Embeddings between Triebel-Lizorkin Spaces on Metric Spaces Associated with Operators

We consider the general framework of a metric measure space satisfying the doubling volume property, associated with a non-negative self-adjoint operator, whose heat kernel enjoys standard Gaussian localization. We prove embedding theorems between Triebel-Lizorkin spaces associated with operators. Embeddings for non-classical Triebel-Lizorkin and (both classical and non-classical) Besov spaces are proved as well. Our result generalize the Euclidean case and are new for many settings of independent interest such as the ball, the interval and Riemannian manifolds.