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.

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.

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.

Under the assumption that the Radon measure μ on Rd satisfies only some growth condition, the authors prove that, for the maximal singular integral operator associated with a singular integral whose kernel only satisfies a standard size condition and the Hormander condition, its boundedness in Lebesgue spaces Lp(μ) for any p ∈ (1,∞) is equivalent to its boundedness from L1(μ) into weak L1(μ). As an application, the authors verify that if the truncated singular integral operators are bounded on L2(μ) uniformly, then the associated maximal singular integral operator is also bounded on Lp(μ) for any p ∈ (1,∞).

Let (X,Э,μ)d,θ be a space of homogeneous type, i.e. X is a set, Эis a quasi-metric on X with the property that there are constants θ∈ (0，1] and C0 > 0 such that for all x; x1; y∈X, ︱Э(x,y)- Э(x1, y)︱≤C0Э(x,x1) θ[Э(x,y) +Э(x1, y)]1-Э, and μ is a nonnegative Borel regular measure on X such that for some d > 0 and all x ∈ X, μ(｛y∈X: Э(x,y)＜r｝～rd. LetЭ∈ (0,θ], ︱s︱ <ε and max｛d/(d + ε); d/(d + s + ε)｝ < q ≤∞. The author introduces new inhomogeneous Triebel-Lizorkin spaces Fs∞q(X) and establishes their frame characterizations by first establishing a Plancherel-Polya-type inequality related to the norm ‖·‖Fs∞q (X), which completes the theory of function spaces on spaces of homogeneous type. Moreover, the author establishes the connection between the space Fs∞q (X) and the homogeneous Triebel-Lizorkin space Fs∞q (X). In particular, he proves that bmo(X) coincides with F F0∞q(X).