Publication Cover
Applicable Analysis
An International Journal
Volume 101, 2022 - Issue 16
76
Views
0
CrossRef citations to date
0
Altmetric
Research Article

Multivariate multifractal formalism for simultaneous pointwise (Tuipi)i regularities

, , &
Pages 5824-5849 | Received 27 Jun 2019, Accepted 17 Mar 2021, Published online: 05 Apr 2021
 

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 (Tuipi)i=1,,L regularities. The Tup regularity for 1p< was first introduced in order to better study elliptic partial differential equations where the natural function space setting is Lp 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,,LBtisi,ri(Rd), si,ti,ri>0,1pi such that sidti>dpi. We therefore extend previous results where only cases (pi= and sidti>0 for all i) and (pi=, si=dti and ri 1 for all i) for simultaneous pointwise Hölder regularities have been proved.

2010 Mathematics Subject Classifications:

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).

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.