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Let $$\left( {\mathcal{Y},d,d\lambda } \right)$$ be (ℝ, |·|, µ), where |·| is the Euclidean distance, µ is a nonnegative Radon measure on ℝ satisfying the polynomial growth condition, or the Gauss measure metric space (ℝ, |·|, d), or the space ( S, d, ρ), where S ≡ ℝ ⋉ ℝ is the ( ax + b)-group, d is the left-invariant Riemannian metric and ρ is the right Haar measure on S with exponential growth. In this paper, the authors introduce and establish some properties of the atomic Hardy-type spaces $$\left\{ {X_s \left( \mathcal{Y} \right)} \right\}_{0 < s \leqslant \infty }$$ and the BMO-type spaces $$\left\{ {BMO\left( {\mathcal{Y}, s} \right)} \right\}_{0 < s \leqslant \infty }$$. Let H $$\left( \mathcal{Y} \right)$$ be the known atomic Hardy space and L $$\left( \mathcal{Y} \right)$$ the subspace of f ∈ L $$\left( \mathcal{Y} \right)$$ with integral 0. The authors prove that the dual space of X $$\left( \mathcal{Y} \right)$$ is $$BMO\left( {\mathcal{Y},s} \right)$$ when s ∈ (0,∞), X $$\left( \mathcal{Y} \right)$$ = H $$\left( \mathcal{Y} \right)$$ when s ∈ (0, 1], and X $$\left( \mathcal{Y} \right)$$ = L $$\left( \mathcal{Y} \right)$$ (or L $$\left( \mathcal{Y} \right)$$). As applications, the authors show that if T is a linear operator bounded from H $$\left( \mathcal{Y} \right)$$ to L $$\left( \mathcal{Y} \right)$$ and from L $$\left( \mathcal{Y} \right)$$ to L $$\left( \mathcal{Y} \right)$$, then for all r ∈ (1,∞) and s ∈ ( r,∞], T is bounded from X $$\left( \mathcal{Y} \right)$$ to the Lorentz space L $$\left( \mathcal{Y} \right)$$, which applies to the Calderón-Zygmund operator on (ℝ, |·|, µ), the imaginary powers of the Ornstein-Uhlenbeck operator on (ℝ, |·|, d) and the spectral operator associated with the spectral multiplier on ( S, d, ρ). All these results generalize the corresponding results of Sweezy, Abu-Shammala and Torchinsky on Euclidean spaces. [ABSTRACT FROM AUTHOR]

Under the assumption that the underlying measure is a non-negative Radon measure which only satisfies some growth condition and may not be doubling, we define the product of functions in the regular BMO and the atomic block H in the sense of distribution, and show that this product may be split into two parts, one in L and the other in some Hardy-Orlicz space. [ABSTRACT FROM AUTHOR]

Let μ be a non-negative Radon measure on ℝ d which satisfies only some growth conditions. Under this assumption, the boundedness in some Hardy-type spaces is established for a class of maximal Calderón–Zygmund operators and maximal commutators which are variants of the usual maximal commutators generated by Calderón–Zygmund operators and RBMO( μ) functions, where the Hardytype spaces are some appropriate subspaces, associated with the considered RBMO( μ) functions, of the Hardy space H 1( μ) of Tolsa. [ABSTRACT FROM AUTHOR]

The paper deals with multifractal quantities for some types of Radon measures, especially self–similar probability measures, and their relations to Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper, we establish the relationship between Hausdorif measures and Bessel capacities on any nilpotent stratified Lie group G (i.e., Carnot group). In particular, as a special corollary of our much more general results, we have the following theorem (see Theorems A and E in Section 1): Let Q be the homogeneous dimension of G. Given any set E ⊂ G, Bα,p(E) = 0 implies &hamlit;Q-αp+&epsiv;(E)= 0 for all &epsiv; > 0. On the other hand, &hamlit;Q-αp+&epsiv; (E) < ∞ implies Bα,p(E) = 0. Consequently, given any set E ⊂ G of Hausdorif dimension Q - d, where 0 > d > Q, Bα,p(E) = 0 holds if and only if αp ≤ d. A version of O. Frostman's theorem concerning Hausdorff measures on any homogeneous space is also established using the dyadlic decomposition on such a space (see Theorem 4.4 in Section 4). [ABSTRACT FROM AUTHOR]