The role of restriction theorems in harmonic analysis on harmonic \(NA\) groups (Q2269689)

In the paper the authors consider issues related to harmonic analysis on harmonic NA groups. The main object investigated is the Fourier transform (an analogue of the Helgason Fourier transform on symmetric spaces) defined by means of the Poisson kernel. The paper was inspired by a recent paper of Bray and Pinsky on growth properties of the Fourier transform on the Euclidean space, or on a noncompact rank one Riemannian symmetric space. The main result of the paper is a restriction type theorem for the Fourier transform. The authors also formulate and prove a Hausdorff-Young type inequality with mixed norms involved. These tools are then used to state and prove results on the growth of a Fourier transform in terms of moduli of continuity. Finally, restriction theorems related to the Kunze-Stein phenomenon are discussed.