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Fractal surfaces in Hölder and Sobolev spaces

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Abstract

Following the construction of fractal surfaces due to Ruan and Xu (Bulletin of the Australian Mathematical Society 91:435–446, 2015) and the theory of \(\alpha\)-fractal functions due to Navascués (Zeitschrift fur Analysis und ihre Anwendungen 25:401–418, 2005), we show that bivariate \(\alpha\)-fractal functions (or fractal surfaces) belong to the well-known Hölder spaces and Sobolev spaces under certain assumptions on the base function and scaling function used in the construction. We also define a fractal operator on the Hölder spaces and then study some properties of it. In the end, we provide an alternate approach for defining a fractal operator on Sobolev spaces.

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Funding

The first author thanks IIIT Allahabad (Ministry of Education, India) for financial support in the form of a Junior Research Fellowship.

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Authors and Affiliations

  1. Department of Applied Sciences, IIIT Allahabad, Prayagraj, 211015, India

    Ekta Agrawal & Saurabh Verma

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  1. Ekta Agrawal
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  2. Saurabh Verma
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All authors contributed equally to this manuscript.

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Correspondence to Saurabh Verma.

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Communicated by S Ponnusamy.

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Agrawal, E., Verma, S. Fractal surfaces in Hölder and Sobolev spaces. J Anal 32, 1161–1179 (2024). https://doi.org/10.1007/s41478-023-00672-6

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  • DOI: https://doi.org/10.1007/s41478-023-00672-6

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