  In this paper, we extend the definition of fractional gradients found in
Mazowiecka-Schikorra to tempered distributions on $\R^n$, introduce associated
regularisation procedures and establish some first regularity results for
distributional fractional gradients in $L^{1}_{od}$. The key feature is the
introduction of a suitable space of off-diagonal Schwarz functions
$\mathcal{S}_{od}(\R^{2n})$, allowing for a dual definition of the fractional
gradient on an appropriate space of distributions
$\mathcal{S}^{\prime}_{od}(\R^{2n})$ by means of fractional divergences defined
on $\mathcal{S}_{od}(\R^{2n})$. In the course of the paper, we make a first
attempt to define Sobolev spaces with negative exponents in this framework and
derive a result reminiscent of Bourgain-Brezis and Da Lio-Rivi\`ere-Wettstein
in the form of a fractional Bourgain-Brezis inequality for this kind of
gradient.
