Besov spaces in multifractal environment and the Frisch-Parisi conjecture

Abstract

In this article, a solution to the so-called Frisch-Parisi conjecture is brought. This achievement is based on three ingredients developed in this paper. First almost-doubling fully supported Radon measures on ${\mathbb{R}}^d$ with a prescribed singularity spectrum are constructed. Second we define new \textit{heterogeneous} Besov spaces $B_q^{\mu ,p}$ and find a characterization using wavelet coefficients. Finally, we fully describe the multifractal nature of typical functions in the function spaces $B_q^{\mu ,p}$. Combining these three results, we find Baire function spaces in which typical functions have a prescribed singularity spectrum and satisfy a multifractal formalism. This yields an answer to the Frisch-Parisi conjecture.

77 pages, 6 Figures; we added Theorem 2.29

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