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, $\omega$ 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 ${\mathrm{VMO}}_{\rho ,L}({\mathbb{R}}^n)$ 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 $({\mathrm{VMO}}_{1,L}({\mathbb{R}}^n){)}^{\ast }=H_{L^{\ast }}^1({\mathbb{R}}^n)$, where $H_{L^{\ast }}^1({\mathbb{R}}^n)$ was the Hardy space introduced by Hofmann and Mayboroda.

Integral Equations Operator Theory (to appear)

References in corpus (1)

• Correction to ``Hardy and BMO spaces associated to divergence form elliptic operators"