Paley-Wiener Type Theorems associated to Dirac Operators of Riesz-Feller type

Abstract

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 $\mathbf{D}_\theta^{\alpha}$ of order $\alpha$ and skewness $\theta$. The pseudo-differential reformulation of $\mathbf{D}_\theta^{\alpha}$ in terms of the Riesz derivative $(-\Delta)^{\frac{\alpha}{2}}$ and the so-called {\textit Riesz-Hilbert transform} $H$, allows for the description of generalized Hardy spaces on the upper and lower half-spaces of $\mathbf{R}^{n+1}$, $\mathbf{R}^{n+1}_+$ resp. $\mathbb{R}^{n+1}_-$, using L\'evy-Feller type semigroups generated by $-(-\Delta)^{\frac{\alpha}{2}}$, and the boundary values $\mathbf{f}_\pm=\frac{1}{2}\left(\mathbf{f}\pm H\mathbf{f}\right)$. Subsequently, we employ a proof strategy rooted in {\textit real Paley-Wiener methods} to demonstrate that the growth behavior of the sequences of functions $\left(\left(\mathbf{D}_\theta^{\alpha}\right)^k\mathbf{f}_{\pm}\right)_{k\in \mathbb{N}_0}$ effectively captures the relationship between the support of the Fourier transform $\widehat{\mathbf{f}}$ of the $L^p-$function $\mathbf{f}$, in the case where $\mathrm{supp}\widehat{\mathbf{f}}\subseteq \overline{B(0,R)}$, and the solutions of Cauchy problems equipped with the space-time operator $\partial_{x_0} + \mathbf{D}_\theta^{\alpha}$, 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. Specifically, leveraging the established Stein-Kolmogorov inequalities for hypercomplex variables enables us to accurately determine the maximum radius $R$ for which $\operatorname{supp}\widehat{\mathbf{f}} \subseteq \overline{B(0, R)}$ holds.