New Orlicz–Hardy spaces associated with divergence form elliptic operators

Let L be the divergence form elliptic operator with complex bounded measurable coefficients, ω the positive concave function on (0,∞) of strictly critical lower type pω ∈ (0, 1] and ρ(t) = t−1/ω−1(t−1) for t ∈ (0,∞). In this paper, the authors study the Orlicz–Hardy space Hω,L(Rn) and its dual space BMOρ,L∗ (Rn), where L∗ denotes the adjoint operator of L in L2(Rn). Several characterizations of Hω,L(Rn), including the molecular characterization, the Lusin-area function characterization and the maximal function characterization, are established. The ρ-Carleson measure characterization and the John– Nirenberg inequality for the space BMOρ,L(Rn) are also given. As applications, the authors show that the Riesz transform ∇L−1/2 and the Littlewood–Paley g-function gL map Hω,L(Rn) continuously into L(ω). The authors further show that the Riesz transform ∇L−1/2 maps Hω,L(Rn) into the classical Orlicz–Hardy space Hω(Rn) for pω ∈ ( n n+1 , 1] and the corresponding fractional integral L−γ for certain γ >0 maps Hω,L(Rn) continuously into H ω,L(Rn), where ω is determined by ω and γ , and satisfies the same property as ω. All these results are new even when ω(t) = tp for all t ∈ (0,∞) and p ∈ (0, 1). © 2009 Elsevier Inc. All rights reserved.