Let μ be a non-negative Radon measure on Rd which only satisfies some growth condition. In this paper, the authors obtain the boundedness of Calderon-Zygmund operators in the Hardy space H1(μ).

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).

Suppose that μ is a Radon measure on Rd, which may be nondoubling. The only condition on μ is the growth condition, namely, there is a constant C0 > 0 such that for all x ∈ supp (μ) and r > 0,μ(B(x, r)) ≤ C0rn,where 0 < n ≤ d. In this paper, the authors establish a theory of Besov spaces Bspq(μ) for 1 ≤ p, q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure μ, C0, n and d. The method used to define these spaces is new even for the classical case. As applications, the lifting properties of these spaces by using the Riesz potential operators and the dual spaces are obtained.

In this paper, we prove the product Hp boundedness of Calderon-Zygmund operators which were considered by Fefferman and Stein. The methods used in this paper are new even for the classical Hp boundedness of Calderon-Zygmund operators, namely, using some subtle estimates together with the Hp−Lp boundedness of product vector valued Calderon-Zygmund operators.

In this paper, the authors establish the inhomogeneous Plancherel-Polya inequalities on spaces of homogeneous type by use of the inhomogeneous discrete Calderon reproducing formulas. As an application, the authors prove that the Lebesgue norms of the inhomogeneous Littlewood-Paley g-function and S-function on spaces of homogeneous type are equivalent. All results are new even for Rn.