LAUSR

Abstract Let $\ell \in \mathbb{N}$ and $\alpha \in (0,2\ell )$ . In this article, the authors establish equivalent characterizations of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaces via the sequence ${{\{f-{{B}_{\ell ,{{2}^{-k}}}}f\}}_{k}}$ consisting of the difference between $f$ and the ball average ${{B}_{\ell ,{{2}^{-k}}}}f$ . These results lead to the introduction of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaceswith any positive smoothness order onmetricmeasure spaces. As special cases, the authors obtain a new characterization of Morrey–Sobolev spaces and ${{\text{Q}}_{\alpha }}$ spaces with $\alpha \in (0,1)$ , which are of independent interest.