Besov spaces in multifractal environment and the Frisch-Parisi conjecture

Julien Barral (1, 2) , Stephane Seuret (3)

1 Université Sorbonne Paris Nord
2 LAGA - Laboratoire Analyse, Géométrie et Applications
3 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We give a solution to the so-called Frisch-Parisi conjecture by constructing a Baire functional space in which typical functions satisfy a multifractal formalism, with a prescribed singularity spectrum. This achievement combines three ingredients developed in this paper. First we prove the existence of almost-doubling fully supported Radon measure on $\R^d$ with a prescribed multifractal spectrum. Second we define new \textit{heterogeneous} Besov like spaces possessing a wavelet characterization; this uses the previous doubling measures. Finally, we fully describe the multifractal nature of typical functions in these functional spaces.

Domains: Mathematics [math]; Classical Analysis and ODEs [math.CA]; Mathematics [math]; Functional Analysis [math.FA]; Mathematics [math]; Metric Geometry [math.MG]; Mathematics [math]; Mathematical Physics [math-ph]

Julien Barral, Stephane Seuret. Besov spaces in multifractal environment and the Frisch-Parisi conjecture. 2020. ⟨hal-02899957⟩