Applications of Orlicz-Hardy spaces associated with operators satisfying Poisson estimates

Abstract

Let L be a linear operator in L ²(ℝⁿ) and generate an analytic semigroup {e ^−tL} _t⩾0 with kernel satisfying an upper bound estimate of Poisson type, whose decay is measured by θ(L) ∈ (0,∞). Let Φ be a positive, continuous and strictly increasing function on (0,∞), which is of strictly critical lower type p _Φ ∈ (n/(n + θ(L)), 1]. Denote by H _Φ,L (ℝⁿ) the Orlicz-Hardy space introduced in Jiang, Yang and Zhou’s paper in 2009. If Φ is additionally of upper type 1 and subadditive, the authors then show that the Littlewood-Paley g-function gL maps H _Φ,L (ℝⁿ) continuously into L ^Φ(ℝⁿ) and, moreover, the authors characterize H _Φ,L (ℝⁿ) in terms of the Littlewood-Paley g ^* _λ -function with λ ∈ (n(2/p _Φ + 1),∞). If Φ is further slightly strengthened to be concave, the authors show that a generalized Riesz transform associated with L is bounded from H _Φ,L (ℝⁿ) to the Orlicz space L ^Φ(ℝⁿ) or the Orlicz-Hardy space H _Φ(ℝⁿ); moreover, the authors establish a new subtle molecular characterization of H _Φ,L (ℝⁿ) associated with L and, as applications, the authors then show that the corresponding fractional integral L ^−γ for certain γ ∈ (0,∞) maps H _Φ,L (ℝⁿ) continuously into $H_{\tilde \Phi ,L} (\mathbb{R}^n )$, where $\tilde \Phi$ satisfies the same properties as Φ and is determined by Φ and γ, and also that L has a bounded holomorphic functional calculus in H _Φ,L (ℝⁿ). All these results are new even when Φ(t) ≡ t ^p for all t ∈ (0,∞) and p ∈ (n/(n + θ(L)), 1].