On Dirichlet Spaces of Homogeneous Type Via Heat Kernel

TL;DR

This paper develops band limited frames for Dirichlet spaces of homogeneous type, enabling better analysis of function spaces like Besov spaces on various geometric structures such as Lie groups and Riemannian manifolds.

Contribution

It introduces well-localized band limited frames in Dirichlet spaces with heat kernel estimates, extending harmonic analysis tools to new geometric settings.

Findings

Constructed band limited frames with Gaussian heat kernel bounds

Decomposed Besov spaces using the new frames

Applied the framework to Lie groups and Riemannian manifolds

Abstract

This paper considers the properties of Dirichlet Spaces of Homogeneous type which consist of band limited functions that are nearly exponential localizations on $\mathbb{R}^k.$ This is a powerful tool in harmonic analysis and it makes various spaces of functions and distributions more approachable, utilizable and providing non-zero representation of natural function spaces, such as Besov space, on $\mathbb{R}^k$. Spheres and homogeneous spaces can also admit such frames on the intervals and balls. Here, we present mainly the band limited frames that are well-localized in the general setting of Dirichlet spaces of Homogeneous type which have doubling measure and a local scale-invariant Poincare inequality which generates heat kernels through the Gaussian bounds and H$\ddot{o}$lder's continuity. As an application of this build-up, band limited frames are generated in the context of Lie groups which are homogeneous in nature with polynomial volume growth, complete Riemannian manifolds with Ricci curvature bounded from below and admits the volume doubling property, together with other settings. In this general setting, decomposition of Besov spaces was done with the new frames.