Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators

Details Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators

2009-07-15

arxiv.org/abs/0907.2605v2

Mathematics

Functional Analysis

Integral Equations Operator Theory (to appear)

Scientific paper

Let $L$ be a divergence form elliptic operator with complex bounded measurable coefficients, $\omega$ a positive concave function on $(0,\infty)$ of strictly critical lower type $p_\omega\in (0, 1]$ and $\rho(t)={t^{-1}}/\omega^{-1}(t^{-1})$ for $t\in (0,\infty).$ In this paper, the authors introduce the generalized VMO spaces ${\mathop\mathrm{VMO}_ {\rho, L}({\mathbb R}^n)}$ associated with $L$, and characterize them via tent spaces. As applications, the authors show that $(\mathrm{VMO}_{\rho,L} ({\mathbb R}^n))^\ast=B_{\omega,L^\ast}({\mathbb R}^n)$, where $L^\ast$ 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 $\omega(t)=t^p$ 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 $\rho(t)\equiv 1$ and $({\mathop\mathrm{VMO}_{1, L}({\mathbb R}^n)})^\ast=H_{L^\ast}^1({\mathbb R}^n)$, where $H_{L^\ast}^1({\mathbb R}^n)$ was the Hardy space introduced by Hofmann and Mayboroda.

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