Abstract
Recently, a multivariate multifractal analysis for pointwise regularities based on hierarchical multiresolution quantities was developed. General bounds between the Hausdorff dimension of the intersection of single fractal sets and that of the original sets were derived. Equalities were checked for some synthetic signals that include multiplicative cascades. In this paper, we focus on the setting supplied by simultaneous pointwise regularities. The
regularity for
was first introduced in order to better study elliptic partial differential equations where the natural function space setting is
or a Sobolev space which includes unbounded functions. We will prove that both corresponding multivariate multifractal formalism and equalities above hold Baire generically in a given product of Besov spaces
,
such that
. We therefore extend previous results where only cases (
and
for all i) and (
,
and
for all i) for simultaneous pointwise Hölder regularities have been proved.
Acknowledgements
Mourad Ben Slimane would like to thank Stéphane Jaffard for stimulating discussions. The authors would like to thank the referee(s) for his/her (their) comments and remarks that greatly helped to improve the presentation of the paper.
Disclosure statement
No potential conflict of interest was reported by the authors.