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.