On the Frisch–Parisi conjecture

Abstract

We prove several related results concerning the genericity (in the sense of Baire's categories) of multifractal functions. One result asserts that, if s− d/ p>0 , quasi-all functions of the Sobolev space L p,s( R d) (or the Besov space B s,q p( R d) ) are multifractal functions, with a spectrum of singularities supported by the interval [ s− d/ p, s] , on which the spectrum is d( H)= d−( s− H) p . Another result asserts that the Frisch–Parisi conjecture also holds for quasi-all functions, if the range of p s over which one computes the Legendre transform is chosen appropriately.