Atomic and molecular decomposition of homogeneous spaces of distributions associated to non-negative self-adjoint operators

TL;DR

This paper develops atomic and molecular decompositions for homogeneous Besov and Triebel-Lizorkin spaces on metric measure spaces with non-negative self-adjoint operators, extending analysis tools in this setting.

Contribution

It introduces a class of almost diagonal operators and proves their algebra property, enabling atomic and molecular decompositions for these function spaces.

Findings

Established boundedness of spectral multipliers.

Proved that almost diagonal operators form an algebra.

Developed atomic and molecular decomposition theorems.

Abstract

We deal with homogeneous Besov and Triebel-Lizorkin spaces in the setting of a doubling metric measure space in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. The class of almost diagonal operators on the associated sequence spaces is developed and it is shown that this class is an algebra. The boundedness of almost diagonal operators is utilized for establishing smooth molecular and atomic decompositions for the above homogeneous Besov and Triebel-Lizorkin spaces. Spectral multipliers for these spaces are established as well.