Fourier and Radon transform on harmonic 𝑁𝐴 groups

Abstract

In this article we study the Fourier and the horocyclic Radon transform on harmonic N A NA groups (also known as Damek-Ricci spaces). We consider the geometric Fourier transform for functions on L p L^p -spaces and prove an analogue of the L 2 L^2 -restriction theorem. We also prove some mixed norm estimates for the Fourier transform generalizing the Hausdorff-Young and Hardy-Littlewood-Paley inequalities. Unlike Euclidean spaces the domains of the Fourier transforms are various strips in the complex plane. All the theorems are considered on these entire domains of the Fourier transforms. Finally we deal with the existence of the Radon transform on L p L^p -spaces and obtain its continuity property.