Distributional Fractional Gradients and a Bourgain-Brezis-type Estimate

In this paper, we extend the definition of fractional gradients found in Mazowiecka-Schikorra to tempered distributions on ${\mathbb{R}}^n$, introduce associated regularisation procedures and establish some first regularity results for distributional fractional gradients in $L_{od}^1$. The key feature is the introduction of a suitable space of off-diagonal Schwarz functions ${\mathcal{S}}_{od}({\mathbb{R}}^{2n})$, allowing for a dual definition of the fractional gradient on an appropriate space of distributions ${\mathcal{S}}_{od}^{\mathrm{\prime }}({\mathbb{R}}^{2n})$ by means of fractional divergences defined on ${\mathcal{S}}_{od}({\mathbb{R}}^{2n})$. 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-Rivière-Wettstein in the form of a fractional Bourgain-Brezis inequality for this kind of gradient.