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

Abstract

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.