论文标题
Sobolev空间重新审视
Sobolev spaces revisited
论文作者
论文摘要
我们使用合适的差异商的尺寸，描述了$ \ mathbb {r}^n $在$ \ mathbb {r}^n $上的最新一个参数的特征函数。这为Bourgain，Brezis和Mironescu提供了BBM公式的另一种观点，并在BV的情况下进行了补充，Cohen，Dahmen，Daubechies和Devore的一些结果围绕着此类功能的小波系数的大小。然后给出了对Gagliardo-Nirenberg插值不平等的应用。我们还为$ l^p（\ Mathbb {r}^n）$中的$ l^p $功能范围建立了相关的单参数公式家族。
We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on $\mathbb{R}^n$, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo-Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the $L^p$ norm of functions in $L^p(\mathbb{R}^n)$.
