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Riesz transforms for bounded Laplacians on graphs

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Abstract

We study several problems related to the \(\ell ^p\) boundedness of Riesz transforms for graphs endowed with so-called bounded Laplacians. Introducing a proper notion of the gradient of a function, we prove for \(p\in (1,2]\) an \(\ell ^p\) estimate for the gradient of the continuous time heat semigroup, an \(\ell ^p\) interpolation inequality as well as the \(\ell ^p\) boundedness of the modified Littlewood–Paley–Stein function for a graph with bounded Laplacian. This yields an analogue to Dungey’s results in [21] while removing some additional assumptions. Coming back to the classical notion of the gradient, we give a counterexample to the interpolation inequality and hence to the boundedness of Riesz transforms for bounded Laplacians for \(1<p<2\). Finally, we prove the boundedness of the Riesz transform for \(1< p<\infty \) under the assumption of positive spectral gap.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, 341 Mansfield Road, Storrs, CT, 06269-1009, USA

    Li Chen

  2. Université de Cergy-Pontoise, Cergy, France

    Thierry Coulhon

  3. School of Mathematical Sciences, LMNS, Fudan University, Shanghai, 200433, China

    Bobo Hua

  4. Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Bobo Hua

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  1. Li Chen
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  2. Thierry Coulhon
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  3. Bobo Hua
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Correspondence to Bobo Hua.

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Li Chen was supported in part by ICMAT Severo Ochoa project SEV-2011-0087 and she acknowledges that the research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC agreement no. 615112 HAPDEGMT. Bobo Hua is supported by NSFC (China), grant no. 11401106.

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Chen, L., Coulhon, T. & Hua, B. Riesz transforms for bounded Laplacians on graphs. Math. Z. 294, 397–417 (2020). https://doi.org/10.1007/s00209-019-02253-5

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