Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators

Let L be a 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 /ω(t) for t ∈ (0,∞). In this paper, the authors introduce the generalized VMO spaces VMOρ,L(R) associated with L, and characterize them via tent spaces. As applications, the authors show that (VMOρ,L(R)) = Bω,L∗(R), where L∗ denotes the adjoint operator of L in L(R) and Bω,L∗(R) the Banach completion of the Orlicz-Hardy space Hω,L∗(R). Notice that ω(t) = t for all t ∈ (0,∞) and p ∈ (0, 1] is a typical example of positive concave functions satisfying the assumptions. In particular, when p = 1, then ρ(t) ≡ 1 and (VMO1,L(R)) = H L∗(R ), where H L∗(R ) was the Hardy space introduced by Hofmann and Mayboroda.