Distributional Fractional Gradients and a BourgainBrezistype Estimate
Abstract
In this paper, we extend the definition of fractional gradients found in MazowieckaSchikorra 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 offdiagonal 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 BourgainBrezis and Da LioRivièreWettstein in the form of a fractional BourgainBrezis inequality for this kind of gradient.
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 arXiv:
 arXiv:2208.07806
 Bibcode:
 2022arXiv220807806W
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Functional Analysis