Thierry Coulhon

On a complete non-compact Riemannian manifold satisfying the volume doubling property, we give conditions on the negative part of the Ricci curvature that ensure that, unless there are harmonic 1-forms, the Gaussian heat kernel upper estimate on functions transfers to one-forms. These conditions do no entail any constraint on the size of the Ricci curvature, only on its decay at infinity.

Wavelet bases and frames consisting of band limited functions of nearly exponential localization on Rd are a powerful tool in harmonic analysis by making various spaces of functions and distributions more accessible for study and utilization, and providing sparse representation of natural function spaces (e.g. Besov spaces) on Rd. Such frames are also available on the sphere and in more general homogeneous spaces, on the interval and ball. The purpose of this article is to develop band limited well-localized frames in the general setting of Dirichlet spaces with doubling measure and a local scale-invariant Poincar\'e inequality which lead to heat kernels with small time Gaussian bounds and H\"older continuity. As an application of this construction, band limited frames are developed in the context of Lie groups or homogeneous spaces with polynomial volume growth, complete Riemannian manifolds with Ricci curvature bounded from below and satisfying the volume doubling property, and other settings. The new frames are used for decomposition of Besov spaces in this general setting.

Project Euclid - mathematics and statistics online

Let M be a smooth Riemannian manifold that is the union of a compact part and a finite number of Euclidean ends, Rn∖B(0,R) for some R>0, each of which carries the standard metric. Our main result is that the Riesz transform on M is bounded from Lp(M)→Lp(M;T*M) for 12 implies a Hodge–de Rham interpretation of the Lp-cohomology in degree 1 and that the map from L2- to Lp-cohomology in this degree is injective

We prove an integral maximum principle for random walks on graphs, and give several applications to pointwise estimates of their transition probabilities, including the time-dependent case.