Orlicz-Hardy spaces associated with operators

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 of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞). Let ω on (0,∞) be of upper type 1 and of critical lower type p _o (ω) ∂ (n/(n+θ(L)),1] and ρ(t) = t ^t1/ω ⁻¹(t ⁻¹) for t ∈ (0,∞). We introduce the Orlicz-Hardy space H _{ω, L} (ℝⁿ) and the BMO-type space BMO _{ρ, L} (ℝⁿ) and establish the John-Nirenberg inequality for BMO _{ρ, L} (ℝⁿ) functions and the duality relation between H _{ω, L} ((ℝⁿ) and BMO _{ρ, L*} (ℝⁿ), where L* denotes the adjoint operator of L in L ² (ℝⁿ). Using this duality relation, we further obtain the ρ-Carleson measure characterization of BMO _{ρ, L*} (ℝⁿ) and the molecular characterization of H _{ω, L} (ℝⁿ); the latter is used to establish the boundedness of the generalized fractional operator L _ρ ^−γ from H _{ω, L} (ℝⁿ) to H _L ¹ (ℝⁿ) or L ^q (ℝⁿ) with certain q > 1, where H _L (ℝⁿ) is the Hardy space introduced by Auscher, Duong and McIntosh. These results generalize the existing results by taking ω(t) = t ^p for t ∈ (0,∞) and p ∈ (n/(n + θ(L)), 1].