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

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 Dαθ - a space-fractional operator of order α and skewness θ, encompassing the Dirac operator D. We will show that such family of function spaces seamlessly characterizes the interplay between Clifford-valued Lᵖ-functions satisfying the support condition supp\ f⊆ B(0,R), and the solutions of the Cauchy problems endowed by the space-time operator ∂x₀+D_θ^α that are of exponential type R^α. 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 supp\ f is compactly supported in $B(0,R)$ rather explicit.