论文标题
分布分数梯度和波尔加因 - 双梯度估计值
Distributional Fractional Gradients and a Bourgain-Brezis-type Estimate
论文作者
论文摘要
在本文中，我们将Mazowiecka-schikorra中的分数梯度的定义扩展到$ \ r^n $上的钢化分布，引入相关的正则化程序，并为分布分数梯度以$ l^{1} _ {od} $建立一些初步规律性结果。关键功能是引入偏高的Schwarz的合适空间$ \ MATHCAL {s} _ {od}（\ r^{2n}）$，允许在适当的分布空间上对分数梯度进行双重定义。在$ \ Mathcal {s} _ {od}（\ r^{2n}）$上定义的分数差异。在本文的过程中，我们首次尝试在此框架中定义带有负指数的Sobolev空间，并以这种渐变的形式引起了人们对Bourgain-Brezis和Da Lio-Rivière-Wettstein的想法。
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-Rivière-Wettstein in the form of a fractional Bourgain-Brezis inequality for this kind of gradient.
