Hölder exponents of arbitrary functions

Abstract

The functional class of Hölder exponents of continuous function has been completely characterized by P. Andersson, K. Daoudi, S. Jaffard, J. Lévy Véhel andY. Meyer [1, 2, 6, 9]; these authors have shown that this class exactly corresponds to that of the lower limits of the sequences of nonnegative continuous functions. The problem of determining whether or not the H¨older exponents of discontinuous (and even unbounded) functions can belong to a larger class remained open during the last decade. The main goal of our article is to show that this is not the case: the latter H¨older exponents can also be expressed as lower limits of sequences of continuous functions. Our proof mainly relies on a “wavelet-leader” reformulation of a nice characterization of pointwise Hölder regularity due to P. Anderson.