Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators

TL;DR

This paper introduces generalized VMO spaces linked to divergence form elliptic operators, characterizes them via tent spaces, and explores their duality with Banach completions of Orlicz-Hardy spaces, extending classical harmonic analysis results.

Contribution

It defines new VMO spaces associated with elliptic operators and characterizes their duals using tent space techniques, broadening the understanding of function spaces related to PDEs.

Findings

Dual space of VMO_{ρ,L} is the Banach completion of Orlicz-Hardy space.

Special case when p=1 recovers classical Hardy space duality.

Characterization of VMO spaces via tent spaces for 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 ${\mathop\mathrm{VMO}_ {\rho, L}({\mathbb R}^n)}$ 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.