Continuous Translation of Hölder and Lipschitz Functions

Abstract

All functions will be complex, periodic, integrable (on [0, 2π ]) functions of a real variable x . Moreover, we shall require that every function have mean zero on [0, 2π ], so that in particular non-zero constants are excluded. 1. Plessner's characterization of absolutely continuous functions. An old theorem of Plessner (4), generalized to arbitrary compact groups by Bochner (1), can be taken as our starting point. Consider the functions f of bounded variation on [0, 2π ]. These f form a Banach space F when each f is normed by its total variation on [0, 2π ]. And translations define a natural one-parameter group of isometries on F .