VMO spaces associated with Neumann Laplacian

TL;DR

This paper characterizes the VMO space linked to the Neumann Laplacian through various methods, including classical VMO, BMO closures, and operator commutators, and explores duality with Hardy spaces.

Contribution

It provides new characterizations of ${\rm VMO}_{\Delta_N}$, including its relation to BMO closures, commutator characterizations, and duality with Hardy spaces, advancing understanding of Neumann Laplacian-associated function spaces.

Findings

Characterization of ${\rm VMO}_{\Delta_N}$ via classical VMO and half-space VMO.

Demonstration that ${\rm VMO}_{\Delta_N}$ is the BMO closure of smooth compactly supported functions.

Establishment of duality between certain VMO and Hardy spaces on half-spaces.

Abstract

In this paper, we establish several different characterizations of the vanishing mean oscillation space associated with Neumann Laplacian $\Delta_N$, written ${\rm VMO}_{\Delta_N}(\mathbb{R}^n)$. We first describe it with the classical ${\rm VMO}(\mathbb{R}^n)$ and certain ${\rm VMO}$ on the half-spaces. Then we demonstrate that ${\rm VMO}_{\Delta_N}(\mathbb{R}^n)$ is actually ${\rm BMO}_{\Delta_N}(\mathbb{R}^n)$-closure of the space of the smooth functions with compact supports. Beyond that, it can be characterized in terms of compact commutators of Riesz transforms and fractional integral operators associated to the Neumann Laplacian. Additionally, by means of the functional analysis, we obtain the duality between certain ${\rm VMO}$ and the corresponding Hardy spaces on the half-spaces. Finally, we present an useful approximation for ${\rm BMO}$ functions on the space of homogeneous type, which can be applied to our argument and otherwhere.