Hardy spaces H ^p over non-homogeneous metric measure spaces and their applications

Abstract

Let (X, d, μ) be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Let ρ ∈ (1, ∞), 0 < p ⩽ 1 ⩽ q ⩽ ∞, p ≠ q, γ ∈ [1, ∞) and ε ∈ (0, ∞). In this paper, the authors introduce the atomic Hardy space $\tilde H_{atb,\rho }^{p,q,\gamma } (\mu )$ and the molecular Hardy space $\tilde H_{mb,\rho }^{p,q,\gamma ,\varepsilon } (\mu )$ via the discrete coefficient $\tilde K_{B,S}^{(\rho ),p}$, and prove that the Calderón-Zygmund operator is bounded from $\tilde H_{mb,\rho }^{p,q,\gamma ,\delta } (\mu )$ (or $\tilde H_{atb,\rho }^{p,q,\gamma } (\mu )$) into L ^p(μ), and from $\tilde H_{atb,\rho (\rho + 1)}^{p,q,\gamma + 1} (\mu )$ into $\tilde H_{mb,\rho }^{p,q,\gamma ,\tfrac{1} {2}(\delta - \tfrac{v} {p} + v)} (\mu )$. The boundedness of the generalized fractional integral T _β (β ∈ (0, 1)) from $\tilde H_{mb,\rho }^{p_1 ,q,\gamma ,\theta } (\mu )$ (or $\tilde H_{atb,\rho }^{p_1 ,q,\gamma } (\mu )$) into $L^{p_2 } (\mu )$ with 1/p ₂ = 1/p ₁−β is also established. The authors also introduce the ρ-weakly doubling condition, with ρ ∈ (1, ∞), of the measure μ and construct a non-doubling measure μ satisfying this condition. If μ is ρ-weakly doubling, the authors further introduce the Campanato space $\mathcal{E}_{\rho ,\eta ,\gamma }^{\alpha ,q} (\mu )$ and show that $\mathcal{E}_{\rho ,\eta ,\gamma }^{\alpha ,q} (\mu )$ is independent of the choices of ρ, η, γ and q; the authors then introduce the atomic Hardy space $\hat H_{atb,\rho }^{p,q,\gamma } (\mu )$ and the molecular Hardy space $\hat H_{mb,\rho }^{p,q,\gamma ,\varepsilon } (\mu )$, which coincide with each other; the authors finally prove that $\hat H_{atb,\rho }^{p,q,\gamma } (\mu )$ is the predual of $\mathcal{E}_{\rho ,\rho ,1}^{1/p - 1,1} (\mu )$. Moreover, if μ is doubling, the authors show that $\mathcal{E}_{\rho ,\eta ,\gamma }^{\alpha ,q} (\mu )$ and the Lipschitz space Lip _{α, q} (μ) (q ∈ [1, ∞)), or $\hat H_{atb,\rho }^{p,q,\gamma } (\mu )$ and the atomic Hardy space H ^{p, q}_at (μ) (q ∈ (1, ∞]) of Coifman and Weiss coincide. Finally, if (X, d, μ) is an RD-spac (reverse doubling space) with μ(X) = ∞, the authors prove that $\tilde H_{atb,\rho }^{p,q,\gamma } (\mu )$, $\tilde H_{mb,\rho }^{p,q,\gamma ,\varepsilon } (\mu )$ and H ^{p, q}_at (μ) coincide for any q ∈ (1, 2]. In particular, when (X, d, μ):= (ℝ^D, |·|, dx) with dx being the D-dimensional Lebesgue measure, the authors show that spaces $\tilde H_{atb,\rho }^{p,q,\gamma } (\mu )$, $\tilde H_{mb,\rho }^{p,q,\gamma ,\varepsilon } (\mu )$, $\hat H_{atb,\rho }^{p,q,\gamma } (\mu )$ and $\hat H_{mb,\rho }^{p,q,\gamma ,\varepsilon } (\mu )$ all coincide with H ^p(ℝ^D) for any q ∈ (1, ∞).