Some equivalent characterizations for boundedness of maximal singular integral operators on spaces of homogeneous type are given via certain norm inequalities on John-Stromberg sharp maximal functions and without resorting the boundedness of these operators themselves. As a corollary, the results of Grafakos on Euclidean spaces are generalized to spaces of homogeneous type. Moreover, applications to maximal Monge-Ampere singular integral operators and maximal Nagel-Stein singular integral operators on certain specific smooth manifolds are also presented.

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.

Let Γ be a compact d-set in Rn with 0 < d ≤n, which includes various kinds of fractals. The author shows that the Besov spaces Bspq(Γ) defined by two different and equivalent methods, namely, via traces and quarkonial decompositions in the sense of Triebel are the same spaces as those obtained by regarding Γ as a space of homogeneous type when 0 < s < 1, 1 < p <∞　 and 1≤q≤∞.

An RD-space X is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X, or equivalently, that there exists a constant a0 > 1 such that for all x ∈ X and 0 < r < diam (X)=a0, the annulus B(x, a0r) n B(x, r) is nonempty, where diam (X) denotes the diameter of the metric space (X, d). An important class of RD-spaces is provided by Carnot-Caratheodory spaces with a doubling measure. In this paper, the authors introduce some spaces of Lipschitz type on RD-spaces, and discuss their relations with known Besov and Triebel-Lizorkin spaces and various Sobolev spaces.

Let p ∈ (0,1] and s≧[n(1=p-1)], where [n(1=p-1)] denotes the maximal integer no more than n(1=p-1). In this paper, the authors prove that a linear operator T extends to a bounded linear operator from the Hardy space Hp(Rn) to some quasi-Banach space B if and only if T maps all (p; 2; s)-atoms into uniformly bounded elements of B.