On the Frisch–Parisi conjecture

AbstractWe 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 Lp,s(Rd) (or the Besov space Bs,qp(Rd) ) 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