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

Abstract

We study the L p boundedness of 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 ∈ (1, +∞) 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.

About the $L^2$ analyticity of Markov operators on graphs

“…The last characterization of (1) given in Theorem 1.7 is a practical one, which can be used to check quickly the analyticity of a given graph. As an application, we slightly improve a result in [18,11,4] that establishes the L p boundedness of the Riesz transform for 1 < p < 2 on graphs under Gaussian or sub-Gaussian estimates.…”

“…The second section is dedicated to the proof of the main result. Finally, in the last section, we present an application of our main result to weaken the assumptions used in [18,4].…”

“…With Gaussian pointwise estimates on the Markov kernel, the following result is obtained by Russ (see [18,Theorem 1]). The theorem in his complete form can be found in [4,Theorem 4.2] (see also [11]). Theorem 3.3.…”

New Riemannian manifolds with Lp-unbounded Riesz transform for p > 2

“…One is naturally led to consider also the 'reverse' estimate (R R p ) : (R p ) is known to follow from the volume doubling property and Gaussian or sub-Gaussian heat kernel upper estimates [9,10] (see also [15] for examples which do not satisfy such kernel estimates). The volume doubling property and an appropriately scaled L 2 -Poincaré inequality imply (R p ) for some p > 2 [2].…”

Gundy–Varopoulos martingale transforms and their projection operators on manifolds and vector bundles

“…Another widely-adopted approach to study Riesz transform in various geometric settings is to use the Calderón-Zygmund theory for which heat kernel estimates play an essential role. See, for instance, the setting of Riemannian manifolds in [19,27,20,29,28,3,4,18,17] and the references therein, especially [3, Section 1.3] with a quite complete list of previous results, and the setting of Lie groups, or manifolds with sub-elliptic operators in [1,21,12,11]. We note that, unlike the martingale approach, the Calderón-Zygmund theory do not give constants which are independent of dimension.…”