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 and Y. Meyer [Andersson, P.: Wavelets and local regularity. PhD Thesis. Department of Mathematics, G\"oteborg, 1997], [Andersson, P.: Characterization of pointwise Hölder regularity. Appl. Comput. Harmon. Anal. {\bf 4} (1997), 429-443], [Daoudi, K., Lévy-Véhel J. and Meyer, Y.: Construction of continuous functions with prescribed local regularity. Constr. Approx. {\bf 14} (1998), 349-385], [Jaffard, S.: Functions with prescribed Hölder exponent. Appl. Comput. Harmon. Anal. {\bf 2} (1995), 400-401]; 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 Hölder 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 Hölder 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.