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.

We establish the bounds of Marcinkiewicz integrals associated to surfaces of revolution generated by two polynomial mappings on Triebel–Lizorkin spaces and Besov spaces when their integral kernels are given by functions $\unicode[STIX]{x1D6FA}\in H^{1}(\text{S}^{n-1})\cup L(\log ^{+}L)^{1/2}(\text{S}^{n-1})$. Our main results represent improvements as well as natural extensions of many previously known results.

A class of generalized Marcinkiewicz integral operators is introduced, and, under rather weak conditions on the integral kernels, the boundedness of such operators on ${{L}^{p}}$ and Triebel–Lizorkin spaces is established.

New spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel–Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Hajłasz–Sobolev spaces, Korevaar–Schoen–Sobolev spaces, Newtonian Sobolev space and Cheeger–Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy–Littlewood–Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.

AMS 2000 Mathematics subject classification: Primary 42B35. Secondary 46E35; 58J35; 43A99