How smooth is almost every function in a Sobolev space? (English) Zbl 1155.28302

Fraysse, Aurélia; Jaffard, Stéphane

Rev. Mat. Iberoam. 22, No. 2, 663-682 (2006).

Summary: 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.

MSC:

28C20	Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A21	Classification of real functions; Baire classification of sets and functions
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
28A80	Fractals
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems

Keywords:

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

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