A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II

Abstract

We continue the study of the space $BV^\alpha (\mathbb{R}^n)$ of functions with bounded fractional variation in $\mathbb{R}^n$ and of the distributional fractional Sobolev space $S^{\alpha ,p}(\mathbb{R}^n)$, with $p\in [1,+\infty ]$ and $\alpha \in (0,1)$, considered in the previous works [28, 27]. 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 $S^{\alpha ,p}(\mathbb{R}^n)=L^{\alpha ,p}(\mathbb{R}^n)$, where $H^1(\mathbb{R}^n)$ and $L^{\alpha ,p}(\mathbb{R}^n)$ are the (real) Hardy space and the Bessel potential space, respectively. We then prove that the fractional gradient $\nabla ^\alpha $ strongly converges to the Riesz transform as $\alpha \rightarrow 0^+$ for $H^1\cap W^{\alpha ,1}$ and $S^{\alpha ,p}$ functions. We also study the convergence of the $L^1$-norm of the $\alpha $-rescaled fractional gradient of $W^{\alpha ,1}$ functions. To achieve the strong limiting behavior of $\nabla ^\alpha $ as $\alpha \rightarrow 0^+$, we prove some new fractional interpolation inequalities which are stable with respect to the interpolating parameter.