2024/05/08 by Bernstein, Swanhild, Faustino, Nelson
#15A66 #30G35 #35S10 #42B10 #47A11 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2405.04989
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 Dθα of order α and skewness θ. The pseudo-differential reformulation of Dθα in terms of the Riesz derivative (-Δ)^\fracα2 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 Rn+1, Rn+1+ resp. ℝn+1-, using Lévy-Feller type semigroups generated by -(-Δ)^\fracα2, and the boundary values f_±=(1)/(2)(f± Hf). Subsequently, we employ a proof strategy rooted in \textit real Paley-Wiener methods to demonstrate that the growth behavior of the sequences of functions ((Dθα)kf±)k∈ ℕ0 effectively captures the relationship between the support of the Fourier transform \widehatf of the Lp-function f, in the case where supp\widehatf⊆ B(0,R), and the solutions of Cauchy problems equipped with the space-time operator ∂x0 + Dθα, which are of exponential type Rα. Within the developed framework, introducing a hypercomplex analog for the Bernstein spaces BRp 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 supp\widehatf ⊆ B(0, R) holds.