We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space $\dot W^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodecki\u{\i} spaces. We establish the equivalence between Fourier analytic definitions and definitions via difference operators acting on measurable functions. We prove various new results on embeddings and non-embeddings, and give applications to harmonic and caloric extensions. For suitable wavelet bases we obtain a characterization of the approximation spaces for best $n$-term approximation from a wavelet basis via smoothness conditions on the function; this extends a classical result by DeVore, Jawerth and Popov.
翻译:我们研究了Brezis、Van Schaftingen和Yung在Sobolev空间研究中引入的准温度的分数变体,结果的空格被确定为Sobolev-Slobodecki\u\i}空间实际内插空间的特殊类别,我们通过不同操作者在可测量功能上采取行动,在Fourier分析定义和定义之间建立等值。我们证明了嵌入和非编组方面的各种新结果,并应用了调和和热量扩展。对于合适的波盘基,我们通过功能上的平滑条件,从波盘中获得了对近似空间的定性,以便从波盘中得出最佳的一美元-期近差;这扩大了DeVore、Jawerth和Popov的经典结果。