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.

In this article we prove an embedding theorem for inhomogeneous Besov and Triebel–Lizorkin spaces on $\text{RD}$-spaces. The crucial idea is to use the geometric density condition on the measure.

In this article, we establish a new atomic decomposition for $f\,\in \,L_{w}^{2}\,\bigcap \,H_{w}^{p}$, where the decomposition converges in $L_{w}^{2}$-norm rather than in the distribution sense. As applications of this decomposition, assuming that $T$ is a linear operator bounded on $L_{w}^{2}$ and $0\,<\,p\,\le \,1$, we obtain (i) if $T$ is uniformly bounded in $L_{w}^{p}$-norm for all $w-p$-atoms, then $T$ can be extended to be bounded from $H_{w}^{p}$ to $L_{w}^{2}$; (ii) if $T$ is uniformly bounded in $H_{w}^{p}$-norm for all $w-p$-atoms, then $T$ can be extended to be bounded on $H_{w}^{p}$; (iii) if $T$ is bounded on $H_{w}^{p}$, then $T$ can be extended to be bounded from $H_{w}^{p}$ to $L_{w}^{2}$.

We obtain an atomic decomposition for weighted Triebel–Lizorkin spaces on spaces of homogeneous type, using the area function, the discrete Calderón reproducing formula and discrete sequence spaces.

Let (X, ρ, μ)d, θ be a space of homogeneous type with d < 0 and θ ∈ (0, 1], b be a para-accretive function, ε ∈ (0, θ], ∣s∣ > ∈ and a0 ∈ (0, 1) be some constant depending on d, ∈ and s. The authors introduce the Besov space bBspq (X) with a0 > p ≧ ∞, and the Triebel-Lizorkin space bFspq (X) with a0 > p > ∞ and a0 > q ≧∞ by first establishing a Plancherel-Pôlya-type inequality. Moreover, the authors establish the frame and the Littlewood-Paley function characterizations of these spaces. Furthermore, the authors introduce the new Besov space b−1 Bs (X) and the Triebel-Lizorkin space b−1 Fspq (X). The relations among these spaces and the known Hardy-type spaces are presented. As applications, the authors also establish some real interpolation theorems, embedding theorems, T b theorems, and the lifting property by introducing some new Riesz operators of these spaces.