Riesz transforms for bounded Laplacians on graphs

TL;DR

This paper investigates the boundedness of Riesz transforms on graphs with bounded Laplacians, establishing new $ ext{L}^p$ estimates, interpolation inequalities, and boundedness results, while also providing counterexamples and conditions for boundedness.

Contribution

It introduces a proper notion of gradient on edges, proves $ ext{L}^p$ estimates for heat semigroups, and extends boundedness results for Riesz transforms on graphs with bounded Laplacians.

Findings

Established $ ext{L}^p$ estimates for the gradient of heat semigroups for $p o(1,2]$

Proved $ ext{L}^p$ boundedness of Littlewood-Paley-Stein functions for all graphs with bounded Laplacians

Provided counterexamples to the boundedness of Riesz transforms for $1<p<2$

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.