New characterizations of generalized uniform Lipschitz classes via the quaternion Fourier–Mellin transform

Let G=R+∗×S1, where S1 denotes the unit circle in the plane R2, and let H represent the quaternion algebra. For functions f in the space L2(G,H), we study the regularity of f through its quaternion Fourier–Mellin transform, denoted as M{f}. We establish necessary and sufficient conditions, in terms of behaviour of M{f}, to ensure that f belongs to the generalized Lipschitz classes Hα1,α2m1,m2 and hα1,α2m1,m2 for 0<αi<mi, where i=1, 2 and mi∈N.