Revista Matemática Iberoamericana


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Volume 22, Issue 2, 2006, pp. 663–682
DOI: 10.4171/RMI/469

Published online: 2006-08-31

How smooth is almost every function in a Sobolev space?

Aurélia Fraysse[1] and Stéphane Jaffard[2]

(1) Université Paris Est, Créteil, France
(2) Université Paris Est, Créteil, France

We show that almost every function (in the sense of prevalence) in a Sobolev space is multifractal: Its regularity changes from point to point; the sets of points with a given Hölder regularity are fractal sets, and we determine their Hausdorff dimension.

Keywords: Sobolev spaces, Besov spaces, prevalence, Haar-null sets, multifractal functions, Hölder regularity, Hausdorff dimension, wavelet bases

Fraysse Aurélia, Jaffard Stéphane: How smooth is almost every function in a Sobolev space?. Rev. Mat. Iberoam. 22 (2006), 663-682. doi: 10.4171/RMI/469