Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators

Abstract

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 ${t\in (0,\infty).}$ In this paper, the authors introduce the generalized VMO spaces ${{\mathop{\rm VMO}_ {\rho, L}({\mathbb R}^n)}}$ associated with L, and characterize them via tent spaces. As applications, the authors show that ${({\rm VMO}_{\rho,L} ({\mathbb R}^n))^\ast=B_{\omega,L^\ast}({\mathbb R}^n)}$, where L ^* denotes the adjoint operator of L in ${L^2({\mathbb R}^n)}$ and ${B_{\omega,L^\ast}({\mathbb R}^n)}$ the Banach completion of the Orlicz–Hardy space ${H_{\omega,L^\ast}({\mathbb R}^n)}$. Notice that ω(t) = t ^p for all ${t\in (0,\infty)}$ and ${p\in (0,1]}$ is a typical example of positive concave functions satisfying the assumptions. In particular, when p = 1, then ρ(t) ≡ 1 and ${({\mathop{\rm VMO}_{1, L}({\mathbb R}^n)})^\ast=H_{L^\ast}^1({\mathbb R}^n)}$, where ${H_{L^\ast}^1({\mathbb R}^n)}$ was the Hardy space introduced by Hofmann and Mayboroda.