论文标题
与诺伊曼·拉普拉斯（Neumann Laplacian）相关的VMO空间
VMO spaces associated with Neumann Laplacian
论文作者
论文摘要
在本文中，我们建立了与neumann laplacian $Δ_n$相关的消失的平均振荡空间的几种不同特征，书面$ {\ rm vmo} _ {Δ_n}（\ mathbb {r}^n）$。我们首先用经典的$ {\ rm vmo}（\ mathbb {r}^n）$和某些$ {\ rm vmo} $在半空间上进行描述。然后，我们证明$ {\ rm vmo} _ {Δ_n}（\ m athbb {r}^n）$实际上是$ {\ rm bmo} _ {δ_n}（\ mathbb {r}^n）$ - 与压缩支持的平滑功能的关闭。除此之外，它可以以与诺伊曼·拉普拉斯（Neumann Laplacian）相关的Riesz变换和分数积分运算符的紧凑型换向器来表征。此外，通过功能分析，我们获得了某些$ {\ rm vmo} $与半空间上相应的耐寒空间之间的双重性。最后，我们提出了$ {\ rm bmo} $函数在同质类型空间上的有用近似值，可以应用于我们的参数和其他地方。
In this paper, we establish several different characterizations of the vanishing mean oscillation space associated with Neumann Laplacian $Δ_N$, written ${\rm VMO}_{Δ_N}(\mathbb{R}^n)$. We first describe it with the classical ${\rm VMO}(\mathbb{R}^n)$ and certain ${\rm VMO}$ on the half-spaces. Then we demonstrate that ${\rm VMO}_{Δ_N}(\mathbb{R}^n)$ is actually ${\rm BMO}_{Δ_N}(\mathbb{R}^n)$-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.
