In this paper, the author introduces the Morrey-Campanato spaces Lsp(X) and the spaces Csp(X) on spaces of homogeneous type including metric spaces and some fractals, and establishes some embedding theorems between these spaces under some restrictions and the Besov spaces and the Triebel-Lizorkin spaces. In particular, the author proves that Lsp(X) = Bs∞,∞(X) if 0 < s < ∞ and μ(X) < ∞. The author also introduces some new function spaces Asp(X) and Bsp(X) and proves that these new spaces when 0 < s < 1 and 1 < p < ∞ are just the Triebel-Lizorkin space Fsp;∞(X) if X is a metric space, and the spaces A1p(X) and B1p(X) when 1 < p≤∞ are just the Hajlasz-Sobolev spaces W1p (X). Finally, as an application, the author gives a new characterization of the Hajlasz-Sobolev spaces by making use of the sharp maximal function.

By using the discrete Calderón reproducing formulae, the author first establishes the boundedness of the Riesz-potential-type operator in homogeneous Besov and Triebel–Lizorkin spaces over spaces of homogeneous type. Then, by use of the T 1 theorems for these spaces, the author proves that this operator of Riesz potential type can be used as the lifting operator of these spaces.

In this paper, we introduce some new function spaces of Sobolev type on metric measure spaces. These new function spaces are defined by variants of Poincaré inequalities associated with generalized approximations of the identity, and they generalize the classical Sobolev spaces on Euclidean spaces. We then obtain two characterizations of these new Sobolev spaces including the characterization in terms of a variant of local sharp maximal functions associated with generalized approximations of the identity. For the well-known Hajłasz–Sobolev spaces on metric measure spaces, we also establish some new characterizations related to generalized approximations of the identity. Finally, we clarify the relations between the Sobolev-type spaces introduced in this paper and the Hajłasz–Sobolev spaces on metric measure spaces.

Let μ be a Radon measure on Rd which satisfies the growth condition only namely, there is a constant C > 0 such that for all x ∈ Rd,r > 0 and for some fixed 0 < n ≤d, μ(B(x, r)) ≤Crn, where B(x, r) is the ball centered at x and having radius r. In this paper, we first give a new atomic characterization of the Hardy space H1(μ) introduced by X. Tolsa. As applications of this new characterization, we establish the (H1(μ), L1,∞(μ)) estimate of the commutators generated by RBMO(μ) functions with the Calderon–Zygmund operators whose kernels satisfy only the size condition and a certain minimum regularity condition. Using this endpoint estimate and a new interpolation theorem for operators which is also established in this paper and has independent interest, we further obtain the Lp(μ) (1 < p < ∞) boundedness of these commutators.

Let (X, d,μ) be a space of homogeneous type in the sense of Coifman and Weiss. Assuming that μ satisfies certain estimates from below and there exists a suitable Calderon reproducing formula in L2(X), the authors establish a Lusin-area characterization for the atomic Hardy spaces Hpat(X) of Coifman and Weiss for p ∈ (p0, 1], where p0 = n/(n +∈1) depends on the “dimension” n of X and the “regularity” ∈1 of the Calderon reproducing formula. Using this characterization, the authors further obtain a Littlewood–Paley g∗λ-function characterization for Hp(X) when λ > n + 2n/p and the boundedness of Calderon–Zygmund operators on Hp(X). The results apply, for instance, to Ahlfors n-regular metric measure spaces, Lie groups of polynomial volume growth and boundaries of some unbounded model domains of polynomial type in CN.