Compactly Supported Frames for Spaces of Distributions Associated with Non-negative Self-adjoint Operators

A small perturbation method is developed and deployed to the construction of frames with compactly supported elements of small shrinking supports for Besov and Triebel-Lizorkin spaces in the general setting of a doubling metric measure space in presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. This allows, in particular, to develop compactly supported frames for Besov and Triebel-Lizorkin spaces on the sphere, the interval with Jacobi weights as well as on Lie groups, Riemannian manifolds, and various other settings. The compactly supported frames are utilized for the development of atomic Hardy spaces H A in the general setting of this article.