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Pointwise Rectangular Lipschitz Regularities for Fractional Brownian Sheets and Some Sierpinski Selfsimilar Functions

Author

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  • Mourad Ben Slimane

    (Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
    These authors contributed equally to this work.)

  • Moez Ben Abid

    (Ecole Supérieure des Sciences et Technologie de Hammam Sousse, Université de Sousse, Sousse 4011, Tunisia
    These authors contributed equally to this work.)

  • Ines Ben Omrane

    (Department of Mathematics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, Riyadh 11623, Saudi Arabia
    These authors contributed equally to this work.)

  • Mohamad Maamoun Turkawi

    (Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
    These authors contributed equally to this work.)

Abstract

We consider pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities for continuous functions f on the unit cube I 2 in R 2 . Firstly, we provide characterizations by simple estimates on the decay rate of the coefficients (resp. leaders) of the expansion of f in the rectangular Schauder system, near the point considered. We deduce that pointwise rectangular Lipschitz regularity yields pointwise level coordinate axes Lipschitz regularities. As an application, we refine earlier results in Ayache et al. (Drap brownien fractionnaire. Potential Anal. 2002, 17 , 31–43) and Kamont (On the fractional anisotropic Wiener field. Probab. Math. Statist. 1996 , 16 , 85–98), where uniform rectangular Lipschitz regularity of the trajectories of the fractional Brownian sheet over the total I 2 (or any cube) was considered. Actually, we prove that fractional Brownian sheets are pointwise rectangular and level coordinate axes monofractal. On the opposite, we construct a class of Sierpinski selfsimilar functions that are pointwise rectangular and level coordinate axes multifractal.

Suggested Citation

  • Mourad Ben Slimane & Moez Ben Abid & Ines Ben Omrane & Mohamad Maamoun Turkawi, 2020. "Pointwise Rectangular Lipschitz Regularities for Fractional Brownian Sheets and Some Sierpinski Selfsimilar Functions," Mathematics, MDPI, vol. 8(7), pages 1-18, July.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:7:p:1179-:d:386079
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    Cited by:

    1. Ben Slimane, Mourad & Alzughaibi, Imtithal & Algahtani, Obaid, 2024. "On Lp rectangular multifractal multivariate functions," Chaos, Solitons & Fractals, Elsevier, vol. 183(C).

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