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arXiv · 1708.05476

Riesz transforms for bounded Laplacians on graphs

Li Chen·Thierry Coulhon·Bobo Hua
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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 gradient of functions on edges, 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 functions for all graphs with bounded Laplacians. This yields an analogue to Dungey's results in [Dungey08] while removing some additional assumptions. Coming back to the classical notion of gradient, we give a counterexample to the interpolation inequality 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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Li Chen, Thierry Coulhon, Bobo Hua. 2017-08-18. Riesz transforms for bounded Laplacians on graphs. https://arxiv.org/abs/1708.05476

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