New Orlicz-Hardy Spaces Associated with Divergence Form Elliptic Operators

Let L be the divergence form elliptic operator with complex bounded mea-surable coefficients, ω the positive concave function on (0,∞) of strictly critical lowertype pω ∈ (0, 1] and ρ(t) = t/ω(t) for t ∈ (0,∞). In this paper, the authorsstudy the Orlicz-Hardy spaceHω,L(R) and its dual spaceBMOρ,L∗(R), where L∗denotes the adjoint operator of L in L(R). Several characterizations ofHω,L(R),including the molecular characterization, the Lusin-area function characterizationand the maximal function characterization, are established. The ρ-Carleson mea-sure characterization and the John-Nirenberg inequality for the spaceBMOρ,L(R)are also given. As applications, the authors show that the Riesz transform ∇L−1/2and the Littlewood-Paley g-function gL mapHω,L(R) continuously into L(ω). De-noteHω,L(R) by H pL(R ) when p ∈ (0, 1] and ω(t) = t for all t ∈ (0,∞). Theauthors further show that the Riesz transform ∇L−1/2 maps HL(R) into the clas-sical Hardy space H(R) for p ∈ ( nn+1 , 1] and the corresponding fractional integralL for certain γ > 0 maps HL(R ) continuously into HL(R ), where 0 < p < q ≤ 1and n/p− n/q = 2γ. All these results are new even when ω(t) = t for all t ∈ (0,∞)and p ∈ (0, 1).