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

Abstract EN

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.