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Besov spaces in multifractal environment and the Frisch-Parisi conjecture

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Abstract

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.
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hal-02899957 , version 1 (15-07-2020)

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