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 a positive concave function on (0, ∞) of strictly critical lower type p ω ∈(0, 1] and ρ(t) = t ?1/ω ?1(t ?1) for ${tin (0,infty).}Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω a positive concave function on (0, ∞) of strictly critical lower type p ω ∈(0, 1] and ρ(t) = t −1/ω −1(t −1) for t ? (0,￥).{tin (0,infty).} In this paper, the authors introduce the generalized VMO spaces VMOr, L(mathbb Rn){{mathop{rm VMO}_ {rho, L}({mathbb R}^n)}} associated with L, and characterize them via tent spaces. As applications, the authors show that (VMOr,L (mathbb Rn))*=Bw,L*(mathbb Rn){({rm VMO}_{rho,L} ({mathbb R}^n))^ast=B_{omega,L^ast}({mathbb R}^n)}, where L * denotes the adjoint operator of L in L2(mathbb Rn){L^2({mathbb R}^n)} and Bw,L*(mathbb Rn){B_{omega,L^ast}({mathbb R}^n)} the Banach completion of the Orlicz–Hardy space Hw,L*(mathbb Rn){H_{omega,L^ast}({mathbb R}^n)}. Notice that ω(t) = t p for all t ? (0,￥){tin (0,infty)} and p ? (0,1]{pin (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(mathbb Rn))*=HL*1(mathbb Rn){({mathop{rm VMO}_{1, L}({mathbb R}^n)})^ast=H_{L^ast}^1({mathbb R}^n)}, where HL*1(mathbb Rn){H_{L^ast}^1({mathbb R}^n)} was the Hardy space introduced by Hofmann and Mayboroda.