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

Ben Abid, Moez; Ben Slimane, Mourad; Ben Omrane, Ines; Halouani, Borhen

The t-multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce the t-spectrum of a function f from the knowledge of the (p,t)-oscillation exponent of f. The t-spectrum is the Hausdorff dimension of the set of points where f has a given value of pointwise Lt regularity. The (p,t)-oscillation exponent is measured by determining to which oscillation spaces Op,ts (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 the t-multifractal formalism.

BESOV spaces; SOBOLEV spaces; MULTIFRACTALS; FRACTAL dimensions; VALIDITY of statistics

Journal of Function Spaces, 2019, p1

2314-8896

Academic Journal

10.1155/2019/4358261