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Fractional differential operators, fractional Sobolev spaces and fractional variation on homogeneous Carnot groups

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Abstract

We establish a novel definition of the fractional gradient and divergence of order \(\alpha \in (0, 1)\) through the use of Riesz potential on homogeneous Carnot groups. We introduce and investigate the distributional fractional Sobolev space and the space of fractional BV functions in this context. Additionally, we provide a definition of fractional Caccioppoli sets on homogeneous Carnot groups and demonstrate their blow-up property, using similar methods as outlined in [13].

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Notes

  1. Throughout this study, the notation \(a\lesssim b\) means that there exists a universal constant \(C >0\) such that \(a \leqslant Cb \). And we write \(a \sim b\) if both \(a \lesssim b\) and \(b \lesssim a\) hold.

  2. In the sequel, if the identification of the base point is clear or not important, we often omit x in subscripts.

  3. Here and subsequently, \(\chi _A\) denotes the characteristic function of the set A.

  4. Since \(X_j h\) is smooth except at x and locally integrable, we can employ (5.6) through the use of a standard approximation argument.

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Acknowledgements

The authors are grateful to Prof. Hong-Quan Li of Fudan University for his helpful comments. T.Z. would like thank Zhen-Hang Bao for his patient help. T.Z. is partially supported by NSF of China (Grants No. 12271102). J.-X.Z. is partially supported by the National Key R &D Program of China (No. 2022YFA1006000, 2020YFA0712900) and NNSFC (11921001).

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  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Tong Zhang

  2. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    Jie-Xiang Zhu

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  1. Tong Zhang
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Zhang, T., Zhu, JX. Fractional differential operators, fractional Sobolev spaces and fractional variation on homogeneous Carnot groups. Fract Calc Appl Anal 26, 1786–1841 (2023). https://doi.org/10.1007/s13540-023-00173-0

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