Multivariate multifractal formalism for simultaneous pointwise (Tuipi)i regularities. Issue 16 (2nd November 2022) - Ben Slimane, Mourad; Ben Abid, Moez; Ben Omrane, Ines; Halouani, Borhen
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 ( T u i p i ) i = 1, …, L regularities. The T u p regularity for 1 ≤ p < ∞ was first introduced in order to better study elliptic partial differential equations where the natural function space setting is L p 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 ∏ i = 1, …, L B t i s i, r i ( R d ), s i, t i, r i > 0, 1 ≤ p i ≤ ∞ such that s i − d t i > − d p i . We therefore extend previous results where only cases ( p i = ∞ and s i − d t i > 0 for all i ) and ( p i = ∞, s i = d t i and r i ≤   1 for all i ) for simultaneous pointwise Hölder regularities have been proved.