Atomic Hardy-type spaces between \( H^1\) and \( L^1\) on metric spaces with non-doubling measures
Let \(({\mathcal Y}, d, d\lambda)\) be one of the following three metric spaces with non-doubling measures: (1) \((\mathbb{R^N}, |\cdot|, \mu)\) with \( |\cdot|\) the Euclidean distance and \(\mu\) a non-negative Radon measure on \(\mathbb{R^N}\), satisfying a polynomial growth condition. (2) \((\mathbb{R^N}, |\cdot|, d\gamma)\) the Gauss measure metric space. (3) the space \((S,d,\rho)\) where \(S=\mathbb{R^N}\times \mathbb{R^+}\) is the affine group endowed with the product \((x,a)\cdot(x',a')=(x+ax',aa')\), \(d\) is the left-invariant Riemannian metric and \(\rho\) is the right Haar measure on \(S\). The paper deals with the study of some properties of the atomic Hardy spaces \(\{X_s({\mathcal Y})\}_{0<s\leq \infty}\) and the BMO-type spaces \(\{\text{BMO}({\mathcal Y},s)\}_{0<s< \infty}\). In particular, it is proved that the dual space of \(X_s({\mathcal Y})\) is \(\text{BMO}({\mathcal Y},s)\) when \(s\in (0,\infty)\), \(X_s({\mathcal Y})=H^1({\mathcal Y})\) when \(s\in (0,1]\), and \(X_{\infty}({\mathcal Y})=L^1\) or \(L_0^1({\mathcal Y})\), subspace of functions in \(L^1({\mathcal Y})\) with integral 0. As an application, it is shown that a sublinear operator \(T\), which is bounded from \(L^1({\mathcal Y})\) to \(L^{1,\infty}({\mathcal Y})\), is bounded from \(X_s({\mathcal Y})\) to \(L^1({\mathcal Y})\) if and only if \(T\) maps all \((p,s)\)-atoms with \(p\in (1,\infty)\) into uniformly bounded elements of \(L^1({\mathcal Y})\) . Interpolation results in this context are also proved. These results about the boundedness of linear and sublinear operators are applied to the Calderón-Zygmund operators on \((\mathbb{R^N}, |\cdot|, \mu)\), the imaginary powers of the Ornstein-Uhlenbeck operator on \((\mathbb{R^N}, |\cdot|, d\gamma)\) and the spectral operator associated with the spectral multiplier on \((S,d,\rho)\). All these results are generalization of the corresponding results obtained for Euclidean spaces by several authors such as \textit{C. Sweezy} [Proc. Am. Math. Soc. 132, No. 12, 3599--3606 (2004; Zbl 1045.42017)] or \textit{W. Abu-Shammala} and \textit{A. Torchinsky}, [Proc. Am. Math. Soc. 135, No. 9, 2839--2843 (2007; Zbl 1278.42023), Proc. Am. Math. Soc. 136, No. 5, 1743--1748 (2008; Zbl 1138.42009)].
scientific article; zbMATH DE number 6145916
scientific article