Distributional Fractional Gradients and a Bourgain-Brezis-type Estimate

TL;DR

This paper generalizes fractional gradients to tempered distributions, introduces new regularization techniques, and establishes a Bourgain-Brezis-type inequality in this extended framework, advancing the understanding of fractional Sobolev spaces.

Contribution

It extends fractional gradient definitions to distributions, introduces off-diagonal Schwartz functions, and proves a Bourgain-Brezis-type inequality for these fractional gradients.

Findings

Defined fractional gradients for tempered distributions.

Introduced off-diagonal Schwartz function spaces and duality framework.

Proved a Bourgain-Brezis-type inequality in this new setting.

Abstract

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.