Vanishing Mean Oscillation Spaces Associated with Operators Satisfying Davies-Gaffney Estimates

TL;DR

This paper introduces a new vanishing mean oscillation space linked to operators satisfying Davies-Gaffney estimates on metric measure spaces, characterizes it via tent spaces, and explores its duality with Orlicz-Hardy spaces.

Contribution

It defines the generalized VMO space associated with such operators and establishes its characterization and duality properties, advancing the understanding of function spaces in harmonic analysis.

Findings

Defined the generalized VMO space ${ m VMO}_{ ho,L}({ m extbf{X}})$.

Characterized ${ m VMO}_{ ho,L}({ m extbf{X}})$ via tent spaces.

Proved the duality ${ m VMO}_{ ho,L}({ m extbf{X}})^* = B_{ ext{ extit{ extbf{ extPhi}}},L^*}({ m extbf{X}})$.

Abstract

Let $(\mathcal{X}, d, \mu)$ be a metric measure space, $L$ a linear operator which has a bounded $H_\infty$ functional calculus and satisfies the Davies-Gaffney estimate, $\Phi$ a concave function on $(0,\infty)$ of critical lower type $p_\Phi^-\in(0,1]$ and $\rho(t)\equiv t^{-1}/\Phi^{-1}(t^{-1})$ for all $t\in(0,\infty)$. In this paper, the authors introduce the generalized VMO space ${\mathrm {VMO}}_{\rho,L}({\mathcal X})$ associated with $L$, and establish its characterization via the tent space. As applications, the authors show that $({\mathrm {VMO}}_{\rho,L}({\mathcal X}))^*=B_{\Phi,L^*}({\mathcal X})$, where $L^*$ denotes the adjoint operator of $L$ in $L^2({\mathcal X})$ and $B_{\Phi,L^*}({\mathcal X})$ the Banach completion of the Orlicz-Hardy space $H_{\Phi,L^*}({\mathcal X})$.