the space-time operator $\partial_{x_0}+D_\theta^\alpha$
that are of exponential type $R^\alpha$. Such construction allows us to
generalize, in a meaningful way, the results obtained by Kou and Qian (2002)
and Franklin, Hogan and Larkin (2017). Noteworthy, the exploitation of the
well-known Kolmogorov-Stein inequalities to hypercomplex variables permits us
to make the computation of the maximal radius $R$ for which $\mathrm{supp}\
\widehat{f}$ is compactly supported in $B(0,R)$ rather explicit.
