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Hardy and Rellich inequalities for anisotropic p-sub-Laplacians

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Abstract

In this paper we establish the subelliptic Picone type identities. As consequences, we obtain Hardy and Rellich type inequalities for anisotropic p-sub-Laplacians which are operators of the form

$$\begin{aligned} {\mathcal {L}}_{p}f:= \sum _{i=1}^{N} X_i\left( |X_i f|^{p_i-2} X_i f \right) ,\quad 1<p_i<\infty , \end{aligned}$$

where \(X_i\), \(i=1,\ldots , N\), are the generators of the first stratum of a stratified (Lie) group. Moreover, analogues of Hardy type inequalities with multiple singularities and many-particle Hardy type inequalities are obtained on stratified groups.

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Acknowledgements

M. Ruzhansky was supported by the EPSRC Grant EP/R003025/1, by the Leverhulme Research Grant RPG-2017-151, and by the FWO Odysseus Grant. B. Sabitbek was supported by the MESRK target program BR05236656 and the Nazarbayev University SPG Grant SST 2018040. D. Suragan was supported in parts by the Nazarbayev University SPG Grant. No new data was collected or generated during the course of this research.

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Authors and Affiliations

  1. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, 9000, Ghent, Belgium

    M. Ruzhansky

  2. School of Mathematical Sciences, Queen Mary University of London, London, UK

    M. Ruzhansky

  3. Institute of Mathematics and Mathematical Modeling, 125 Pushkin Str., 050010, Almaty, Kazakhstan

    B. Sabitbek

  4. Al-Farabi Kazakh National University, Almaty, Kazakhstan

    B. Sabitbek

  5. Department of Mathematics, Nazarbayev University, 53 Kabanbay Batyr Ave, 010000, Astana, Kazakhstan

    D. Suragan

Authors
  1. M. Ruzhansky
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  2. B. Sabitbek
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  3. D. Suragan
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Correspondence to D. Suragan.

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Communicated by Maria Alessandra Ragusa.

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Ruzhansky, M., Sabitbek, B. & Suragan, D. Hardy and Rellich inequalities for anisotropic p-sub-Laplacians. Banach J. Math. Anal. 14, 380–398 (2020). https://doi.org/10.1007/s43037-019-00011-7

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  • DOI: https://doi.org/10.1007/s43037-019-00011-7

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