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 Lévy-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 $\hat{\mathbf{f}}$ of the $L^p-$function $\mathbf{f}$, in the case where $\mathrm{supp}\hat{\mathbf{f}}\subseteq \overline{B(0,R)}$, 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 $\mathrm{supp}\hat{\mathbf{f}}\subseteq \overline{B(0,R)}$ holds.

33 pages, revised version, some minor changes