The classical Paley-Wiener theory consists of two parts. First, the equivalence of the compact support of the Fourier transform of the boundary values to the exponential growth of the extended function, and second, the equivalence to the Bernstein spaces. This paper explores Paley-Wiener-type theorems within the framework of hypercomplex variables. The investigation focuses on a space-fractional version of the Dirac operator \(\varvec{D}_\theta ^\alpha \) of order \(\alpha \) and skewness \(\theta \) . The pseudo-differential reformulation of \(\varvec{D}_\theta ^\alpha \) in terms of the Riesz derivative \((-\Delta )^{\frac{\alpha }{2}}\) and the so-called Riesz-Hilbert transform \(\mathcal {H}\) , allows for the description of generalized Hardy spaces, using Lévy-Feller type semigroups generated by \(-(-\Delta )^{\frac{\alpha }{2}}\) , and the boundary values. Subsequently, we employ a proof strategy rooted in real Paley-Wiener methods to demonstrate that the growth behavior of the sequences of functions \(\left( \left( ~\varvec{D}_\theta ^\alpha ~\right) ^k\varvec{f}_\pm \right) _{k\in {\mathbb N}_0}\) effectively captures the relationship between the support of the Fourier transform \(\widehat{\varvec{f}}=\mathcal {F}\varvec{f}\) of the \(L^p-\) function \(\varvec{f}\) , in the case where \(\operatorname {supp}\widehat{\varvec{f}}\subseteq \overline{B(0,R)}\) , and the solutions of Cauchy problems equipped with the space-time operator \(\partial _{x_0} + \varvec{D}_\theta ^\alpha \) , which are of exponential type \(R^\alpha \) . Within the developed framework, introducing a hypercomplex analog for the Bernstein spaces \(B_R^p\) arises naturally, allowing for the meaningful extension of the results by Kou and Qian as well as Franklin, Hogan, and Larkin.
