Continuous Translation of Hölder and Lipschitz Functions

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Abstract
All functions will be complex, periodic, integrable (on [0,2π]) functions of a real variablex. 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 ofbounded variationon [0,2π]. These f form a Banach spaceFwhen each f is normed by its total variation on [0,2π]. And translations define a natural one-parameter group of isometries onF.

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