  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 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
$S^{\alpha,p}(\mathbb R^n)=L^{\alpha,p}(\mathbb R^n)$, 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 $\alpha\to0^+$ 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 $\alpha\to0^+$, we prove some new fractional interpolation
inequalities which are stable with respect to the interpolating parameter.
