The paper considers the heat kernel KΩ(t, x, y) of the operator – Δ on a proper Euclidean domain Ω, with Dirichlet boundary conditions. A general pointwise lower bound for KΩ, which is valid for t larger than a suitable t0(x,y), is proved (the short-time behaviour being well understood). The resulting non-Gaussian bounds describe simultaneously both the case of bounded domains and the case, modelled on the half-space example, of domains which satisfy a twisted infinite internal cone condition. Bounds for the Green's function are given as well.

The semi-classical asymptotic behaviour of the Riesz means of a distribution of eigenvalues is investigated at a non-critical energy level. For Schrödinger type operators, the second term related to the periodic trajectories of the classical Hamiltonian is obtained. This oscillating term is explained for Riesz means of order 1 with the aim of discussing a Lieb–Thirring conjecture.

Let {Xi}i[ges ]1 be a sequence of independent random variables taking the values ±1 with the probability ½, and let us set Sn = X1 + X2 +…+ Xn. A classical theorem of Hardy and Littlewood (1914) says that, for any C > 0 and for all n large enough, we have

formula here

with probability 1. In 1924, Khinchin showed that (1) can be replaced by a sharper inequality

formula here

for any ε > 0. In view of Khinchin's result, inequality (1) has long been considered as one of a rather historical value. However, the recent results on Brownian motion on Riemannian manifolds give a new insight into it. In this paper, we show that an analogue of (1), for the Brownian motion on Riemannian manifolds of the polynomial volume growth, is sharp and, therefore, cannot be replaced by an analogue of (2).