On the Baire Generic Validity of the t-Multifractal Formalism in Besov and Sobolev Spaces. (29th August 2019) - Ben Abid, Moez; Ben Slimane, Mourad; Ben Omrane, Ines; Halouani, Borhen
Abstract : Thet - multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce thet - spectrum of a functionf from the knowledge of the( p, t ) - oscillation exponent off . Thet - spectrum is the Hausdorff dimension of the set of points wheref has a given value of pointwiseL t regularity. The( p, t ) - oscillation exponent is measured by determining to which oscillation spacesO p, t s (defined in terms of wavelet coefficients)f belongs. 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.