ON CHARACTERIZATIONS OF $VMO$$_{¥Delta_{N}}$($¥mathbb{R}$$^{n}$)SPACE (Theory of function spaces and related topics)

Abstract

In this note, we shall give a resume of a joint work with Mingming Cao [5]. We state several different characterizations of the vanishing mean oscillation space associated with Neumann Laplacian Δ[N], written VMOΔ[N](ℝ[N]). We first describe it with the classical VMO(ℝ[N]) and certain VMO on the half-spaces. Then we comment that VMOΔ[N](ℝ[N]) is actually BMOΔ[N](ℝ[N])-closure of the space of the smooth functions with compact supports. Beyond that, it can be characterized in terms of the compact commutators of Riesz transforms and fractional integral operators associated to the Neumann Laplacian. Additionally, we by means of the functional analysis obtain the duality between certain VMO and the corresponding Hardy spaces on the half-spaces. Finally, we present an useful approximation for BMO functions on the space of homogeneous type, which can be applied to our argument and otherwhere.