论文标题
分数Sobolev空间和分数变化的分布方法：渐近学II
A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II
论文作者
论文摘要
我们继续研究功能的空间$ bv^α（\ mathbb r^n）$，具有$ \ mathbb r^n $的分数变化以及分布分数sobolev space $ s^{α，p}（\ mathbb r^n）$，以前在[1，+p in]中，$ p。 Arxiv：1809.08575和Arxiv：1910.13419。我们首先定义空间$ bv^0（\ mathbb r^n）$，并建立标识$ bv^0（\ mathbb r^n）= h^1（\ mathbb r^n）$和$ s^{α，p}（\ mathbb r^n）= l^n）= l^n） r^n）$和$ l^{α，p}（\ mathbb r^n）$分别是（真实的）hardy空间和贝塞尔电位空间。然后，我们证明，分数梯度$ \ nabla^α$强烈收敛于riesz转换为$α\ to0^+$ for $ h^1 \ cap w^{α，1} $和$ s^{α，p} $函数。我们还研究了$ l^1 $ norm $ w^{α，1} $ functions的$ l^1 $ norm的收敛性。为了将$ \ nabla^α$的强限制行为作为$α\ to0^+$，我们证明了一些新的分数插值不平等，这些不平等相对于interpolation参数稳定。
We continue the study of the space $BV^α(\mathbb R^n)$ of functions with bounded fractional variation in $\mathbb R^n$ and of the distributional fractional Sobolev space $S^{α,p}(\mathbb R^n)$, with $p\in [1,+\infty]$ and $α\in(0,1)$, considered in the previous works arXiv:1809.08575 and arXiv:1910.13419. We first define the space $BV^0(\mathbb R^n)$ and establish the identifications $BV^0(\mathbb R^n)=H^1(\mathbb R^n)$ and $S^{α,p}(\mathbb R^n)=L^{α,p}(\mathbb R^n)$, where $H^1(\mathbb R^n)$ and $L^{α,p}(\mathbb R^n)$ are the (real) Hardy space and the Bessel potential space, respectively. We then prove that the fractional gradient $\nabla^α$ strongly converges to the Riesz transform as $α\to0^+$ for $H^1\cap W^{α,1}$ and $S^{α,p}$ functions. We also study the convergence of the $L^1$-norm of the $α$-rescaled fractional gradient of $W^{α,1}$ functions. To achieve the strong limiting behavior of $\nabla^α$ as $α\to0^+$, we prove some new fractional interpolation inequalities which are stable with respect to the interpolating parameter.
