VMO spaces associated with Neumann Laplacian (Q2074515)

In this paper the authors establish several characterizations of the vanishing mean oscillation space \(\mathrm{VMO}_{\Delta_{N}}(\mathbb{R}^n)\) associated with the Neumann Laplacian \(\Delta_{N}\). The authors first describe \(\mathrm{VMO}_{\Delta_{N}}(\mathbb{R}^n)\) with the classical \(\mathrm{VMO}(\mathbb{R}^n)\) and certain VMO on the half space and demonstrate that \(\mathrm{VMO}_{\Delta_{N}}(\mathbb{R}^n)\) is the \(\mathrm{BMO}_{\Delta_{N}}(\mathbb{R}^n)\)-closure of the space of smooth functions with compact supports. They also prove that the dual space of \(\mathrm{VMO}_{\Delta_{N}}(\mathbb{R}^n)\) is the Hardy space of \(H^1_{\Delta_{N}}(\mathbb{R}^n)\) associated with \(\Delta_{N}\). Furthermore, \(\mathrm{VMO}_{\Delta_{N}}(\mathbb{R}^n)\) can be characterized in terms of compact commutators of Riesz transforms and fractional integral operators associated with \(\Delta_{N}\). Finally, the authors present an useful approximation for BMO functions on the space of homogeneous type, which can be applied to their argument.