On the Baire Generic Validity of thet-Multifractal Formalism in Besov and Sobolev Spaces

Abstract

Thet-multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce thet-spectrum of a functionffrom the knowledge of the(p,t)-oscillation exponent off. Thet-spectrum is the Hausdorff dimension of the set of points wherefhas a given value of pointwiseLtregularity. The(p,t)-oscillation exponent is measured by determining to which oscillation spacesOp,ts(defined in terms of wavelet coefficients)fbelongs. In this paper, we first prove embeddings between oscillation and Besov-Sobolev spaces. We deduce a general lower bound for the(p,t)-oscillation exponent. We then show that this lower bound is actually equality generically, in the sense of Baire’s categories, in a given Sobolev or Besov space. We finally investigate the Baire generic validity of thet-multifractal formalism.