论文标题
新的Brezis-Van Schaftingen-Yung Sobolev类型的不平等现象与最大不平等和一个操作员的参数家族有关
New Brezis-Van Schaftingen-Yung Sobolev type inequalities connected with maximal inequalities and one parameter families of operators
论文作者
论文摘要
由于Brezis-Van Schaftingen-Yung的近期表征，我们证明了与混合规范不平等相关的操作员的一个参数家族的新弱型不平等。这里的新颖性来自以下事实：基础测量空间将参数作为变量。通过使用广义的Riesz电位空间和Caffarelli-Silvestre扩展原理，显示了与经典和分数Sobolev空间的连接。还考虑了高阶不平等现象。我们向PDE和不同的分析领域指示了许多例子和应用，这表明了未来研究的巨大潜力。在不同的方向上，受到最初是由于Gagliardo和Garsia引起的方法的启发，我们获得了新的最大不平等现象，这些不平等与混合的规范不平等相结合，以获得Brezis-van schaftingen--van schaftingen-yung型不平等，在calderón-campanato空间的背景下。特别地，与Calderón-Campanato空间的相应限制版本相比，引入了Gagliardo-Brezis-Van Schaftingen-Yung空间的日志版本，从而导致由于Crippa-De Lellis Lellis和BruéLellis和Brué-nguyen引起的近期不平等。
Motivated by the recent characterization of Sobolev spaces due to Brezis-Van Schaftingen-Yung we prove new weak-type inequalities for one parameter families of operators connected with mixed norm inequalities. The novelty here comes from the fact that the underlying measure space incorporates the parameter as a variable. The connection to classical and fractional order Sobolev spaces is shown through the use of generalized Riesz potential spaces and the Caffarelli-Silvestre extension principle. Higher order inequalities are also considered. We indicate many examples and applications to PDE's and different areas of Analysis, suggesting a vast potential for future research. In a different direction, and inspired by methods originally due to Gagliardo and Garsia, we obtain new maximal inequalities which combined with mixed norm inequalities are applied to obtain Brezis--Van Schaftingen--Yung type inequalities in the context of Calderón-Campanato spaces. In particular, Log versions of the Gagliardo-Brezis-Van Schaftingen-Yung spaces are introduced and \ compared with corresponding limiting versions of Calderón-Campanato spaces, resulting in a sharpening of recent inequalities due to Crippa-De Lellis and Brué-Nguyen.
