The Frisch-Parisi conjecture I: Prescribed multifractal behavior, and a partial solution

In this work and its companion [1], we construct Baire function spaces in which typical elements share the same prescribed multifractal behavior and obey a multifractal formalism, providing a solution to the so-called Frisch-Parisi conjecture for functions, an inverse problem raised by S. Jaffard. In this first part, a family Ed of almost-doubling fully supported capacities on Rd with prescribed singularity spectra is constructed. With each & mu; & ISIN; Ed we associate a Baire function space B & mu;(Rd) (a generalisation of Holder-Zygmund spaces) in which typical functions share the same singularity spectrum as & mu;. This yields a partial solution to the conjecture. In [1], we introduce and study a family B = {B & mu;,p q (Rd)}& mu;EEd,(p,q)E[1,+oo]2 of heterogeneous Besov spaces that contains {B & mu;(Rd)}& mu;EEd and generalises in a natural direction the family of standard Besov spaces, and we solve the inverse problem exhaustively inside B. & COPY; 2023 Elsevier Masson SAS. All rights reserved.

Automated summary

In this work, Baire function spaces are constructed to address the Frisch-Parisi conjecture for functions, solving an inverse problem raised by S. Jaffard. A family of almost-doubling fully supported capacities is constructed, and each capacity is associated with a Baire function space in which typical functions have the same singularity spectrum. In the companion work, a family of heterogeneous Besov spaces is introduced and studied, providing a complete solution to the inverse problem.