Multivariate multifractal formalism for simultaneous pointwise (Tuipi)i regularities

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 $(T_{u_i}^{p_i}{)}_{i=1,\dots ,L}$ regularities. The $T_u^p$ regularity for $1\le p<\mathrm{\infty }$ 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 $\prod _{i=1,\dots ,L}{B}_{t_i}^{s_i,r_i}\left({\mathbb{R}}^d\right)$, $s_i,t_i,r_i>0,1\le p_i\le \mathrm{\infty }$ such that $s_i-\frac{d}{t_i}>-\frac{d}{p_i}$. We therefore extend previous results where only cases ($p_i=\mathrm{\infty }$ and $s_i-\frac{d}{t_i}>0$ for all i) and ($p_i=\mathrm{\infty }$, $s_i=\frac{d}{t_i}$ and $r_i\le 1$ for all i) for simultaneous pointwise Hölder regularities have been proved.

Keywords:

• Multivariate multifractal analysis
• intersection of fractal sets
• pointwise $T_u^p$ regularity
• Hausdorff dimension
• wavelets
• Besov spaces
• baire genericity

2010 Mathematics Subject Classifications:

• 26A16
• 26B35
• 46E35
• 42C40
• 26A21

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.

Additional information

Funding

Mourad Ben Slimane and Borhen Halouani extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group No (RG-1435-063).