We prove that a two sided sub-Gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.

Abstract not found

Let (M, g) be a complete Riemannian manifold with infinite volume, we denote by ∆ = ∆ g its Laplace operator, it has an unique self-adjoint extension on L 2 (M, dvolg) which is also denoted by ∆. The Green formula and the spectral theorem show that for any ϕ ∈ C ∞ 0 (M):

L p spectral theory and heat dynamics of locally symmetric spaces

We study the L p-spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces M = Γ\X with finite volume and arithmetic fundamental group Γ whose universal covering X is a symmetric space of non-compact type. We also show, how the obtained results for locally symmetric spaces can be generalized to manifolds with cusps of rank one. Keywords: Arithmetic lattices, heat semigroup on L p-spaces, Laplace-Beltrami operator, locally symmetric space, L p-spectrum, manifolds with cusps of rank one.

Abstract. The first purpose of this note is to provide a proof of the usual square function estimate on L p (Ω). It turns out to follow directly from a generic Mikhlin multiplier theorem obtained by Alexopoulos, which mostly relies on Gaussian bounds on the heat kernel. We also provide a simple proof of a weaker version of the square function estimate, which is enough in most instances involving dispersive PDEs. Moreover, we obtain, by a relatively simple integration by parts, several useful L p (Ω; H) bounds for the derivatives of the heat flow with values in a given Hilbert space H. 1.

Dynamics of the heat semigroup on symmetric spaces

A trace on the C ∗-algebra A of quasi-local operators on an open manifold is described, based on the results in [36]. It allows a description à la Novikov-Shubin [31] of the low frequency behavior of the Laplace-Beltrami operator. The 0-th Novikov-Shubin invariant defined in terms of such a trace is proved to coincide with a metric invariant, which we call asymptotic dimension, thus giving a large scale “Weyl asymptotics ” relation. Moreover, in analogy with the Connes-Wodzicki result [7, 8, 45], the asymptotic dimension d measures the singular traceability (at 0) of the Laplace-Beltrami operator, namely we may construct a (type II1) singular trace which is finite on the ∗-bimodule over A generated by ∆ −d/2. 1 Asymptotic dimension and Novikov-Shubin invariants 2 0 Introduction. The inspiration of this paper came from the idea of Connes ’ [8] of defining the dimension of a noncommutative compact manifold in terms of the Weyl asymptotics,