Riesz Transform for $$1\le p \le 2$$ 1 ≤ p ≤ 2 Without Gaussian Heat Kernel Bound

Abstract

We study the \(L^p\) boundedness of the Riesz transform as well as the reverse inequality on Riemannian manifolds and graphs under the volume doubling property and a sub-Gaussian heat kernel upper bound. We prove that the Riesz transform is then bounded on \(L^p\) for \(1<p<2\), which shows that Gaussian estimates of the heat kernel are not a necessary condition for this. In the particular case of Vicsek manifolds and graphs, we show that the reverse inequality does not hold for \(1<p<2\). This yields a full picture of the ranges of \(p\in (1,+\infty )\) for which respectively the Riesz transform is \(L^p\)-bounded and the reverse inequality holds on \(L^p\) on such manifolds and graphs. This picture is strikingly different from the Euclidean one.