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

Abstract

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 μ∈Ed we associate a Baire function space Bμ(Rd) (a generalisation of Hölder-Zygmund spaces) in which typical functions share the same singularity spectrum as μ. This yields a partial solution to the conjecture. In [1], we introduce and study a family B={Bqμ,p(Rd)}μ∈Ed,(p,q)∈[1,+∞]2 of heterogeneous Besov spaces that contains {Bμ(Rd)}μ∈Ed and generalises in a natural direction the family of standard Besov spaces, and we solve the inverse problem exhaustively inside B.